ADVANCED CALCULUS A TEXT UPON SELECT PARTS OF DIFFERENTIAL CAL- CULUS, DIFFERENTIAL EQUATIONS, INTEGRAL CALCULUS, THEORY OF FUNCTIONS, WITH NUMEROUS EXERCISES BY EDWIN BIDWELL WILSON, Ph.D. PKOFESSOR OF MATHEMATICAL PHYSICS IX THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY GINN AND COMPANY BOSTON • NEW YORK • CHICAGO • LONDON ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO COPYRIGHT, 1911, 1912, BY EDWIN BIDWELL WILSON ALL RIGHTS RESERVED 121.6 G1NN AND COMPANY • PRO- PRIETORS • BOSTON • U.S.A. < L1BBARV r4 ,- M . «? COLLEGE ?/• N \ t •' " ; " '' *>> - ^1 v.j 5 PREFACE It is probable that almost every teacher of advanced calculus feels the need of a text suited to present conditions and adaptable to his use. To write such a book is extremely difficult, for the attainments of students who enter a second course in calculus are different, their needs are not uniform, and the viewpoint of their teachers is no less varied. Yet in view of the cost of time and money involved in producing an Advanced Calculus, in proportion to the small number of students who will use it, it seems that few teachers can afford the luxury of having their own text ; and that it consequently devolves upon an author to take as un- selfish and unprejudiced a view of the subject as possible, and, so far as in him lies, to produce a book which shall have the maximum flexibility and adaptability. It was the recognition of this duty that has kept the present work in a perpetual state of growth and modification during five or six years of composition. Every attempt has been made to write in such a manner that the individual teacher may feel the minimum embarrassment in picking and choosing what seems to him best to meet the needs of any particular class. As the aim of the book is to be a working text or laboratory manual for classroom use rather than an artistic treatise on analysis, especial attention has been given to the preparation of numerous exercises which should range all the way from those which require nothing but substi- tution in certain formulas to those which embody important results withheld from the text for the purpose of leaving the student some vital bits of mathematics to develop. It has been fully recognized that for the student of mathematics the work on advanced calculus falls in a period of transition, — of adolescence, — in which he must grow from close reliance upon his book to a large reliance upon himself. More- over, as a course in advanced calculus is the ultima Thule of the mathematical voyages of most students of physics and engineering, it is appropriate that the text placed in the hands of those who seek that goal should by its method cultivate in them the attitude of courageous iii iv PREFACE explorers, and in its extent supply not only their immediate needs, but much that may be useful for later reference and independent study. With the large necessities of the physicist and the growing require- ments of the engineer, it is inevitable that the great majority of our students of calculus should need to use their mathematics readily and vigorously rather than with hesitation and rigor. Hence, although due attention has been paid to modern questions of rigor, the chief desire has been to confirm and to extend the student's working knowledge of those great algorisms of mathematics which are naturally associated with the calculus. That the compositor should have set "vigor" where "rigor" was written, might appear more amusing were it not for the suggested antithesis that there may be many who set rigor where vigor should be. As I have had practically no assistance with either the manuscript or the proofs, I cannot expect that so large a work shall be free from errors ; I can only have faith that such errors as occur may not prove seriously troublesome. To spend upon this book so much time and energy which could have been reserved with keener pleasure for vari- ous fields of research would have been too great a sacrifice, had it not been for the hope that I might accomplish something which should be of material assistance in solving one of the most difficult problems of mathematical instruction, — that of advanced calculus. EDWIN BIDWELL WILSON Massachusetts Institute of Technology CONTENTS INTRODUCTORY REVIEW CHAPTER I REVIEW OF FUNDAMENTAL RULES SECTION 1. On differentiation ....... 4. Logarithmic, exponential, and hyperbolic functions 0. Geometric properties of the derivative 8. Derivatives of higher order 10. The indefinite integral 13. Aids to integration .... 16. Definite integrals .... PAGE 1 4 7 11 15 18 24 CHAPTER II REVIEW OF FUNDAMENTAL THEORY 18. Numbers and limits 21. Theorems on limits and on sets of points 23. Real functions of a real variable 26. The derivative .... 28. Summation and integration 33 37 40 45 50 PART I. DIFFERENTIAL CALCULUS CHAPTER III TAYLORS FORMULA AND ALLIED TOPICS 31. Taylor's Formula ........ 3o. Indeterminate forms, infinitesimals, infinites 36. Infinitesimal analysis ....... 40. Some differential geometry ...... oo 61 68 78 VI CONTENTS CHAPTER IV PARTIAL DIFFERENTIATION ; EXPLICIT FUNCTIONS SECTION 43. Functions of two or more variables ..... 46. First partial derivatives ........ 50. Derivatives of higher order ....... 54. Taylor's Formula and applications ...... CHAPTER V PARTIAL DIFFERENTIATION; IMPLICIT FUNCTIONS 56. The simplest case ; F (x,y) = Q 59. More general cases of implicit functions 62. Functional determinants or Jacobians 65. Envelopes of curves and surfaces 68. More differential geometry PAGE 87 93 102 112 117 122 129 135 143 CHAPTER VI COMPLEX NUMBERS AND VECTORS 70. Operators and operations 71. Complex numbers . 73. Functions of a complex variable 75. Vector sums and products 77. Vector differentiation 149 153 157 103 170 PART II. DIFFERENTIAL EQUATIONS CHAPTER VII GENERAL INTRODUCTION TO DIFFERENTIAL EQUATIONS 81. Some geometric problems ........ S3. Problems in mechanics and physics ...... 85. Lineal element and differential equation ..... 87. The higher derivatives ; analytic approximations .... CHAPTER VIII THE COMMONER ORDINARY DIFFERENTIAL EQUATIONS 89. Integration by separating the variables ...... 91. Integrating factors ......... 95. Linear equations with constant coefficients ..... 98. Simultaneous linear equations with constant coefficients 179 184 191 197 203 207 211 CONTENTS vii CHAPTER IX ADDITIONAL TYPES OF ORDINARY EQUATIONS SECTION PAGE 100. Equations of the first order and higher degree .... 228 102. Equations of higher order ........ 234 104. Linear differential equations ....... 240 107. The cylinder functions 247 CHAPTER X DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES 109. Total differential equations ........ 254 111. Systems of simultaneous equations ...... 200 113. Introduction to partial differential equations .... 267 116. Types of partial differential equations ...... 273 PART III. INTEGRAL CALCULUS CHAPTER XI ON SIMPLE INTEGRALS 118. Integrals containing a parameter ....... 281 121. Curvilinear or line integrals ........ 288 124. Independency of the path ........ 298 127. Some critical comments ........ 308 CHAPTER XII ON MULTIPLE INTEGRALS 129. Double sums and double integrals ...... 315 133. Triple integrals and change of variable ..... 326 135. Average values and higher integrals ...... 332 137. Surfaces and surface integrals ....... 338 CHAPTER XIII ON INFINITE INTEGRALS 140. Convergence and divergence ........ 352 142. The evaluation of infinite integrals ...... 360 144. Functions defined by infinite integrals ...... 368 viii CONTEXTS CHAPTER XIV SPECIAL FUNCTIONS DEFINED BY INTEGRALS SECTION' PAGE 147. The Gamma and Beta functions ....... 378 150. The error function , ...... 386 153. Bessel functions .......... 393 CHAPTER XV THE CALCULUS OF VARIATIONS 155. The treatment of the simplest case ...... 400 157. Variable limits and constrained minima ..... 404 159. Some generalizations ......... 409 PART IV. THEORV OF FUNCTIONS CHAPTER XVI INFINITE SERIES 162. Convergence or divergence of series ...... 419 105. Series of functions ......... 430 168. Manipulation of series ......... 440 CHAPTER XVII SPECIAL INFINITE DEVELOPMENTS 171. The trigonometric functions ........ 453 173. Trigonometric or Fourier series ....... 458 175. The Theta functions ......... 467 CHAPTER XVIII FUNCTIONS OF A COMPLEX VARIABLE 178. General theorems . . . . . . . . . .476 ISO. Characterization of some functions ...... 482 183. Conformal representation ........ 490 185. Integrals and their inversion ....... 496 CONTEXTS ix CHAPTER XIX ELLIPTIC FUNCTIONS AND INTEGRALS SECTION PAGE 187. Legendre's integral I and its inversion ...... 503 190. Legendre's integrals II and III ....... 511 192. Weierstrass's integral and its inversion ...... 517 CHAPTER XX FUNCTIONS OF REAL VARIABLES 194. Partial differential equations of physics ..... 524 196. Harmonic functions; general theorems ..... 530 198. Harmonic functions ; special theorems ...... 537 201. The potential integrals ......... 546 BOOK LIST 555 INDEX ............ 557 ADVANCED CALCULUS INTRODUCTORY REVIEW CHAPTER I REVIEW OF FUNDAMENTAL RULES 1. On differentiation. If the function f(x) is interpreted as the curve y=f(x),* the quotient of the increments Ay and Ax of the dependent and independent variables measured from (x Q , t/ ) is y-//o = ±i = 4/^0 = /K + Ax)-/(x ) m x — x fl Ax A/' Ax- ^ ; and represents the slope of the secant through the points P(x , y ) and P' (x -f Ax, y + A//) on the curve. The limit approached by the quo- tient Ay/Ax when P remains fixed and Ax = is the slope of the tangent to the curve at the point /'. This limit, AiiO *A-' AiiO L± x is called the derivative of f(x) for the value x = x . As the derivative may be computed for different points of the curve, it is customary to speak of the derivative as itself a function of x and write A// /(x + Ax)-/(x) llm X^ = llm _ ~A^ --=•/ (*)• ( 3 ) AiiO <± J AiiO i*33 There are numerous notations for the derivative, for instance /'(•') = W- - 1 = fl - /= ^ - »■ - B/= ^ * Here and throughout the work, where figures are not given, the reader should draw graphs to illustrate the statements. Training in making one's own illustrations, whether graphical or analytic, is of great value. 1 2 INTRODUCTORY REVIEW The first five show distinctly that the independent variable is x, whereas the last three do not explicitly indicate the variable and should not be used unless there is no chance of a misunderstanding. 2. The fundamental formulas of differential calculus are derived directly from the application of the definition (2) or (3) and from a few fundamental propositions in limits. Eirst may be mentioned •% = %%: where z = * ^ and v = f w- w dx ^ (If' 1 (if) _ 1 _ 1 dy dy df(x) dy dx dx (5) D(u ± v) = Du ± Dv, D{uv) = uDv + vDu. (6) D (-) = ? ,* D (*■) = nx-\ (7) It may be recalled that (4), which is the rule for differentiating a function of a function, follows from the application of the theorem that the limit of a product is Az Az Ay the product of the limits to the fractional identity — = ; whence Ax Ay Ax Az .. Az Ay x Az Ay hm — = lim — • lim — = hint lim ■ — » Ax = o Ax Ax = o Ay Ax = o Ax A?/ = o Ay Ax = o Ax which is equivalent to (4). Similarly, if y =/(x) and if x, as the inverse function of y, be written x=f~ l (y) from analogy with y = smx and x = sin _1 i/, the relation (5) follows from the fact that Ax/Ay and Ay/ Ax are reciprocals. The next three result from the immediate application of the theorems concerning limits of sums, products, and quotients (§ 21). The rule for differentiating a power is derived in case n is integral by the application of the binomial theorem. Ay — ■'' (r* _|_ ft— ■'' sinh ./• = > cosh x == ; (22) and the related functions tanh x, coth x, sech x, csch x, derived from them by the same ratios as those by which the corresponding trigono- metric functions are derived from sin a? and cos x. From these defini- tions in terms of exponentials follow the formulas : coslr .y — sinlr.r = 1, tank 2 a; + sech 2 a; = 1, (23) sinh (x ± y) = sinh x cosh y ±_ cosh x sinh y, (24) cosh (x ± y) = cosh x cosh y ± sinh x sinh y, (25) . x cosh.r+1 . . x , Icoshsr— 1 .„_. cosh- = + -J > sinh-=±J > (26) D sinh .'■ = cosh x, J) cosh ./■ = sinh x, (27) D tanh x — sech 2 a*, D coth x = — csch 2 a", (28) D sech x = — sech ./• tanh x, 1) csch x = — csch ./• coth x. (29) The inverse functions are expressible in terms of logarithms. Thus e 2v — 1 ?/ = sinh l x, x = sinh ■?/ = — ? e 2y — 2xe«—l=0, e v = x± Vcc 2 -f 1. * The treatment of this limit is far from complete in the majority of texts. Reference for a careful presentation may, however, he made to Granville's " Calculus," pp. 'M-'M, and Osgood's " Calculus," pp. 78~82. See also Ex. 1, (/3), in § 1G5 below. 6 INTRODUCTORY REVIEW Here only the positive sign is available, for e y is never negative. Hence sinh -1 x = log (a; + Vx 2 +l), any x, (30) cosh -1 a = log (a- ± Vas 2 — l), x > 1, (31) tanh- 1 ^ = - log — '—- ? cc 2 < 1, (32) coth- 1 x = ^ log ^— , > 1, (33) e \as seek- 1 a- = log ( - ± -» — - 1 1 , CI, (34) any x, x > 1, * 2 <1, a' 2 > 1, a; < 1, csch -1 a = log I - + -xH + 1 ) ' any a, (35) Z> sinh" 1 as = . > Z> cosh" 1 x = ^ , (36) Va/ 2 +l Va; 2 -1 V 7 D tanh- x ic = = = D coth- 1 a = v (37) 1—x 1 — x- v ' D sech- 1 x = = — — , D csch" 1 x = ~ • (38) x VI— x' 2 a-VT + ar EXERCISES 1. Show by logarithmic differentiation that 1) (uvw ...) = (— + _ + _ + ... (uvw ■ ■ •), and hence derive the rule : To differentiate a product differentiate each factor alone and add all the results thus obtained. 2. Sketch the graphs of the hyperbolic functions, interpret the graphs as those of the inverse functions, and verify the range of values assigned to x in (30)-(35). 3. Prove sundry of formulas (23)-(29) from the definitions (22). 4. Prove sundry of (30)-(38), checking the signs with care. In cases where double signs remain, state when each applies. Note that in (31) and (34) the double sign may be placed before the log for the reason that the two expressions are reciprocals. 5. Derive a formula for sinhw ± sinhw by applying (24) ; find a formula for tanh l x analogous to the trigonometric formula tan \x = sinx/(l + cosx). 6. The gudermannian. The function = gdx, defined by the relations sinh x = tan 0, = gd x = tan -1 sinh x, — \ it <

and negative when x < 0, and yet log x has no graph when x < and is not considered as decreasing. Thus the formal derivative may be real when the function is not real, and it is therefore best to make a rough sketch of the function to corroborate the evidence furnished by the examination of /'(x). If x Q is a value of x such that immediately t upon one side of x = x the function f(x) is increasing whereas immediately upon the other side it is decreasing, the ordinate y Q =:f(x Q ) will be a maximum or minimum or f(x) will become positively or negatively infinite at x Q . If the case where f(x) becomes infinite be ruled out, one may say that the function will have a minimum ov maximum at x according as the derivative changes from negative to positive or from positive to negative when x, moving in the positive direction, passes through the value x Q . Hence the usual rule for determining maxima and minima is to find the roots o//'(ar) = 0. This rule, again, must not be applied blindly. For first, /'(x) may vanish where there is no maximum or minimum as in the case y = x 3 at x = where the deriva- tive does not change sign; or second, /'(x) may change sign by becoming infinite as in the case y = x^ at x = where the curve has a vertical cusp, point down, and a minimum ; or third, the function /(x) may be restricted to a given range of values a == x =§ b for x and then the values /(a) and/(6) of the function at the ends of the interval will in general be maxima or minima without implying that the deriva- tive vanish. Thus although the derivative is highly useful in determining maxima and minima, it should not be trusted to the complete exclusion of the corroborative evidence furnished by a rough sketch of the curve y =/(x). * The construction of illustrative figures is again left to the reader. f The word "immediately" is necessary because the maxima or minima may be merely relative; in the case of several maxima and minima in an interval, some of the maxima may actually be less than some of the minima. INTRODUCTORY REVIEW 7. The derivative may be used to express the equations of the tangent and normal, the values of the subtangent and subnormal, and so on. Equation of tangent, y - y Q = y'^x - x Q ), (39) Equation of normal, (y — y ) y' -f- (x — xA = 0, (40) TM = subtangent = yjy' , MN = subnormal = y y' , (41) OT — ^-intercept of tangent = x — y /y' , etc. (42) The derivation of these results is sufficiently evi- dent from the figure. It may be noted that the subtangent, subnormal, etc., are numerical values for a given point of the curve but may be regarded as functions of x like the derivative. In geometrical and physical problems it is frequently necessary to apply the definition of the derivative to finding the derivative of an unknown function. For instance if A denote the area under a curve and measured from a fixed ordinate to a variable ordinate, A is surely a func- tion A (x) of the abscissa x of the variable ordinate. If the curve is rising, as in the figure, then MPQ'M' < AA < MQP'M', or yAx < AA < (y + Ay) Ax Divide by Ax and take the limit when Ax = 0. There results Hence lim y == lim ^x = o' Ax = o Ax .. A A hm Aj±0 Ax lim (// + Ay). Ax = dA Tx = y - (43) Halle's Theorem and the Theorem of the Mean are two important theorems on derivatives which will be treated in the next chapter but may here be stated as evident from their geometric interpretation. Rolle's Theorem states that: If a function bus a derivative at every r VL -^ 1 ,r n /,' o Fig. 1 Fig. 2 Fi< point of an interval and if the function vanishes at the ends of the in- terval, then there is at least one point within the interval at which the derivative vanishes. This is illustrated in Fig. 1, in which there are two such points. The Theorem of the Mean states that: If a function FUNDAMENTAL EULES 9 has a derivative at each point of an interval, there is at least one point in the interval such that the tangent to the carve y=f(x) is pjarallel to the chord of the interval. This is illustrated in Fig. 2 in which there is only one such point. Again care must be exercised. In Fig. 3 the function vanishes at A and B but there is no point at which the slope of the tangent is zero. This is not an excep- tion or contradiction to Rolle's Theorem for the reason that the function does not satisfy the conditions of the theorem. In fact at the point P, although there is a tangent to the curve, there is no derivative ; the quotient (1) formed for the point P becomes negatively infinite as Ac = from one side, positively infinite as Ax = from the other side, and therefore does not approach a definite limit as is required in the definition of a derivative. The hypothesis of the theorem is not satisfied and there is no reason that the conclusion should hold. EXERCISES 1. Determine the regions in which the following functions are increasing or decreasing, sketch the graphs, and find the maxima and minima : (a) lx 3 -x- + 2. (/3) (x + 1)* (x - 5) 3 , ( 7 ) log (x 2 - 4), (5) (x - 2) Vx - 1, (0 - (x + 2)V12 - x-. (f) x> + ax + b. 2. The ellipse is r = Vx'- 2 + y 2 = e (d + x) referred to an origin at the focus. Find the maxima and minima of the focal radius r, and state why DjV = does not give the solutions while D^r = does [the polar form of the ellipse being r = k(l— ecos^)- 1 ]. 3. Take the ellipse as x-/a- + y"/b- = 1 and discuss the maxima and minima of the central radius r = Vx- + y-. Why does D x r = give half the result when r is expressed as a function of x. and why will ]) K r = give the whole result when X = a cosX, y = bsinX and the ellipse is thus expressed in terms of the eccentric angle ? 4. If y — P{x) is a polynomial in x such that the equation P(x) = has multiple roots, show that P'(x) = for each multiple root. What more complete relationship can be stated and proved ? 5. Show that the triple relation 27 b- + 4 a 3 J§ determines completely the nature of the roots of x 3 + ax + b = 0, and state what corresponds to each possibility. 6. Define the angle 6 between two intersecting curves. Show that tan 9 = [/'(x ) - r/'(x )] h- [1 +/'(x )y'(x )] if y = fix) and y = g (x) cut at the point (x , y ). 7. Find the subnormal and subtangent of the three curves {a) y- = Apx, (P) x- = 4py, (y) x- + if- = a". 8. The pedal curve. The locus of the foot of the perpendicular dropped from a fixed point to a variable tangent of a given curve is called the pedal of the given curve with respect to the given point. Show that if the fixed point is the origin, the pedal of y =f(x) may be obtained by eliminating x^. y , 2/6 from the equations y — y n = V'o ^ - ^ Wf + x = 0. y = f(x ), y' = /'(x ). 10 INTRODUCTORY REVIEW Find the pedal {a) of the hyperbola with respect to the center and (/3) of the parabola with respect to the vertex and (7) the focus. Show (5) that the pedal of the parabola with respect to any point is a cubic. 9. If the curve y =f(x) be revolved about the x-axis and if V(x) denote the volume of revolution thus generated when measured from a fixed plane perpen- dicular to the axis out to a variable plane perpendicular to the axis, show that D X V = mj-. 10. More generally if A (x) denote the area of the section cut from a solid by a plane perpendicular to the x-axis, show that D X V = A (x). 11. If A (4>) denote the sectorial area of a plane curve r = f( x > — 1, show that - x 2 < x — log (1 + x) < — 2 1 + x P. Derivatives of higher order. The derivative of the derivative 'regarded as itself a function of x) is the second derivative, and so on ;o the nth derivative. Customary notations are : /"(■>0 = "gr 2 = -£* = Dif= T>iy = y" = D'f= ffy, The nth derivative of the sum or difference is the sum or difference of the nth derivatives. For the nth derivative of the product there is a special formula known as Leibniz's Theorem. It is ix ( n — 1 ) D n (uv) = D n u ■ v + nD n ~ l uDv + v 9| ' D n - - ulfv -\ \- ulTv. (44) This result may be written in symbolic form as Leibniz's Theorem D n (uv) = (Du + Dv) n , (44') where it is to be understood that in expanding (Du -f- /)/•)" the term ; (Dn) k is to be replaced by D k u and (Du)° by D°u = v. In other words | the powers refer to repeated differentiations. A proof of (44) by induction will be found in § 27. The following proof is interesting on account of its ingenuity. Xote first that from I) (uv) = uDv + vDu, If- (uv) = I) (uDv) + D (vDu), and so on, it appears that D 2 (uv) consists of a sum of terms, in each of which there are two differentiations, with numerical coefficients independent of u and v. In like manner it is clear that I)" (uv) = C D»u ■ v + CJ)" -i uDv + ... + C n _i DuT)» -i v + C n uB"v is a sum of terms, in each of which there are n differentiations, with coefficients C independent of u and v. To determine the C'a any suitable functions u and v, say, u = e x , v = e**, uv = eO+ a ) x , D k e ax = a k e ax , may be substituted. If the substitution be made and e (l+a)x be canceled, e-a+a)xDn( uv ) = (i + a )n = C + C x a + ■■■ + C n _ia»-i + C n a n , and hence the C's are the coefficients in the binomial expansion of (1 + a) n . 12 INTRODUCTORY REVIE 1 function may be Formula (4) for the derivative of a function ,.- generally extended to higher derivatives by repeated applic ' , j f t\\ any desired change of variable may be made by the functions and (5). For if x and y be expressed in terms of , .i deriva- of new variables u and v, it is always possible to obta „ ress j on tives D x y, D*y, ■ • • in terms of D u v, D%v, ■ ■ ■, and thus an> oreS sion F(x, y, y', y", ■ • ■) may be changed into an equivalent , es the $>(», r, v', r", • • •) in the new variables. In each case that a.p n\ transformations should be carried out by repeated application and (5) rather than by substitution in any general formulas. The following typical cases are illustrative of the method of change of variab Suppose only the dependent variable y is to be changed to z defined a.ay=f(z). Th d 2 y _ d fdy\ _ d /dz dy\ _ d-z dy dz I d dy' dx 2 dx \dxj dx \dx dz) dx 2 dz dx \dx d _ d 2 z dy dz Id dy dz\ _ d 2 z dy /^z\ 2 d-y dx" dz dx \dz dz dx] dx 2 dz \dx) dz 2 As the derivatives of y =f(z) are known, the derivative d 2 y/dx 2 has been expr in terms of z and derivatives of z with respect to x. The third derivative won found by repeating the process. If the problem were to change the indepe variable x to 2, defined by x =f{z), dy _ dy dz _ dy /dxW d 2 y _ d Vdy (dx\~n dx dz dx dz\dz) dx 2 dx[_dz\dz/ J d 2 y _ d 2 y dz /dxX- 1 dy /dx\~ 2 dz d 2 x _ Yd 2 y dx d 2 x dyl A dx 2 dz 2 dx\dz/ dz\dz/ dx dz 2 \_dz 2 dz dz 2 dz\ \< The change is thus made as far as derivatives of the second order are coiicen the change of both dependent and independent variables was to be made, th would be similar. Particularly useful changes are to find the derivatives of when y and x are expressed parametrically as functions of t, or when both ; pressed in terms of new variables r,

, y = r sin . For these see the exercises. 9. The concavity of a carve //=,/'(■'') is given by the table: if /"(.t* ) > 0, the curve is concave up at ,r = .r , if /"(?■) < 0, the curve is concave down at x =x Q , if f"( : '\) = 0, an inflection point at x = x Q . (?) Hence the criterion for distinguishing between maxima and minium if f'(,r ( ) = and /"(.r) > 0, a minimum at x = x Q , if f'(.r ( ) = and /" (r ) < 0, a maximum at x = .r Q , if f'(x ) = and /"(•'' ) = 0, neither max. nor min. (? FUNDAMENTAL RULES 13 The question points are necessary in the third line because the state- ments are not always true unless /"'(a' ) =£ (see Ex. 7 under § 39). It may be recalled that the reason that the curve is concave up in case/"(x ) > is because the derivative f'{x) is then an increasing function in the neighborhood of x — x ; whereas if f"{x ) < 0, the derivative /'(x) is a decreasing function and the curve is convex up. It should be noted that concave up is not the same as concave toward the x-axis, except when the curve is below the axis. With regard to the use of the second derivative as a criterion for distinguishing between maxima and minima, it should be stated that in practical examples the criterion is of rela- tively small value. It is usually shorter to discuss the change of sign of /'(x) directly, — and indeed in most cases either a rough graph of /(x) or the physical conditions of the problem which calls for the determination of a maximum or minimum will immediately serve to distinguish between them (see Ex. 27 above). The second derivative is fundamental in dynamics. By definition the average velocity v of a particle is the ratio of the space traversed to the time consumed, v = s/t. The actual velocity v at any time is the limit of this ratio when the interval of time is diminished and approaches zero as its limit. Thus _ As . As els v = — and v = lim ■ — ■ = — - • (45) At At = oAt dt In like manner if a particle describes a straight line, say the cr-axis, the average acceleration f is the ratio of the increment of velocity to the increment of time, and the actual acceleration f at any time is the limit of this ratio as At = 0. Thus /=— and / = Inn — = — = — ■ (46) t\t A( = o^£ at at By Newton's Second Law of Motion, the force acting on the particle is equal to the rate of change of momentum with the time, momentum being defined as the product of the mass and velocity. Thus tl(niv) dv d-x dt dt J dt 2 where it has been assumed in differentiating that the mass is constant, as is usually the case. Hence (47) appears as the fundamental equa- tion for rectilinear motion (see also §§ 79, 84). It may be noted that (47') where T = i mv 2 denotes by definition the kinetic energy of the particle. For comments see Ex. 6 following. dv d n o\ dr F = m v — — - m v = — — ax dx \2 / dx 14 INTRODUCTORY REVIEW EXERCISES 1. State and prove the extension of Leibniz's Theorem to products of three or more factors. Write out the square and cube of a trinomial. 2. Write, by Leibniz's Theorem, the second and third derivatives : (a) e*sinx, (/3) cosh x cos x, (7) x^logx. 3. Write the nth derivatives of the following functions, of which the last three should first be simplified by division or separation into partial fractions. (a) Vx + 1, (j8) log(ax + 6), (7) (x 2 + 1) (x + l)-», (5) cosax, (e) e x sinx, (f) (1 — x)/(l + x), / \ 1 5. Find the inflection points of the curve x = 4

. 6. Prove (47'). Hence infer that the force which is the time-derivative of the momentum mv by (47) is also the space-derivative of the kinetic energy. 7. If A denote the area under a curve, as in (43), find dA/dO for the curves (a) y = a (1 — cos 0), x = a {6 — sin 6), (/3) x = a cos 6, y = b sin 6. 8. Make the indicated change of variable in the following equations: , , d 2 y 2x dy y . . d 2 y , (a) — H H = 0, x = tan z. Ans. — - + y = 0. dx 2 1 + x 2 dx (1 4- x 2 ) 2 dz 2 d 2 » ylns. — + 1 = 0. dw 2 9. Transformation to polar coordinates. Suppose thatx = r cos ,y=r sin — r sin 0, — = — sin

, d<£ d0 d0 d<£ , , • , , • t . ,,. ,*y , <* 2 2/ r 2 +2(Z>^r) 2 -r7;, 2 r and so on for higher derivatives. I ind — and = dx dx 2 (cos

) 3 10. Generalize formula (5) for the differentiation of an inverse function. Find d 2 x/dy 2 and d s x/dy 3 . Note that these may also be found from Ex. 4. 11. A point describes a circle with constant speed. Find the velocity and acceleration of the projection of the point on any fixed diameter. i« -.^ d 2 w „ „ . . /duN- 1 r d 2 v /dv\~ 3 .. 1 12. Prove — - = 2 wu 3 + 4 u 4 ( — ) — v 5 ( — if x = -. y - uv. dx 2 \du/ du 2 \duf v FUNDAMENTAL RULES 15 10. The indefinite integral. To integrate a function f(x) is to find a function F(x) the derivative of which is f(x). The integral F(x) is not uniquely determined by the integrand f(x) • for any two functions which differ merely by an additive constant have the same derivative. In giving formulas for integration the constant may be omitted and understood ; but in applications of integration to actual problems it should always be inserted and must usually be determined to fit the requirements of special conditions imposed upon the problem and known as the initial conditions. It must not be thought that the constant of integration always appears added to the function F(x). It may be combined with F(x) so as to be somewhat disguised. Thus logx, logx + C, logCx, log(x/C) are all integrals of 1/x, and all except the first have the constant of integration C, although only in the second does it appear as formally additive. To illustrate the determination of the constant by initial conditions, consider the problem of finding the area under the curve y = cosx. By (43) D X A = y = cos x and hence A = sin x + C. If the area is to be measured from the ordinate x = 0, then A = when x = 0, and by direct substitution it is seen that C = 0. Hence A = sin x. But if the area be measured from x= — iir, then A=0 when x= — \tr and C = 1. Hence ^4 = 1 + since. In fact the area under a curve is not definite until the ordinate from which it is measured is specified, and the constant is needed to allow the integral to fit this initial condition. 11. The fundamental formulas of integration are as follows: J- -logx, f-'^nh"""' 1 *- 1 ' (48) I e x = e x , JV^yiog*. (49) I sin x = — cos x, I cos x ■=■ sin x, (50) J tan x = — log cos x, I cot x = log sin x, (51) I sec".r = tan x, J csc 2 x = — cot X, (52) I tan x sec x = sec x, I cot X CSC X = — CSC X, (53) with formulas similar to (50)-(53) for the hyperbolic functions. Also /- :, = tan _1 x or — cot -1 ./', | ; = tanh -1 cc or coth -1 .r, (54) 1 + Z 2 'J l-.r~ 16 INTRODUCTORY REVIEW /l r ± l — = sin -1 Q5 or —cos 1 x, l — = + sinh -1 05, (oo) Vl-o- 2 J Vl + x 1 K ' c i r ± i J — = sec -1 05or — esc \r, | — gp secli" 1 .;-, (56) r ±i f ±i ,- — 7 = ±cosh- 1 j-, — =qFcscli- 1 .r, (57) J war — 1 J a? VI + x~ — = vers -1 a:, I see 05 = gd -1 05 = log tan ( — -f- - J • (58) V2o5 — x- J \ 4 -/ For the integrals expressed in terms of the inverse hyperbolic functions, the logarithmic equivalents are sometimes preferable. This is not the case, however, in the many instances in which the problem calls for immediate solution with regard to x. Thus if y = I (1 + x 2 )~ z = sinh -1 x + C, then x = sinh ((/ — C), and the solution is effected and may be translated into exponentials. This is not so easily accomplished from the form y = log (x + V 1 + x 2 ) + C. For this reason and because the inverse hyperbolic functions are briefer and offer striking analogies with the inverse trigonometric functions, it has been thought better to use them in the text and allow the reader to make the necessary substitutions from the table (30)-(35) in case the logarithmic form is desired. 12. In addition to these special integrals, which are consequences of the corresponding formulas for differentiation, there are the general rules of integration which arise from (4) and (G). j( /( _|_ v _ Ir) = i„ + jr - f ,/•, (60) uv= fuo'+ fu'c. (CI) Of these rules the second needs no comment and the third will be treated later. Especial attention should be given to the first. For instance suppose it were re- quired to integrate 2 logx/x. This does not fall under any of the given types ; but 2 _ d (log .r) 2 d log x _ dz dy x d logx dx dy dx Here (logx) 2 takes the place of z and logx takes the place of y. The integral is therefore (logx) 2 as may be verified by differentiation. In general, it may be possible to see that a given integrand is separable into two factors, of which one is integrable when considered as a function of some function of x. while the other is the derivative of that function. Then (59) applies. Other examples arc : ft sin 'cosx, rtan-ix/(l + x 2 ), fx- sin (x"). FUNDAMENTAL RULES 17 In the first, z = e'J is integrable and as y = sinx, y' = cosx ; in the second, z = y is integrable and as ?/ = tan _1 x, ?/ = (1 + x 2 ) -1 ; in the third z = sin?/ is integrable and as y = x 3 , y' = 3x 2 . The results are e sin *, I (tan- 1 x) 2 , — J cos (x 3 ). This method of integration at sight covers such a large percentage of the cases that arise in geometry and physics that it must be thoroughly mastered.* EXERCISES 1. Verify the fundamental integrals (48)-(58) and give the hyperbolic analogues of (50)-(53). 2. Tabulate the integrals here expressed in terms of inverse hyperbolic func- tions by means of the corresponding logarithmic equivalents. 3. Write the integrals of the following integrands at sight: (a) sin ax, (/3) cot (ax + 6), (7) tanh3x, 1 1 ,.,1 (5) (V) a 2 + x 2 1 x logx (k) x 3 vox 2 + b, (p) a 1 + sinx cosx, V-) Vx 2 — a 2 1 (0) X 2 (*) tanx sec 2 x, (0) tanh _1 x 1-x 2 M sinx Vcos X vv V2ox — X 2 (0 X x 2 + a 2 ' (M) cot x log sinx, (») 2 + logx X > (r) 1 Vl — x 2 sin-ix 4. Integrate after making appropriate changes such as sin 2 x = \ — \ cos 2x or sec 2 x = 1 + tan 2 x, division of denominator into numerator, resolution of the product of trigonometric functions into a sum, completing the square, and so on. (a) cos 2 2 x, x 2 + 3x + 25 x + 3 (/3) sin 4 x. (7) tan 4 x, (0 2X + 1 . w x + 2 1 — sin as vers x e 2 * + 1 (0 x . \ 2«x + x 2 (X) sinli ?nx sin ll »IX, (fi) c< is x cos 2 x cos 3 x, ,, cx + d 7»i —l (v) K ' 4x 2 - 5x + 1 (k) sin 5x cos 2x + 1, ( y) sec 5 x tan x — V2 x, x 2 + ax + /, (ax™ + &)>' * The use of differentials (§ 35) is perhaps more familiar than the use of derivatives. /dz C dz dv r ilz Then I -\ogxdx = I 2 log x (Hog x = (logs;) 2 . The use of this notation is left optional with the reader; it has some advantages and some disadvantages. The essential thing is to keep clearly in mind the fact that the problem is to be inspected with a view to detecting the function which will differentiate into the given integrand. 18 INTRODUCTORY REVIEW 5. How are the following types integrated ? (a) sin m x cos n x, m or n odd, or m and n even, (£) tan n x or eot"x when n is an integer, (7) secx or csc"x when n is even, (5) tan m x sec"x or cot m x csc"x, n even. 6. Explain the alternative forms in (54)-(56) with all detail possible. 7. Find (a) the area under the parabola y- = 4px from x = to x = a ; also (j3) the corresponding volume of revolution. Find (7) the total volume of an ellip- soid of revolution (see Ex. 9, p. 10). 8. Show that the area under y = sin ?nx sin nx or y = cos mx cos nx from x = to x = 7r is zero if m and n are unequal integers but | ir if they are equal. 9. Find the sectorial area of r = a tan between the radii = and

— n r -r~z — — rr + • • • + —— ; : r • • • , x l + mx + n (x- + nix + n) 2 (x- + mx + n)M where there is for each irreducible factor of F a term corresponding to the highest power to which that factor occurs in F and also a term corresponding to every lesser power. The coefficients A, B, • • •, M, N, ■ ■ ■ may be obtained by clearing of fractions and equating coefficients of like powers of x, and solving the equations ; or they may be obtained by clearing of fractions, substituting for x as many dif- ferent values as the degree of F, and solving the resulting equations. When f/F has thus been resolved into partial fractions, the problem has been reduced to the integration of each fraction, and this does not present serious difficulty. The following two examples will illustrate the method of resolution into partial fractions and of integration. Let it be required to integrate x 2 + 1 r ^ x 3 + <; r x- + 1 r I and I J x (x - 1) (x - 2) (x 2 + x + 1 ) J ■ ( X _ i) ( X _ 2) (x 2 + x + 1) J (x - l) 2 (x - 3) 3 The first fraction is expansible into partial fractions in the form x 2 + 1 A B C J)x + E = - + -+ ^ + x (x — 1 ) (x — 2) (x 2 + x + 1) x x — 1 x — 2 x 2 + x + 1 Hence x' 2 + 1 = A (x - 1) (x - 2) (x 2 + x + 1) + Bx (x - 2) (x 2 + x + 1) + Cx (x - 1) (x 2 + x + 1) + {Bx + E) x (x - 1) (x - 2). Rather than multiply out and equate coefficients, let 0, 1. 2. — 1. — 2 be substi- tuted. Then 1 = 2 A, 2 =-3 7.', 5 = 14C, D-E = 1/21, E- 2D = 1/7, r x 2 + 1 _ r 1 r 2 r 5 r 4x + 5 J x{x — l)(x — 2)(x 2 + x + l) ~ J 2x J 3(x — 1) J 14(x — 2) J 21(x 2 + x + l) = I logx - I log(x -1) + ~ log(x - 2)- 1 log(x 2 + x + 1)- '■: tan-i ~ x + T 2 3 14 21 ; A ;> V3 FUNDAMENTAL RULES 21 In the second case the form to be assumed for the expansion is 2 x 3 + 6 A B C J) E (x - l) 2 (x - 3) 3 x - 1 (x - l) 2 (x - 3) (x - 3) 2 (x - 3) 3 2 x 3 + 6 = A (x - 1) (x - 3) 3 + B(x- 3) 3 + C (x - l) 2 (x - 3) 2 + B (x - l) 2 (x - 3) + E (x - l) 2 . The substitution of 1, 3, 0, 2, 4 gives the equations 8=-8 7>, 60 = 4£, 0A + 3C -D + 12 = 0, A-C + D + G = 0, ^L + 3C + 3X> = 0. The solutions are — 9/4, — 1, + 9/4, — 3/2, 15, and the integral becomes ./' 2a; 3 + 6 9, . ,. 1 9, . - - log (x - 1) + + - log (x - 3) (x - l) 2 (x - 3) 3 4 ' ° ' ' x - 1 4 3 15 + 2 (x - 3) 2 (x - 3) 2 The importance of the fact that the method of partial fractions shows that any rational fraction may be integrated and, moreover, that the integral may at most con- sist of a rational part plus the logarithm of a rational fraction plus the inverse tangent of a rational fraction should not be overlooked. Taken with the method of substitution it establishes very wide categories of integrands which are inte- grate in terms of elementary functions, and effects their integration even though by a somewhat laborious method. 15. The metliod of substitution depends on the identity f/(*) = ffl(y)1~ ^ » = *(?), (59') J x J u dy which is allied to (59). To show that the integral on the right with respect to y is the integral of f(x) with respect to x it is merely necessary to show that its derivative with respect to x is fix). By definition of integration, dyJy dy dy and '/ fflJ x 2 K, J { l + x) 2 K * } K{x*-a*)i 2. If P(x) is a polynomial and P'(x), P"{x), • ■ • its derivatives, show (a) fp (x) <*>* = 1 e»* \p (x) - - P'(x) + - P"(x) 1 , (j8) Cr (x) cos fix = - sin ax \P (x) - — P"(x) + - P iv (x) 1 J a a 2 a; 1 + - cos ax \- P'(x) - \ P'"{x) + \ Pv(x) 1, a i_a a- a" and (7) derive a similar result for the integrand P(x) -in ax. FUNDAMENTAL RULES 23 3. By successive integration by parts and subsequent solution, show inbx = e ax cos bx = t c " (a sin bx — b cos bx) a 2 -f b' 1 e ax (b sin bx + a cos bx) (a)jVsi (0) f «/ a- + 6- (7) / xe 2x cosx = fee' 2x [5x(sinx + 2 cosx) — 4 sinx — 3 cosx]. 4. Prove by integration by parts the reduction formulas . . c ■ sin m + 1 x cos"- 1 x n — lr. (a) I sin»'x cos"x = - -| I sin m xcos" _2 x, J in + n m + n J ,„. r i tan"'- 1 x sec"x m — 1 r (/3) I tan m xsec»x = j tan" - 2 x sec" x, «/ ?/i + n — 1 »i + n — 1 «/ (7) f — x —= — - 1 r x ~ +(2»-8) r - - J (x 2 + a")« 2 (n - I) a 2 [_ (x 2 + a' 2 )"- 1 J (x 2 + a 2 )''- 1 (5) f-^-= ^ mH^f^" 'J (10gX)» (?l- l)(l0gX)»-! ?1-1J (lOgX)"- 1 5. Integrate by decomposition into partial fractions: v ' J (x - 1) (x - 2) VH ' J a 4 - x 4 l Jl+ x 4 (5) r — % — , (.) r 4j2 - 3x + 1 , (f ) r_i v 'J (x + 2)-(x + 1) w J 2x 5 + x 3 w ii(l + x-) 2 6. Integrate by trigonometric or hyperbolic substitution : (a) fVa? - x 2 , (/3) jVx 2 - a 2 , (7) JV« 2 + x 2 , J («.-x 2 )t J x J X J 7. Find the areas of these curves and their volumes of revolution : (a) xf + yl = «?, ($ a *y 2 = a 2 x* - x 6 , (7) Q" + (| 8. Integrate bv converting to a rational algebraic fraction : /* sin3x ,„, /» cos3x , r sin2x (a) I — » (,3) |— ; , (7) I , J a- c< is 2 x + b- sin- x •/ a'- cos- x + 6- sin' 2 x J a- cos- x + b'- sin 2 x («) r i— , (o r ! , to f- J a + h cosx J a + b cosx + c sinx ./ 1 -f sinx 9. Show that I R (x, \if-f V + ex'-') may be treated by trigonometric substitu- tion : distinguish between b- — 4 ac ~%_ 0. 10. Show that (e(x, -%/— — I is made rational by y n = — — • Hence infer J \ ' \ ex + dl ex + d that I K(x, V(j: — or) (x — j3)) is rationalized bv y' 2 = - — ■ This accomplishes J ' x — tr the integration of 11 (x. va + frx + ex' 2 ) when the roots of a + bx + ex' 2 = are real, that is, when b' 2 — 4 ac > 0. 24 INTRODUCTORY REVIEW 1 1 ™ , C r, T /az + ^\ m /a* + 6\" 1 1 1 . Show that I K x, , , • • • , where the exponents m. n, J L \cx + d) \cx + d) J • ■ • are rational, is rationalized by y k = if k is so chosen that km, kn, • • • are ex + d integers. 12. Show that J (a + by)pyi may be rationalized if p or g or p + q is an integer. By setting x" = y show that I x m (a + bx n ) p may be reduced to the above type and . . , , , in + 1 ??i + l . . hence is integrable when or p or 1- p is integral. 71 n 13. If the roots of a + bx + ex" = are imaginary, / E (x, Va + bx + ex' 2 ) may be rationalized by y = va + 6x + ex 2 =p x V c. 14. Integrate the following. J Vx-1 J l+Vx ^Vl + x-Vl + x J vV + i J Va-x 2 ) 3 J i v / e* : + 1 J V(l — x 2 ) 3 J (x — d) Va + bx + ex' 2 J X(l + X 2 )2 ^ X ^ VI -X 3 * 15. In view of Ex. 12 discuss the integrability of : , , c . , / _ , r x m r let x = aw 2 , (cr) I sm'"xcos n x, let sinx = Vvy, (/3) I — == -j . ^ J Vox — x 2 t or v «x — x 2 = xy. 16. Apply the reduction formulas, Table, p. 66, to show that the final integral for r x m . r 1 r x r 1 I — - - is I — or | — ====; or I J VI — x 2 ^ x 7 ! — x 2 J Vl — x 2 ^ x Vl — x 2 according as ?>i is even or odd and positive or odd and negative. 17. Prove sundry of the formulas of Peirce's Table. 18. Sho w that if B (x, Va 2 — x 2 ) contains x only to odd powers, the substitu- tion z = Va 2 — x 2 will rationalize the expression. Use Exs. 1 (f) and 6 (e) to compare the labor of this algebraic substitution with that of the trigonometric or hyperbolic. 16. Definite integrals. If an interval from x = a to x = b be divided into n successive intervals A.r 1; A.a>, • • ■, A.r„ and the value f(i { ) of a function /(.v) be computed from some point £■ in each interval Ax ( and be multiplied by A,/-., then the limit of the sum lim [/(&) A.r t +/(&) A.r 2 + • • • + /(£„) A.r„] = fj(x) dx, (62) FUNDAMENTAL RULES 25 when each interval becomes infinitely short and their number n be- comes infinite, is known as the definite integral ol fix) from a to b, and is designated as indicated. If y =f(x) be graphed, the sum will be represented by the area under a broken line, and it is clear that the limit of the sum, that is, the integral, will be repre- sented by 'the area under the curve y =f(x) and between the ordinates x = a and x — b. Thus the definite integral, de- fined arithmetically by (62), may be connected with a geo- metric concept which can serve to suggest properties of the integral much as the interpretation of the derivative as the slope of the tan- gent served as a useful geometric representation of the arithmetical definition (2). For instance, if a, b, c are successive values of x, then f a mdx + f b C f(x)dx = fj(x)dx (63) is the equivalent of the fact that the area from a to c is equal to the sum of the areas from a to b and b to e. Again, if A.r be considered positive when x moves from a to b, it must be considered negative when x moves from b to a and hence from (G2) fj(x)dx = -f b a f(x)dx. (64) Finally, if M be the maximum of f(x) in the interval, the area under the curve will be less than that under the line y = M through the highest point of the curve ; and if in be the minimum of f(x), the area under the curve is greater than that under y = in. Hence m (b -«)/«• Choose the points & in the intervals Ax,- as the initial points of the intervals. Then Axi Ax 2 Ax„ _ a (r - 1) ar(r-l) «r»-i(r-1) _ — (- — h • • • H — r~ — 1 r • • • -\ : — ny — 1;. £1 £2 L a ar ar"- 1 But r = Vb/a or n = log (b/a) -f- logr. Axi Ax 2 Ax„ , , , . b r — 1 , 6 /t Hence — H -\ 1 = n(r — 1) = log - • = log £1 £2 £» a logr °a log(l + A) Now if n becomes infinite, r approaches 1, and h approaches 0. But the limit of log (1 + h)/h as h^zO is by definition the derivative of log (1 + x) when x = and is 1. Hence r b dx ,. TAct Ax 2 Ac„~| , b . . , I — = hm — - H -J 1 = log- = logb — loga. J a x « = *> |_ £1 £ 2 £n J ~ a As another illustration let it be required to evaluate the integral of cos 2 x from to \ tt. Here let the intervals Ar,- be equal and their number odd. Choose the £'s as the initial points of their intervals. The sum of which the limit is desired is a- = cos 2 ■ Ac + cos 2 Ac • Ax + cos 2 2 Ax ■ Ax + • • • + cos 2 ()i — 2) Ac ■ Ac + cos 2 (n — 1) Ax • Ax. But nAc = 1 7T, and (n — 1) Ac = I it — Ax, (?i - 2) Ac = \ -k - 2 Ax, • • -, and cos (\ -k — y) = sin y and sin 2 y + cos 2 y = \. Hence b + \b 3>(ft + A&) — b + Aft f») I A/, A/, ~A&VW^» where £ is intermediate between /; and & + A/y. Let A5 = 0. Then £ approaches & and/(£) approaches f(b). Hence $, ^ = l r a*) **=/(*)■ ( 66 > U a If preferred, the variable & may be written as ,r, and *(*)= f /(*)<**> ^'(•'■) = ^ f'f(x)dx=f(x). (66') This equation will establish the relation between the definite integral and the indefinite integral. For by definition, the indefinite integral ]'(■>■) of /(.#•) is any function such that /<"(./•) equals /(.r). As $'(•'') =/(*) it follows that r ,■ / /( 3 -)rfa J = F(x)+C. (67) Hence except for an additive constant, the indefinite integral of f is the definite integral of f from a fixed lower limit to a variable upper limit. As the definite integral vanishes when the upper limit coincides with the lower, the constant C is — F(«) and f f(x)dx = F(b)-F(a). (67') Hence, the definite integral of f(x) from a to l> is the difference between the values of any indefinite integral Fix) taken for the upper and lower limits of the definite integral; and if the indefinite integral of f is known, the definite integral may be obtained without applying the definition (62) to/ 28 INTRODUCTORY REVIEW The great importance of definite integrals to geometry and physics lies in that fact that many quantities connected with geometric figures or physical bodies may be expressed si?nj)ly for small portions of the figures or bodies and may then be obtained as the sum of those quanti- ties taken over all the small portions, or rather, as the limit of that sum when the portions become smaller and smaller. Thus the area under a curve cannot in the first instance be evaluated ; but if only that portion of the curve which lies over a small interval Ax be considered and the rectangle corresponding to the ordinate f(i~) be drawn, it is clear that the area of the rectangle is f(£) Ax, that the area of all the rectangles is the sum 2/(£) Ax taken from a to b, that when the intervals Ax approach zero the limit of their sum is the area under the curve ; and hence that area may be written as the definite integral oif(x) from a to b* In like manner consider the mass of a rod of variable density and suppose the rod to lie along the x-axis so that the density may be taken as a function of x. In any small length Ax of the rod the density is nearly constant and the mass of that part is approximately equal to the product pAx of the density p(x) at the initial point of that part times the length Ax of the part. In fact it is clear that the mass will be intermediate between the products mAx and MAx, where m and M are the minimum and maximum densities in the interval Ax. In other words the mass of the section Ax will be exactly equal to p (£) Ax where £ is some value of x in the interval Ax. The mass of the whole rod is therefore the sum 2p(£)Ax taken from one end of the rod to the other, and if the intervals be allowed to approach zero, the mass may be written as the integral of p(x) from one end of the rod to the other. t Another problem that may be treated by these methods is that of finding the total pressure on a vertical area submerged in a liquid, say, in water. Let w be the weight of a column of water of cross section 1 sq. unit and of height 1 unit. (If the unit is a foot, iv = 62.5 lb.) At a point h units below the surface of the water the pressure is ivh and upon a small area near that depth the pressure is approximately vohA if A be the area. The pressure on the area A is exactly equal to w%A if £ is some depth interme- diate between that of the top and that of the bottom of the area. Now let the finite area be ruled into strips of height Ah. Consider the product whb (h) Ah where b(h) =f(h) is the breadth of the area at the depth h. This * The £'s may evidently be so chosen that the finite sum 1f(^)Ax is exactly equal to the area under the curve ; but still it is necessary to let the intervals approach zero and thus replace the sum by an integral because the values of £ which make the sum equal to the area are unknown. t This and similar problems, here treated by using the Theorem of the Mean for integrals, may lie treated from the point of view of differentiation as in § 7 or from that of DuhamePs or Osgood's Theorem as in §§ 34, 35. It should be needless to state that in any particular problem some one of the three methods is likely to be somewhat preferable to either of the others. The reason for laying such emphasis upon the Theorem of the Mean here and in the exercises below is that the theorem is in itself very important and needs to be thoroughly mastered. FUNDAMENTAL RULES 29 is approximately the pressure on the strip as it is the pressure at the top of the strip multiplied by the approximate area of the strip. Then io£6 (|)AA, where £ is some value between h and h + Ah, is the actual pressure on the strip. (It is sufficient to write the pressure as approximately whb(h)Ah and not trouble with the |.) The total pressure is then 2w£&(£) Ah or better the limit of that sum. Then P = lim Vwt&(^)dA = f ivhb(h)dh, where a is the depth of the top of the area and b that of the bottom. To evaluate the pressure it is merely necessary to find the breadth b as a function of h and integrate. EXERCISES Jib pb kf(x) dx = k i f(x) dx. a v a J^b /16 p b (u ± v) dx = I udx ± I vdx. a J a J (t Jib p b pb \p (x) dx < I f(x) dx < I

(x) be positive so that m < Q (x) = -'-^- < M and iu

{x)dx / (x) dx < I f(x) 4> (x) dx < M I {x) dx — fj. j (x)dx. Note that the integrals of [3/" — /(x)] (x) and [/(x) — m] (x) are positive and apply Ex. 2. 6. Evaluate the following by the direct application of (62) : Jib ])" — a 1 pb xdx = , (/3) I t x dx = & — e a . a 2 J a Take equal intervals and use the rules for arithmetic and geometric progressions. Jib 1 pb I x m dx = (b»> + 1 — a m + 1 ), (/3) / cdx = (c h — c a ). a m + 1 J, i log c In the first the intervals should be taken in geometric progression with r n = b/a. 30 INTRODUCTORY REVIEW 8. Show directly that (a) ( sin 2 xdx = I 7r, (/3) I cos n xdx = 0, if n is odd. «/0 Jo 9. With the aid of the trigonometric formulas cosx + cos2x + • • ■ + cos (n — l)x = \ [sin nx cot \ x — 1 — eos7ix], sinx + sin2x + • • • + sin (n — 1) x — \ [(1 — cosnx) cot | x — sinnx], Ji h p b cos xdx = sin o — sin a, (/3) J sin xdx = cos a — cos 6. a v a 10. A function is said to be even if /(— x) =/(x) and odd if /(— x) = — /(x) Show (or) f /(x) cZx = 2 T /(x) cZx, / even, (/3) f "f{x)dx = 0, /odd. 11. Show that if an integral is regarded as a function of the lower limit, the upper limit being fixed, then *'(«) = -£- f f(x)dx = - f(a) , if # (a) = f /(x) '(£) ' which states that the quotient of the increments AF and A«i> of two functions, in any interval in which the derivative <&'(x) does not vanish, is equal to the quotient of the derivatives of the functions for some interior point of the interval. What would the application of the Theorem of the Mean for derivatives to numerator and denominator of the left-hand fraction give, and wherein does it differ from Cauchy's Formula ? 14. Discuss the volume of revolution of y =f(x) as the limit of the sum of thin cylinders and compare the results with those found in Ex. 9, p. 10. 15. Show that the mass of a rod running from a to b along the x-axis is \ k(b 2 — a 2 ) if the density varies as the distance from the origin (k is a factor of proportionality). 16. Show («) that the mass in a rod running from a to b is (he same as the area under the curve y — p(x) between the ordinates x = a and x = b. and explain why this should be seen intuitively to be so. Show (/3) that if the density in a plane slab bounded by the x-axis, the curve y = /(x), and the ordinates x = ". and x = b is a r h function p (x) of x alone, the mass of the slab is I yp (x) dx ; also (7) that the mass of the corresponding volume of revolution is / iry"p(x)dx. 17. An isosceles triangle has the altitude a and the base 2b. Find (a) the mass on the assumption that the density varies as the distance from the vertex (meas- ured along the altitude). Find (/3) the mass of the cone of revolution formed by revolving the triangle about its altitude if the law of density is the same. FUNDAMENTAL RULES 31 18. In a plane, the moment of inertia I of a particle of mass m with respect to a point is denned as the product mr 2 of the mass by the square of its distance from the point. Extend this definition from particles to bodies. (a) Show that the moments of inertia of a rod running from a to b and of a circular slab of radius a are respectively x 2 p (x) dx and 1=1 2 irr z p (/•) dr, p the density, if the point of reference for the rod is the origin and for the slab is the center. (j8) Show that for a rod of length 21 and of uniform density, 1 = \MP with respect to the center and I = j Ml 2 with respect to the end, M being the total mass of the rod. (7) For a uniform circular slab with respect to the center I = \Ma 2 . (5) For a uniform rod of length 21 with respect to a point at a distance d from its center is / = M (I I" + d' 1 )- Take the rod along the axis and let the point be {a, /3) with d 2 = a 2 + /3 2 . 19. A rectangular gate holds in check the water in a reservoir. If the gate is submerged over a vertical distance II and has a breadth 7> and the top of the gate is a units below the surface of the water, find the pressure on the gate. At what depth in the water is the point where the pressure is the mean pressure over the gate ? 20. A dam is in the form of an isosceles trapezoid 100 ft. along the top (which is at the water level) and (SO ft. along the bottom and 30 ft. high. Find the pres- sure in tons. 21. Find the pressure on a circular gate in a water main if the radius of the circle is r and the depth of the center of the circle below the water level is d=j=r. 22. In space, moments of inertia are defined relative to an axis and in the for- mula I = mr 2 , for a single particle, r is the perpendicular distance from the particle to the axis. (a) Show that if the density in a solid of revolution generated by ?/ = /(.<_•) varies only with the distance along the axis, the moment of inertia about the axis of r b revolution is I = I I ■7ry i p(x)dx. Apply Ex. 18 after dividing the solid into disks. (j3) Find the moment of inertia of a sphere about a diameter in case the density is constant ; I — i Ma' 1 = T s - irpiv'. (7) Apply the result to find the moment of inertia, of a spherical shell with external and internal radii a and b ; 1=1 M(a 5 — ¥ J )/(a s — b s ). Let b = a and thus find I = I Ma 2 as the moment of inertia of a spherical surface (shell of negli- gible thickness). (5) For a cone of revolution I = j- 3 5 Ma 2 where a is the radius of the base. 23. If the force of attraction exerted by amass m upon a point is kmf(r) where r is the distance from the mass to the point, show that the attraction exerted at the origin by a rod of density p(x) running from a to b along the x-axis is A = C kf(x) p (x) dx, and that A = JcM/ab, M = p(b- a), is the attraction of a uniform rod if the law is the Law of Nature, that is, 32 INTRODUCTORY REVIEW 24. Suppose that the density p in the slab of Ex. 16 were a function p (x, y) of both x and y. Show that the mass of a small slice over the interval Ax,- would be of the form Xv=f(£) /•* n b V r v=/ " (r > 1 p (x, y) dy = (|) Ax and that / (x)Ax=J J p(x,y)dy\dx would be the expression for the total mass and would require an integration with respect to y in which x was held constant, a substitution of the limits f(x) and for y, and then an integration with respect to x from a to 6. 25. Apply the considerations of Ex. 24 to finding moments of inertia of (a) a uniform triangle y = mx, y = 0, x — a with respect to the origin, (/3) a uniform rectangle with respect to the center, (7) a uniform ellipse with respect to the center. 26. Compare Exs. 2-4 and 10 to treat the volume under the surface z = p (x, y) and over the area bounded by y = /(x), y = 0, x = a, x = b. Find the volume (a) under z = xy and over ?/'- = 4px. ?/ = 0. x = 0. x = 6, (/3) under z = x 2 + y' 2 and over x 2 + y 2 = a 2 , y = 0. x = 0. x = Q, x 2 y 2 z 2 X 2 if 2 (7) under ' 1 1 — = 1 and over — -\ = 1, y = 0, x = 0, x = a. a 2 6'- c 2 «' J 6 2 27. Discuss sectorial area -| | r 2 fZ0 in polar coordinates as the limit of the sum of small sectors running out from the pole. 28. Show that the moment of inertia of a uniform circular sector of angle a and radius a is \ pan 4 . Hence infer I =. \ p I i A d(p in polar coordinates. 29. Find the moment of inertia of a uniform (a) lemniscate r" = a- cos 2 2

and (e) the rose >• = a sin 3 (p. CHAPTER II REVIEW OF FUNDAMENTAL THEORY* 18. Numbers and limits. The concept and theory of real number, integral, rational, and irrational, will not be set forth in detail here. Some matters, however, which are necessary to the proper understand- ing of rigorous methods in analysis must be mentioned ; and numerous points of view which are adopted in the study of irrational number will be suggested in the text or exercises. It is taken for granted that by his earlier work the reader has become familiar with the use of real numbers. In particular it is assumed that he is accustomed to represent numbers as a scale, that is, by points on a straight line, and that he knows that when a line is given and an origin chosen upon it and a unit of measure and a positive direction have been chosen, then to each point of the line corre- sponds one and only one real number, and conversely. Owing to this correspond- ence, that is, owing to the conception of a scale, it is possible to interchange statements about numbers with statements about points and hence to obtain a more vivid and graphic or a more abstract and arithmetic phraseology as may be desired. Thus instead of saying that the numbers x\, x 2 , ■ ■ ■ are increasing algebra- ically, one may say that the points (whose coordinates are) x u x 2 , ■ ■ ■ are moving in the positive direction or to the right ; with a similar correlation of a decreasing suite of numbers with points moving in the negative direction or to the left. It should be remembered, however, that whether a statement is couched in geometric or algebraic terms, it is always a statement concerning numbers when one has in mind the point of view of pure analysis, t It may be recalled that arithmetic begins with the integers, including 0, and with addition and multiplication. That second, the rational numbers of the form p/q are introduced with the operation of division and the negative rational numbers with the operation of subtraction. Finally, the irrational numbers are introduced by various processes. Thus V2 occurs in geometry through the necessity of expressing the length of the diagonal of a square, and V3 for the diagonal of a cube. Again, it is needed for the ratio of circumference to diameter in a circle. In algebra any equation of odd degree has at least one real root and hence may be regarded as defining a number. But there is an essential difference between rational and irrational numbers in that any rational number is of the * The object of this chapter is to set forth systematically, witli attention to precision of statement and accuracy of proof, those fundamental definitions and theorems which lie at the basis of calculus and which have been given in the previous chapter from an intuitive rather than a critical point of view. t Some illustrative graphs will be given; the student should make many others. 3-1 INTRODUCTORY REVIEW form ± p/q with q ^ and can therefore be written down explicitly ; whereas the irrational numbers arise by a variety of processes and, although they may be represented to any desired accuracy by a decimal, they cannot all be written down explicitly. It is therefore necessary to have some definite axioms regulating the essential properties of irrational numbers. The particular axiom upon which stress will here be laid is the axiom of continuity, the use of which is essential to the proof of elementary theorems on limits. 19. Axiom of Continuity. If all the points of a line are dividedinto two classes such that every point of the first class rtrecedes every point of the second class, there must he a point C such that any point preceding C is in the first class and any point succeeding C is in. the second class. This principle may be stated in terms of numbers, as : If all real num- bers he assorted into tiro classes such that every number of the first class is algebraically less than every number of the second class, there mast he a number X such that any number less than X is in the first class and any number greater than X is in the second. The number N (or point C) is called the frontier number (or point), or simply the frontier of the two classes, and in particular it is the upper frontier for the first class and the lower frontier for the second. To consider a particular case, let all the negative numbers and zero constitute the first class and all the positive numbers the second, or let the negative numbers alone be the first class and the positive numbers with zero the second. In either case it is clear that the classes satisfy the conditions of the axiom and that zero is the frontier number such that any lesser number is in the first class and any greater in the second. If. however, one were to consider the system of all positive and negative numbers but without zero, it is clear that there would be no number X which would satisfy the conditions demanded by the axiom when the two classes were the negative and positive numbers ; for no matter how small a posi- tive number were taken as X. there would be smaller numbers which would also be positive and would not belong to the first class ; and similarly in case it were attempted to find a negative X. Thus the axiom insures the presence of zero in the system, and in like manner insures the presence of every other number — a matter which is of importance because there is no way of writing all (irrational) numbers in explicit form. Further to appreciate the continuity of the number scale, consider the four significations attributable to the phrase ''the interval from a to b." They are a = x =s h. it < x = h. a =§ x < b. a < z < b. That is to say. both end points or either or neither may belong to the interval. In the case a is absent, the interval has no first point ; and if b is absent, there is no last point. Thus if zero is not counted as a positive number, there is no least positive number ; for if any least number were named, half of it would surely lie less, and hence the absurdity. The axiom of continuity shows that if all numbers lie divided into two classes as required, there must be either a greatest in the first class or a least in the second — the frontier — but not both unless the frontier is counted twice, once in each class. FUNDAMENTAL THEORY 35 20. Definition of a Limit. If x is a variable which takes on succes- sive values x v x, y , ■ ■ ■, x t , Xj, ■ • ■, the variable x is said to approach, the con- stant I as a limit if the numerical difference between x and I ultimately becomes, and for all succeeding values of x remains, less titan ana preassiqned number no matter how k \ ^ ^ ''"'"' ' ^ small. The numerical difference between x and I is denoted by \x — l\ or \l — x\ and is called the absolute value of the difference. The fact of the approach to a limit may be stated as |.'' — /| < e for all a;'s subsequent to some x or x = / + n, \r}\ < e for all x's subsequent to some x, where e is a. positive number which may be assigned at pleasure and must be assigned before the attempt be made to find an x such that for all subsequent x's the relation \x — /] < c holds. So long as the conditions required in the definition of a limit are satisfied there is no need of bothering about how the variable approaches its limit, whether from one side or alternately from one side and the other, whether discontinuously as in the case of the area of the polygons used for computing the area of a circle or continuously as in the case of a train brought to rest by its brakes. To speak geometrically, a point x which changes its position upon a line approaches the point I as a limit if the point x ultimately conies into and remains in an assigned interval, no matter how small, surrounding /. A variable is said to become infinite if the numerical value of the variable ultimately becomes and remains greater than any preassigned number K, no matter how large.* The notation is x = oo, but had best be read " x becomes infinite," not "' x equals infinity." Theorem 1. If a variable is always increasing, it either becomes infinite or approaches a limit. That the variable may increase indefinitely is apparent. But if it does not become infinite, there must be numbers K which are greater than any value of the variable. Then any number must satisfy one of two conditions : either there are values of the variable which are greater than it or there are no values of the variable greater than it. Moreover all numbers that satisfy the first condition are less than any number which satisfies the second. All numbers are therefore divided into two classes fulfilling the requirements of the axiom of continuity, and there must be a number N such that there are values of the variable greater than any number N — e which is less than A'. Hence if e be assigned, there is a value of the variable which lies in the interval J\ r — e and c > b, then _ > - and - < - , a a c b (p) \a + b + c+ ...|s|a| + |6| + |c|+ •••, (7) \abc ■ ■ .\ = \a\-\b\-\c\- ■ ■, where the equality sign in (/3) holds only if the numbers a, b, c, • • • have the same sign. By these relations and the definition of a limit prove the fundamental theorems : If lim x = X and lim y = Y, then lim (x ± y) = -3T ± Y and lim xy = XY. 4. Prove Theorem 1 when restated in the slightly changed form : If a variable x never decreases and never exceeds K, then x approaches a limit N and N ==j K. Illustrate fully. State and prove the corresponding theorem for the case of a variable never increasing. 5. If Xi, x 2 , • • ■ and j/i, y 2 , • ■ ■ are two suites of which the first never decreases and the second never increases, all the ?/'s being greater than any of the x's, and if when e is assigned an n can be found such that y n — x n < e, show that the limits of the suites are identical. 6. If xi, x 2 , • • • and y\, y 2 , ■ ■ ■ are two suites which never decrease, show by Ex. 4 (not by Ex. 3) that the suites X\ + y\, x 2 + y 2 , • • • and X{yi, x 2 y 2 , • ■ • approach limits. Note that two infinite decimals are precisely two suites which never de- crease as more and more figures are taken. They do not always increase,for some of the figures may be 0. 7. If the word "all "in the hypothesis of the axiom of continuity be assumed to refer oidy to rational numbers so that the statement becomes : If all rational numbers be divided into two classes • • • , there shall be a number 2V (not neces- sarily rational) such that • • • ; then the conclusion may be taken as defining a number as the frontier of a sequence of rational numbers. Show that if two num- bers X, Y be defined by two such sequences, and if the sum of the numbers be defined as the number defined by the sequence of the sums of corresponding terms as in Ex. 6, and if the product of the numbers be defined as the number defined by the sequence of the products as in Ex. 0, then the fundamental rules X +Y=Y+ X, XY = YX, (X + Y) Z = XZ + YZ of arithmetic hold for the numbers X, Y, Z defined by sequences. In this way a complete theory of irrationals may be built up from the properties of rationals combined witli the principle of continuity, namely, 1° by defining irrationals as frontiers of sequences of rationals, 2° by defining the operations of addition, multi- plication, • • • as operations upon the rational numbers in the sequences, 3° by showing that the fundamental rules of arithmetic still hold for the irrationals. FUNDAMENTAL THEORY 37 8. Apply the principle of continuity to show that there is a positive number x such that x 2 = 2. To do this it should be shown that the rationals are divisible into two classes, those whose square is less than 2 and those whose square is not less than 2 ; and that these classes satisfy the requirements of the axiom of conti- nuity. In like manner if a is any positive number and n is any positive integer, show that there is an x such that x" = a. 21. Theorems on limits and on sets of points. The theorem on limits which is of fundamental algebraic importance is Theorem 2. If R (x, >/, z, ■ • •) be any rational function of the variables x, y, z, •••. and if these variables are approaching limits A*, Y, Z, ■■■, then the value of R approaches a limit and the limit is R(X, Y, Z, ■ ■ •), provided there is no division by zero. As any rational expression is made up from its elements by combinations of addition, subtraction, multiplication, and division, it is sufficient to prove the theorem for these four operations. All except the last have been indicated in the above Ex. 3. As multiplication has been cared for, division need be considered only in the simple case of a reciprocal 1/x. It must be proved that if lira x = A', then lira (1/x) = 1/X. Now i - X I 1 1 x _ X X by Ex. 3 (7) above. This quantity must be shown to be less than any assigned e. As the quantity is complicated it will be replaced by a simpler one which is greater, owing to an increase in the denominator. Since x = A", x — A' may be made numerically as small as desired, say less than e', for all x*s subsequent to some particular x. Hence if e' be taken at least as small as I \X\, it appears that |x| must be greater than \\X\. Then |x — X\ |x — XI e' , _ „ . , . !f the axiom of continuity. First note that as \x„4. p — x„ | < e for all x's subsequent to x„, the x's cannot become infinite. Suppose 1° that there is some number I such that no matter how remote x n is in the suite, there are always subsequent values of x which are greater than I and others which are less than /. As all the x's after x n lie in the interval 2e and as I is less than some x*s and greater than others, I must lie in that interval. Hence \l — x n+p \ < 2 e fur all 38 INTRODUCTORY REVIEW x's subsequent to x n . But now 2 e can be made as small as desired because e can be taken as small as desired. Hence the definition of a limit applies and the x*s approach I as a limit. Suppose 2° that there is no such number I. Then every number k is such that either it is possible to go so far in the suite that all subsequent numbers x are as great as k or it is possible to go so far that all subsequent x's are less than k. Hence all numbers & are divided into two classes which satisfy the requirements of the axiom of continuity, and there must be a number X such that the x*s ultimately come to lie between X — e" and X + e, no matter how small e is. Hence the x*s approach Jasa limit. Thus under either supposition the suite approaches a limit and the theorem is proved. It may be noted that under the second supposition the x's ultimately lie entirely upon one side of the point y and that the condition \x n + jl — x n \ < e is not used except to show that the x's remain finite. 22. Consider next a set of points (or their correlative numbers) without any implication that they form a suite, that is, that one may be said to be subsequent to another. If there is only a finite number of points in the set, there is a point farthest to the right and one farthest to the left. If there is an infinity of points in the set, two possibilities arise. Either 1° it is not possible to assign a point A' so far to the right that no point of the set is farther to the right — in which case the set is said to be unlimited chore — or 2° there is a point K such that no point of the set is beyond A' — and the set is said to be limited above. Similarly, a set may be limited below or un- limited below. If a set is limited above and below so that it is entirely contained in a finite interval, it is said merely to be limited. If there is a point C such that in any interval, no matter how small, surround- ing C there are points of the set, then C is called a point of condensa- tion of the set (C itself may or may not belong to the set). Theorem 4. Any infinite set of points which is limited lias an upper frontier (maximum ?), a lower frontier (minimum ?), and at least one point of condensation. Before proving this theorem, consider three infinite sets as illustrations : (a) 1, 1.9, 1.99, 1.999, • • •, (/3) - 2, • • •, - 1.99, - 1.9, - 1, ( 7 ) — i, — -1, — i, •■•,!, M- In (a) the element 1 is the minimum and serves also as the lower frontier ; it is clearly not a point of condensation, but is isolated. There is no maximum ; but 2 is the upper frontier and also a point of condensation. In (/3) there is a maximum — 1 and a minimum — 2 (for — 2 has been incorporated with the set). In (7) there is a maximum and minimum; the point of condensation is 0. If one could be sure that an infinite set had a maximum and minimum, as is the case with finite sets, there would be no need of considering upper and lower frontiers. It is clear that if the upper or lower frontier belongs to the set. there is a maximum or minimum and the frontier is not necessarily a point of condensation ; whereas FUNDAMENTAL THEORY- 39 if the frontier does not belong to the set, it is necessarily a point of condensation and the corresponding extreme point is missing. To prove that there is an upper frontier, divide the points of the line into two classes, one consisting of points which are to the left of some point of the set, the other of points which are not to the left of any point of the set — then apply the axiom. Similarly for the lower frontier. To show the existence of a point of con- densation, note that as there is an infinity of elements in the set, any point p is such that either there is an infinity of points of the set to the right of it or there is not. Hence the two classes into which all points are to be assorted are suggested, and the application of the axiom offers no difficulty. EXERCISES 1. In a manner analogous to the proof of Theorem 2, show that , s ,. x — 1 1 ,„. .. 3x — 1 5 . , ,. x 2 + 1 (a) Inn - = -, £ Inn — = -, ( 7 lim - = -1. x = ox — 2 2 x = i x + 5 i /i-ir- 1 2. Given an infinite series S = «i + » 2 + »3 + • • • • Construct the suite Si = Ml, So = «i + « 2 , S 3 = «i + « 2 + u s , ■ ■ -, Si = U X +M 2 H r Ui, ■ • -, where Si is the sum of the first i terms. Show that Theorem 3 gives : The neces- sary and sufficient condition that the series S converge is that it is possible to find an n so large that \S n + p — S„\ shall be less than an assigned e for all values of p. It is to be understood that a series converges when the suite of S's approaches a limit, and conversely. 3. If in a series n x — u 2 + u 3 — 114 + • • • the terms approach the limit 0, are alternately positive and negative, and each term is less than the preceding, the series converges. Consider the suites Si, .S3, S5, • • • and So, >'.(. >'«. • • • . 4. Given three infinite suites of numbers Xi, x 2 , •• •, x n , •••; 2/1. y- 2 . ■ ■ ■. y n . ■ ■ • : z x , z 2 , ■ ■ -, z n , ■ ■ ■ of which the first never decreases, the second never increases, and the terms of the third lie between corresponding terms of the first two, x„ ~ z„ ~ y„. Show that the suite of z's has a point of condensation at or between the limits approached by the x's and by the y"s ; and that if lim x = lim y = I, then the z's approach I as a limit. 5. Restate the definitions and theorems on sets of points in arithmetic terms. 6. Give the details of the proof of Theorem 4. Show that the proof as outlined gives the least point of condensation. How would the proof be worded so as to give the greatest point of condensation'/ Show that if a set is limited above,it has an upper frontier but need not have a lower frontier. 7. If a set of points is such that between any two there is a third, the set is said to be dense. Show that the rationals form a dense set ; also the irrationals. Show that any point of a dense set is a point of condensation for the set. 8. Show that the rationals p/q where q < K do not form a dense set — in fact are a finite set in any limited interval. Hence in regarding any irrational as the limit of a set of rationals it is necessary that the denominators and also the numer- ators should become infinite. 40 INTRODUCTORY REVIEW 9. Show that if an infinite set of points lies in a limited region of the plane, say in the rectangle a =§ x =i= b. c =§ y = d, there must be at least one point of condensation of the set. Give the necessary definitions and apply the axiom of continuity successively to the abscissas and ordinates. 23. Real functions of a real variable. If x be a variable which takes on a cert" in set of values of which the totality may be denoted by [./■] and if y is a second variable the value of which is uniquely determined fur each x of the set [./•], then y is said to be a function of x defined over the set [.r]. The terms " limited," " unlimited," " limited above," " unlimited below," • • • are applied to a function if they are applicable to the set [?/] of values of the function. Hence Theorem 4 has the corollary : Theorem 5. If a function is limited over the set [>], it has an upper frontier M and a lower frontier m for that set. If the function takes on its upper frontier M, that is, if there is a value x Q in the set [./•] such that f(x^) = M, the function has the abso- lute maximum M at x ; and similarly with respect to the lower frontier. In any case, the difference M—m between the upper and lower frontiers is called the oscillation of the function for the set [./•]. The set \x~\ is generally an interval. Consider some illustrations of functions and sets over which they are denned. The reciprocal 1/x is defined for all values of x save 0. In the neighborhood of the function is unlimited above for positive x*s and unlimited below for negative x's. It should be noted that the function is not limited 'in the interval < x = a but is limited in the interval eS /S a where e is any assigned positive number. The function + vk is defined for all positive x's including and is limited below. It is not limited above for the totality of all positive numbers ; but if K is assigned, the function is limited in the interval = x = K. The factorial function x ! is de- fined only for positive integers, is limited below by the value 1. but is not limited above unless the set [x] is limited above. The function E (x) denoting the integer not greater than x or "the integral part of x" is defined for all positive numbers — for instance E (?,) = E (ir) = 3. This function is not expressed, like the elemen- tary functions of calculus, as a " formula ; it is defined by a definite law. however, and is just as much of a function as x- + 3x + 2 or \ sin 2 2x + logx. Indeed it should be noted that the elementary functions themselves are in the first instance defined by definite laws and that it is not until after they have been made 'the subject of considerable study and have been largely developed along analytic lines that they appear as formulas. The ideas of function and formula are essentially distinct and the latter is essentially secondary to the former. The definition of function as given above excludes the so-called multiple-valued functions such as Vx and sin -1 x where to a given value of x correspond more than one value of the function. It is usual, however, in treating multiple-valued func- tions to resolve the functions into different parts or branches so that each branch is a single-valued function. Thus — ^ x is one branch and — vx the other branch FUNDAMENTAL THEORY 41 of Vx ; in fact when x is positive the symbol Vx is usually restricted to mean merely + Vx and thus becomes a single-valued symbol. ( )ne branch of sin- 1 x con- sists of the values between — i tt and + \ ir, other branches give values between \ it and | 7r or — \ir and — | w, and so on. Hence the term "function" will be restricted in this chapter to the single-valued functions allowed by the definition. 24. Ifx = a is tiny point of an interval on'/- which f(x\ is defined, the function f (x) is said to be continuous at the point x = a if \imf(x) =f(a\ no matter how x == a. x = a The function is said to be continuous in the interval if it is continuous at every point of the interval. If the function is not continuous at the point a, it is said to be discontinuous at a ; and if it fails to be con- tinuous at any one point of an interval, it is said to be discontinuous in the interval. Theorem 6. If any finite number of functions are continuous (at a point or over an interval), any rational expression formed of those functions is continuous (at the point or over the interval) provided no division by zero is called for. Theorem 7. If y = /'(') is continuous at x Q and takes the value y o =f(x Q ) and if z = <}>(y) is a continuous function of y at y = y , then z = <£[/(•'•)] will be a continuous function of x at x Q . In regard to the definition of continuity note that a function cannot be con- tinuous at a point unless it is defined at that point. Thus c-- 1 /- 1 ' is not continuous atx = because division by is impossible and the function is undefined. If. how- ever, the function be defined at as/(0) = 0, the function becomes continuous at x = 0. In like manner the function 1/x is not continuous at the origin, and in this case it is impossible to assign to/(0) any value which will render the function continuous; the function becomes infinite at the origin and the very idea of be- coming infinite precludes the possibility of approach to a definite limit. Again, the function E (x) is in general continuous, but is discontinuous for integral values of x. "When a function is discontinuous at x = a. the amount of tlte discontinuity is the limit of the oscillation M — m of the function in the interval a — 5 < x < a + 5 surrounding the point a when 5 approaches zero as its limit. The discontinuity of E (x) at each integral value of x is clearly 1 ; that of 1/x at the origin is infi- nite no matter what value is assigned to/(0). In case the interval over which /(x) is defined has end points, say « =x ^6, the question of continuity at x = a must of course be decided by allowing x to approach a from the right-hand side only ; and similarly it is a question of left- handed approach to b. In general, if for any reason it is desired to restrict the approach of a variable to its limit to being one-sided, the notations x = a+ and x == b~ respectively are used to denote approach through greater values (right- handed) and through lesser values (left-handed). It is not necessary to make this specification in the case of the ends of an interval ; for it is understood that x shall take on only values in the interval in question. It should be noted that 42 INTRODUCTORY REVIEW lim f(x) = /(x ) when x = x + in no wise implies the continuity of f(x) at x ; a simple example is that of E (x) at the positive integral points. The proof of Theorem is an immediate corollary application of Theorem 2. For lim R [/(x), (lim x), • • ■], and the proof of Theorem 7 is equally simple. Theorem 8. If ,/%?) is continuous at x = l, when x > 1, x" < 1, when 0=i=xx" if m>n and x>l, x'"n and 0 and cannot be negative when A<0. Hence the right-hand derivative cannot be positive and the left-hand derivative cannot be negative. As these two must be equal if the function has a derivative, it follows that they must lie zero, and the derivative is zero. The second theorem is an immediate corollary. For as the function is continuous it must have a maximum and a minimum (Theorem 11) both of which cannot be zero unless the function is always zero in the interval. Now if the function is identically zero, the derivative is identically zero ami the theorem is true ; whereas if the function is not identically zero, either the maximum or minimum must be at an interior point, and at that point the derivative will vanish. * That the theorem is true for any part of the interval from a to h if it is true for the whole interval follows from the fact that the conditions, namely, that/' lie continuous and that/" exist, hold for any part of the interval if they hold for the whole. FUNDAMENTAL THEORY 47 To prove the last theorem construct the auxiliary function * (X, =/(,, -,(.) - „ - ., m=m , f w =m _ m . o —a o — « As ^ («) = i/' (6) = 0, Rolle's Theorem shows that there is some point for which i//(£) = 0, and if this value be substituted in the expression for \p' (x) the solution for /'(I) gives the result demanded by the theorem. The proof, however, requires the use of the function \j/ (x) and its derivative and is not complete until it is shown that \p (x) really satisfies the conditions of Kolle*s Theorem, namely, is continuous in the interval a^x^b and has a derivative for every point a uD" + * - 'v in the expansion of the (n + 1) st derivative of uv by Leibniz's Theorem. With regard to this rule and the other elementary rides of operation (4)-(7) of the previous chapter it should be remarked that a theorem as well as a rule is in- volved — thus: If two functions u and v are differentiate at x , then the product uv is differentiate at x , and the value of the derivative is u (x ) v' (x ) + u' (x ) v (x ) . And similar theorems arise in connection with the other rules. As a matter of fact the ordinary proof needs only to be gone over with care in order to convert it into a rigorous demonstration. But care does need to be exercised both in stating the theorem and in looking to the proof. For instance, the above theorem concerning a product is not true if infinite derivatives are allowed. For let u be — 1, 0, or + 1 according as x is negative, 0, or positive, and let v = x. Xow v has always a deriva- tive which is 1 and u has always a derivative which is 0, -f =o, or according as x is negative, 0, or positive. The product uv is |x|, of which the derivative is — 1 for negative x's, + 1 for positive x's, and nonexistent for 0. Here the product has no derivative atO, although each factor has a derivative, and it would be useless to have a formula for attempting to evaluate something that did not exist. EXERCISES 1. Show that if at a point the derivative of a function exists and is positive, the function must be increasing at that point. 2. Suppose that the derivatives f'{a) and f'(b) exist and are not zero. Show that /(a) and /(b) are relative maxima or minima of/ in the interval a =l=x === b, and determine the precise criteria in terms of the signs of the derivatives /'(a) and /'(b). 3. Show that if a continuous function has a positive right-hand derivative at every point of the interval a^.r^h, then /(b) is the maximum value of/. Simi- larly, if the right-hand derivative is negative, show that /(b) is the minimum of/. 4. Apply the Theorem of the Mean to show that if /'(.'") is continuous at a, then f(x')-f(x") Inn -> — '- x — L—f'^a), y' ', x" == a X — X x' and x" being regarded as independent. 5. Form the increments of a function/ for equicrescent values of the variable : A 1 / = /(o + h) -/(a), A,f = f(a + 2 h) -/(a + h), A 3 f = f(a + 3h)-f{a + 2h), ■••. FUNDAMENTAL THEORY 49 These are called first differences ; the differences of these differences are A 1 2 / = /(a + 2 A) - 2/(a + h) +/(«), A.?/ =/(« + 3 A) - 2/(a + 2 /;) + /(a + h), ■ ■ ■ which are called the second differences ; in like manner there are third differences Af / = f(a + 3 h) - 3/(a + 2h)+ 3/(a + fl) - /(a), ■ . . and so on. Apply the Law of the Mean to all the differences and show that Af/= h°f"{a + 9 x h + 2 h), \\f = h 3 f'"(a + 9 x h + 2 h 4- B z h), ■■■. Hence show that if the first n derivatives of /are continuous at a, then .„. , .. A-/ A 3 / A"/ /"(a) = Inn — f , / (") = hin — f- , • • • , /°'>(«) = Inn -— . »i0 ft" 7i=0 ft" A = ft" 6. Cauchy's Theorem. If /(x) and 4>(x) are continuous over a == j =s />, have derivatives at each interior point, and if $' (x) does not vanish in the interval, f(b)-f(a) = /'(g) or /( a + /,)-/(«.) = /(r t + (9/Q (6) - ^ (a) 0'(£) (a + /j) - (a) '(". + 9h)' Prove that this follows from the application of Rolle's Theorem to the function (x) =f(x)-f(a) - [4>(x) - *(«)] {fl'i^l ■ 7. One application of Ex. <> is to the theory of indeterminate forms. Show that if f(a) = — a) n + - t Jr-p/*- 1 ^) + { — ^-/<»>(0. (n — 1 ) ! ?i ! What are the restrictions that must be imposed on the function and its derivatives '.' 9. If a continuous function over a=x = b has a right-hand derivative at each point of the interval which is zero, show that the function is constant. Apply Ex. 2 to the functions f(x) + e (x — a) and/(x) — e(x — a) to show that the maximum difference between the functions is 2 e (b — a) and that / must therefore be constant. 50 INTRODUCTORY REVIEW 10. State and prove the theorems implied in the formulas (4)-(6), p. 2. 11. Consider the extension of Ex. 7, p. 44, to derivatives of functions defined over a dense set. If the derivative exists and is uniformly continuous over the dense set, what of the existence and continuity of the derivative of the function when its definition is extended as there indicated ? 12. If f(x) has a finite derivative at each point of the interval a =§i x === b, the derivative f'(x) must take on every value intermediate between any two of its values. To show this, take first the case where f'(a) and/'(6) have opposite signs and show, by the continuity of / and by Theorem 13 and Ex. 2, that /'(f) = 0. Next if f'{a)i I ) X m (b - a) == ^mfr =E 5)/(^)8i = %V& = M ( h ') (A) will also hold. Let * = 2 »»,-$, ( f and /,f are continuous, and if f(x) is continuous, they are differentiable and have the common derivative f(fi)- To prove that L c b t = i a e + L'.'. consider c as one of the points of division of the interval from a to b. Then the sums S will satisfy .S ( '' = S a c + iS c 6 , and as the limit of a sum is the sum of the limits, the corresponding relation must hold for the frontier L. To show that L% —— L" it is merely necessary to note that S% =— Sg because in passing from b to a the intervals 5,- must be taken with the sign opposite to that which they have when the direction is from a to b. From (J.) it appears that m (b — a) =E S* s 3/ (1, — (( j and hence in the limit m (b — <() g L tt h s M (I, _ a ). 52 INTRODUCTORY REVIEW Hence there is a value /*, m =j= yu == M, such that L* = fi(b — a). To show that Lff is a continuous function of /3, take Jv >|-M| and \m\, and consider the relations L3 + A _ if = £f + Z | + » _ L B = L | + a = ^ H and never decreases, show that the function is integrable. This follows from the fact that 20; == O is finite. 7. More generally, let/(x) be such a function that 20,- remains less than some number K, no matter how the interval be divided. Show that/ is integrable. Such a function is called & function of limited variation (§ 127). 8. Change of variable. Let f(x) be continuous over a === x =§ b. Change the variable to x = (t x ) and b = (t), <£'(£), and/[0 (t)] are continuous in t over t x =§= t =s t 2 . Show that f b f{x)dx= f'/IXWWtB or f* (,) f(x)dx= f /[> (W(') dt. Do this by showing that the derivatives of the two sides of the last equation with respect to t exist and are equal over t x ^t^t.,, that the two sides vanish when t = t x and are equal, and hence that they must be equal throughout the interval. 9. Osgood's Theorem. Let ai be a set of quantities which differ uniformly from f{ki) 8* by an amount &5i, that is, suppose ai =f{£;) Si + fr5;, where | ft | < e and a =§= £ === b. Trove that if /is integrable, the sum So:, approaches a limit when 5,- = and that r b the limit of the sum is I f(x)dx. J a . 10. Apply Ex. 9 to the case A/ = /'Ax + fAx where/' is continuous to show directly that f (b) - f (a) = / f'(x)dx. Also by regarding Ax = ) + ■ • . + V^)T / >W + i? - (1) Such an expansion is necessarily true because the remainder R may be considered as defined by the equation ; the real significance of the formula must therefore lie in the possibility of finding a simple ex- pression for R, and there are several. Theorem. On the hypothesis that f(x) and its first n derivatives exist and are continuous over the interval a =§ .'• =2 b, the function may be expanded in that interval into a polynomial in x — a, /(•<•) =/('<) + (■'' - ")/'(") + ^ff^7>) + • ' • (x — aY- 1 + V=T)" r/< "" >( " ) + '''' (1) with the remainder 7? expressible in any one of the forms (x — aY J/"(l — 0)"- 1 nl J w (n-l)\ J K - } ~yf' l t n -\r n Kn. + h-t)dt, (2) where h = x — a and a < $ < x or £ = a -\- Oh where < 6 < 1. A first, proof may be made to depend on Rolle's Theorem as indicated in Ex. 8, p. 49. Let x be regarded for the moment as constant, say equal to b. Construct 05 56 DIFFERENTIAL CALCULUS the function \p (x) there indicated. Note that \p (a) = \p (b) = and that the deriva- tive ^'(x) is merely *'&)=--. ^-/ ( ' (« + n\ L—\ /('>)-/(«) - (h - a)f'{a) (n— 1)! (b — a)" L ^- a >"-> -i)(a)l. (n-1)!'' K, \ By llolle's Theorem ^'(f) = 0. Hence if £ be substituted above, the result is /(ft) =/(a) + (6 - «)/'(«■) + • • • + (t ' " a) *~* f<*-»(a) + i^L-^/(»>(£), (n — ] ) ! n ! after striking out the factor — (b — £)" _1 , multiplying by (b — a) a /n, and transposing f(b). The theorem is therefore proved with the first form of the remainder. This 'proof does not require the continuity of the nth derivative nor its existence at a and at b. The second form of the remainder may be found by applying Rolle's Theorem to * (x) = m - /(•<•) - (6 - *)/'(*) (&-x)"-i /(w _ x _ p ^ (n-1) ! where P is determined so that Ii = (b— a) P. Note that \p (b) = and that by Taylor's Formula \p (a) = 0. Now (n-l)! ("-5)! Hence if £ be written % = a + 6h where h = l>—a, then 6— £=6— a— 6h—{b— a)(l— #). And K = (&- a P = (6- a) K —^- \. ° } /<">(£) = ^ , * , / ( '° (*)• (n — 1) ! (u — i) ! The second form of R is thus found. In this work as before, the result is proved for x = 6, the end point of the interval a .=s x = b. But as the interval could be considered as terminating at any of its points, the proof clearly applies to any x in the interval. A second proof of Taylor's Formula, and the easiest, to remember, consists in integrating the nth derivative n times from a to x. The successive results are f >0(x)dx =/»-i(x)T=/o-i>(a;) -/<«-« (a). Jn J« f X f /<"> (.'•) (a) - (x - re)/(»- 1 )(a). f* • • f ><>(x)cZx« =/(x) -/(a) - (x - a)/' (a) J a J a 2 ! (n — 1) ! The formula is therefore proved with 7? in the form / ■■■ ( f(")(x)dx n . To trans- form this to the ordinary form, the Law of the Mean may be applied ((05), § 10). For \x - a) < f )'(")(.r) dx < M(x - a), m {X — '-'■- <['■■■ f'f0»(x)dx»< J u n ! J a J a (X _ a)» ^ f* r ',y„w.a,7„„ ^ M ( X ~ a) " TAYLOR'S FORMULA; ALLIED TOPICS 57 where m is the least and M the greatest value of /W(x) from a to x. There is then some intermediate value /(">(£) = (J. such that J a J a 111 This proof requires that the nth derivative be continuous and is less general. The third proof is obtained by applying successive integrations by parts to the J^ h f'(a + h — t) dt to make the integrand contain o higher derivatives. f(a + h) - /(«) = f */'(« + h - dt = if (a + h - *)] ' + f ' */''(« + ft - t) dt = hf'{a) + I t-f"(a + ft- t) 1 ' + f 1 i 2 /"'(a + ft - Q ) + *f'(0) + §/"(<>) +• • • + ^~[j]f Cn ^ (°) + A> > ( 3 ) a = ^/<"W = — ~y { (i-ey-\r»xe.r)= ^^J^-\foo (x - t )dt. 32. Both Taylor's Formula and its special case, Maclaurin's, express a function as a polynomial in h = x — a, of which all the coefficients except the last are constants while the last is not constant but depends on h both explicitly and through the unknown fraction 6 which itself is a function of h. If, however, the nt\\ derivative is continuous, the coeffi- cient /^{a + Oh)/ ?i ! must remain finite, and if the form of the deriva- tive is known, it may be possible actually to assign limits between which f 0,) (a + Oh)/n ! lies. This is of great importance in making approximate calculations as in Exs. 8 if. below ; for it sets a limit to the value of 11 for any value of n. Theorem. There is only one possible expansion of a function into a polynomial in h = x — a of which all the coefficients except the last are constant and the last finite ; and hence if such an expansion is found in any manner, it must be Taylor's (or Maclaurin's). To prove this theorem consider two polynomials of the nth order c + cji + c.M + h Cn-ih"- 1 + c n h> 1 = G + C x h + CJi 2 -\ + (J n -ih> 1 - 1 + C n h n , which represent the same function and hence are equal for all values of h from to b — a. It follows that the coefficients must be equal. For let ft approach 0. 68 DIFFERENTIAL CALCULUS The terms containing h will approach and hence c and C may be made as nearly equal as desired ; and as they are constants, they must be equal. Strike them out from the equation and divide by h. The new equation must hold for all values of h from to b — a with the possible exception of 0. Again let h = and now it follows that c x = C v And so on, with all the coefficients. The two devel- opments are seen to be identical, and hence identical with Taylors. To illustrate the application of the theorem, let it be required to find the expan- sion of tan j- about when the expansions of sinx and cosx about are given. sin x = x — -J x 3 + T \ - - x 5 + Px 7 , cos x = 1 — \ x 2 + jfj x 4 + Qx c , where P and Q remain finite in the neighborhood of x = 0. In the first place note that tanx clearly has an expansion ; for the function and its derivatives (which are combinations of tan x and sec x) are finite and continuous until x approaches 1 it. By division, X + \ x 3 + T 2 T X 5 1 - \ x 2 + - 2 \x 4 + Qx«)x - i x :i + T ^ x 5 j + Px 7 x - i x 3 + sV x 5 ; + Qx~> 1 X 3_ ^ X 5: + ( p_Q) X 7 1 j3_ 1 x 5| + J 3 j7 + 1 QjO T^ x5 \ Hence tan x = x + i x 3 + T 2 5 x 5 -I — ■ — x 7 , where S is the remainder in the division cos x and is an expression containing P, Q, and powers of x ; it must remain finite if P and Q remain finite. The quotient S/cos x which is the coefficient of x" therefore remains finite near x = 0, and the expression for tan x is the Maclaurin expansion up to terms of the sixth order, plus a remainder. In the case of functions compounded from simple functions of which the expan- sion is known, this method of obtaining the expansion by algebraic processes upon the known expansions treated as polynomials is generally shorter than to obtain the result by differentiation. The computation may be abridged by omitting the last terms and work such as follows the dotted line in the example above ; but if this is done, care must be exercised against carrying the algebraic operations too far or not far enough. In Ex. 5 below, the last terms should be put in and carried far enough to insure that the desired expansion has neither more nor fewer terms than the circumstances warrant. EXERCISES 1. Assume E = (b - a) k P; show R = — (l ~ 0) "~ V ( '° (£)• (n- l)\k 2. Apply Ex. 5, p. 29, to compare the third form of remainder with the first. 3. Obtain, by differentiation and substitution in (1), three nonvanishing terms: (a) sin- 1 x. (i — 0. (p) tanhx, a — 0, (7) tanx, a = \v, (5) cscx, a -- I 7T, (e) e 8inz , a — 0, (f) log sin x. a = \ir. 4. Find the xth derivatives in the following cases and write the expansion: (a) sin x, a = 0, (p) sin x. a — \ ir, (7) c x , a = 0. (5) c* a = 1, (e) logx, a = 1, (f) (1 + x)t, a = 0. TAYLOR'S FORMULA; ALLIED TOPICS 59 5. By algebraic processes find the Maclaurin expansion to the term in x 5 : (a) sec x, (j3) tanh x, (7) — Vl — x 2 , (5) e^sinx, (e) [log (1 — x)] 2 , (f) + Vcosh x, (77) e« nx , (6) log cos x, (1) log Vl + x 2 . The expansions needed in this work may be found by differentiation or taken from B. O. Peirce's "Tables." In (7) and (f) apply the binomial theorem of Ex. 4 (f). In (77) let y = sin x, expand e'J, and substitute for y the expansion of sin x. In (d) let cosx = 1 — y. In all cases show that the coefficient of the term in x 6 really remains finite when x = 0. 6. If f(a + It) = c + c x h + c. 2 h" + • • • + Cn-x/c- 1 + c n h n , show that in f f(a + h) dh = cji + - 1 If- + ^ ft 3 4- . ; . + Hizl A" + f cW/t J o 2 3 n Ju the last term may really be put in the form Ph n +1 with P finite. Apply Ex. 5, p. 29. _ r x dx 7. Apply Ex. G to sin-ix = | , etc., to find developments of «/o Vl - x 2 (a) sin _1 x, (j3) tan- 1 x, (7) sinh -1 x, . .. . 1 + x , „ /* x , r x sin x , (5) log-i-, (e) / e-^dx, (f) / -dx. 1 — X t/o t/o X In all these cases the results may be found if desired to n terms. 8. Show that the remainder in the Maclaurin development of e r is less than x n e x /n ! ; and hence that the error introduced by disregarding the remainder in com- puting e r is less than x»c r /n '.. How many terms will suffice to compute e to four decimals ? How many for e 5 and for c 0A ? 9. Show that the error introduced by disregarding the remainder in comput- ing log(l+ x) is not greater than x"/n if x > 0. How many terms are required for the computation of log If to four places ? of log 1.2 ? Compute the latter. 10. The hypotenuse of a triangle is 20 and one angle is 31°. Find the sides by expanding sinx and cosx about a — \iv as linear functions of x — \ it. Examine the term in (x — \ 7r) 2 to find a maximum value to the error introduced by neglecting it. 11. Compute to 6 places: (a) ei, ((3) log 1.1, (7) sin 30'. (5) cos 30'. During the computation one place more than the desired number should be carried along in the arithmetic work for safety. 12. Show that the remainder for log (1 + x) is less than x"/n (1 + x) n if x < 0. Compute (a) log 0.9 to 5 places, (/3) log 0.8 to 4 places. 13. Show that the remainder for tan _1 x is less than x"/n where n may always be taken as odd. Compute to 4 places tan -1 1. 14. The relation \ir = tan- 1 1 = 4 tan -1 1 — tan- 1 2 ^ g enables -]- ir to be found easily from the series for ta.\\- l x. Find \ir to 7 places (intermediate work carried to 8 places). 15. Computation of logarithms, (a) If a = log ijfl, b = log \\, c = log |J-, then log2 = 7 a — 2 b + 3 c, log 3 = 11 a — 3 b + 5 c, log 5 = 16 a — 46 + 7 c. GO DIFFERENTIAL CALCULUS Now a = — log (1 — y 1 ,,), b = — log (1 — T ^y), c = log (1 + /,,) are readily computed and hence log 2, log 3, logo may be found. Carry the calculations of a, b, c to 10 places and deduce the logarithms of 2, 3, 5, 10, retaining only 8 places. Com- pare Peirce's "Tables," p. 109. 1 + x 2 x n (8) Show that the error in the series for log is less than — — Com- V ' ' 1 - X H(l- X)» pute log 2 corresponding to x = | to 4 places, log If to 5 places, log It to 6 places. w show ,„^ = 2r^ + i(^y + ... + -i-(^y" Li + ^ + , 7 LP + 7 3 \p + qj 2n-l \p + qj give an estimate of 2?o n+1 , and compute to 10 figures log 3 and log 7 from log 2 and log 5 of Peirce's "Tables " and from 41og3- 41og2- log5 = log 81 , 41og7- 51o-2- log3 - 2 logo = log 80 ' 7 1 - 1 16. Compute Ex. 7 (e) to 4 places for x = 1 and to G places for x — \. 17. Compute sin -1 0.1 to seconds and sin -1 \ to minutes. 18. Show that in the expansion of (1 + a;)* the remainder, as x is > or < 0, is R l fc-(fc-l)---(fc-n + l) r?i 1 • 2 • • • n or /?„< fc.(fc-l).-.(fc — n + 1) x» | 1 • 2 • • • ji (l+z)"- Ilcnce compute to 5 figures Vl03, V98, V28, V250, VlOOO. 19. Sometimes the remainder cannot be readily found but the terms of the expansion appear to be diminishing so rapidly that all after a certain point appear negligible. Thus use Peirce's "Tables," Xos. 774-789, to compute to four places (estimated) the values of tan 6°, log cos 10°, esc 3°, sec 2°. 20. Find to within 1% the area under cos (./•-) and sin (x-) from to \ it. 21. A unit magnetic pole is placed at a distance L from the center of a magnet of pole strength M and length 2/. where l/L is small. Find the force on the pole if (a) the pole is in the line of the magnet and if (3) it is in the perpendicular bisector. 4 3// //Y 2 , . 2 Ml 3/T Ans. (a) — -— (1 + e) with e about 2 1 -1 , (3) — (1 — e) with e about - 22. The formula for the distance of the horizon is T) =VT/t where I) is the distance in miles and h is the altitude of the observer in feet. Prove the formula and show that the error is about }',' for heights up to a few mih s. Take the radius of the earth as 3900 miles. 23. bind an approximate formula for the dip of the horizon in minutes below the horizontal if h in feet is the height of the observer. 24. If S is a circular arc and C its chord and c the chord of half the arc, prove S = ] (8 c - C) (1 + e) where e is about .S 4 /7080 A' 4 if Ii is the radius. 25. If two quantities differ from each other by a small fraction e of their value show that their geometric mean will differ from their arithmetic mean by about J e- of its value. 26. The algebraic method may be applied to finding expansions of some func- tions which become infinite. (Thus if the series for cos.;- and sin./- be divided to find cot x, the initial term is 1/x and becomes infinite at x = just as cotx does. TAYLOR'S FORMULA; ALLIED TOPICS 61 Such expansions are not Maclaurin developments but are analogous to them. The function xcotx would, however, have a Maclaurin development and the expansion found for cot x is this development divided by x.) Find the develop- ments about x = to terms in x 4 for (a) cotx, (0) cot 2 x, (7) cscx, (5) csc 3 x, (e) cotx cscx, (f) l/(ta,n~ 1 x) 2 , (7?) (sinx — tanx) -1 27. Obtain the expansions : (a) log sin x = log x - | x 2 - T ^ x 4 + R, (/3) log tan x = log x + 1 x 2 + ^x 4 + • • • , (7) likewise for log versx. 33. Indeterminate forms, infinitesimals, infinites. If two functions /'(•'') and <£(•'') are defined for x = a and if <£(") =h 0, the quotient // is defined for x = a. But if (f> (") = 0, the quotient f/ is not defined for a. If in this case /'and are defined and continuous in the neighborhood of a and /'(") =£ 0, the quotient will become infinite as ./■ = « ; whereas if f(a) = 0, the behavior of the quotient f/ is not immediately appar- ent but gives rise to the indeterminate form 0/0. In like manner if/ and become infinite at a, the quotient /'/<£ is not defined, as neither its numerator nor its denominator is defined : thus arises the indeter- minate form cc/cc. The question of determining or evaluating an indeterminate form is merely the question of finding out whether the quotient /'/ approaches a limit (and if so, what limit) or becomes positively or negatively infinite when x approaches ". Theorem. IJHospitaVs Rule. If the functions /(.r) and <£(.r), which give rise to the indeterminate form 0/0 or x>/x> when .'• = a, are con- tinuous and differentiable in the interval a < x === b and if h can be taken so near to a that <£'(.'") does not vanish in the interval and if the quotient f'/' of the derivatives approaches a limit or becomes posi- tively or negatively infinite as x = a, then the quotient f/ will ap- proach that limit or become positively or negatively infinite as the case may be. Hence an indeterminate, form 0/0 or x/x may he replaced by tin 1 quotient of the derivatives of numerator and denominator. Cask I. f(a) =

' with the understanding that proper restrictions were satisfied by/', (a) = oo. Apply Cauchy's Formula as follows : f(x)-f(b ) = f{x) l-f(b)/f(x) = f'(Z) > a(x)- (x) are as near zero as desired. The second equation above then shows that f(x)/

must approach the same limit a.sf'/' is sure to be indeterminate. The advantage of being able to differentiate therefore lies wholly in the possibility that the new form be more amenable to algebraic transformation than the old. The other indeterminate forms • cc, 0°, 1", do , ao — oo may be reduced to the foregoing by various devices which may be indicated as follows : 0-oo=- = -, Oo = ei°soo = e oio g o = e o-oo ) ..., oc-3o = loge»-» = lo- e — . 1 1 ' e°° oo The case where the variable becomes infinite instead of approaching a finite value a is covered in Ex. 1 below. The theory is therefore completed. Two methods which frequently may be used to shorten the work of evaluating an indeterminate form are the method of E -functions and the application of Taylor's Formula. By definition an E-f unction for the point x — a is any continuous function which approaches a finite limit other than when x = a. Suppose then that/(x) or

) _l + (P - Q)x 2 x 2 log (1 + x) ~ x- (x - \ x 2 ' + Ex 3 ) ~ 1 - I x + Ex- ' and now if x = 0, the limit is at once shown to be simply J. When the functions become infinite atx = a, the conditions requisite for Taylor's Formula are not present and there is no Taylor expansion. Nevertheless an expan- sion may sometimes be obtained by the algebraic method (§ 32) and may frequently be used to advantage. To illustrate, let it be required to evaluate cot x — 1/x which is of the form x — x when x = 0. Here cos x 1 + I x- + Px 4 ll-i x 2 + Px 4 1 / 1 „ cot x = = '—A Z = _ ~ = - ( 1 x 2 + Sx'- sin x x — J x- J + Qx u x 1 — I x- + ( t >x 4 x V 3 TAYLOR'S FORMULA; ALLIED TOPICS 63 where S remains finite when x = 0. If this value be substituted for cot x, then lini [ cot a; ) = lim ( x + -S'x 3 ) = lim ( x + Sx 3 \ = 0. x = o\ x) x = o\x 3 x) x=0\ 3 / 34. An infinitesimal is a variable which is ultimately to approach the limit zero ; an infinite is a variable which is to become either positively or negatively infinite. Thus the increments A// and A.r are finite quan- tities, but when they are to serve in the definition of a derivative they must ultimately approach zero and hence may be called infinitesimals. The form 0/0 represents the quotient of two infinitesimals ; * the form oc/cc, the quotient of two infinites; and Ox, the product of an infin- itesimal by an infinite. If any infinitesimal a is chosen as the primary infinitesimal, a second infinitesimal ft is said to be of the same order as a if the limit of the quotient ft/a exists and is not zero when a = ; whereas if the quotient ft/a becomes zero, ft is said to be an infinites- imal of higher order than a, but of lower order if the quotient becomes infinite. If in particular the limit ft/a n exists and is not zero when a = 0, then ft is said to be of the nth order relative to a. The deter- mination of the order of one infinitesimal relative to another is there- fore essentially a problem in indeterminate forms. Similar definitions may be given in regard to infinites. Theorem. If the quotient ft/a of two infinitesimals approaches a limit or becomes infinite when a = 0, the quotient ft' /a' of two infin- itesimals which differ respectively from ft and a by infinitesimals of higher order will approach the same limit or become infinite. Theorem. Duhamel's Theorem. If the sum %a t = a + a -\ (- a n of n positive infinitesimals approaches a limit when their number n becomes infinite, the sum 2/3,- = ft l -f- ft., + • • • + ft,,, where each ft ; differs uniformly from the corresponding a t by an infinitesimal of higher order, will approach the same limit. As a' — a is of higher order than a and j3' — j3 of higher order than /3, l im e^ = , lim^iUo or £ = 1 + ,, £ = l+fi a /3 a p where i) and t are infinitesimals. Now a' = a (1 + rj) and /3 r = /3 (1 + f), Hence — = — and Inn — = lim — , a' a I + rj a' a provided /3/a: approaches a limit ; whereas if p/a becomes infinite, so will ji'/a'. In a more complex fraction such as (/3 — y)/a it is not permissible to replace /3 * It cannot he emphasized too strongly that in the symbol O/0 the 0's arc merely sym- bolic for a mode of variation just as r. is: they are not actual 0's and some other nota- tion would be far preferable, likewise for 0- x, 11 , etc. 64 DIFFERENTIAL CALCULUS and 7 individually by infinitesimals of higher order ; for /3 — 7 may itself be of higher order than /3 or 7. Thus tan x — sin x is an infinitesimal of the third order relative to x although tanx and sin x are only of the first order. To replace tan x and sin x by infinitesimals which differ from them by those of the second order or even of the third order would generally alter the limit of the ratio of tanx — sinx to x 3 when x = 0. To prove Duhamel's Theorem the /3's may be written in the form ft= «i(l + Vi ), * = 1| 2, •••, n, hf|,>/■ ■'■ h by virtue of (4), § 2. From this appears tin; important theorem : The quotient dy/dx is the derivative of y with respect to x no matter what the independent rar'adde, may be. It is tins theorem which really justifies writing tin; derivative as a fraction and treating the component differentials according to the rules of ordinary fractions. For higher derivatives this is not so, as may be seen by reference to Ex. 10. As Ay and Ax are regarded as infinitesimals in defining the deriva- tive, it is natural to regard dy and dx as infinitesimals. The difference Ay — dy may be put in the form f(r + Ax)-f(x) Ay — dy = Ax -fV) Ax, 0>) wherein it appears that, when Ax == 0, the bracket approaches zero. Hence arises the theorem: If x is tin- independent, variable and if Ay and dy are regarded as infinitesimals, tin' difference Ay — dy is an infin- itesimal of higher order than A.r. This lias an application to the TAYLOR'S FORMULA; ALLIED TOPICS 65 subject of change of variable in a definite integral. For if x = (t'), then dx — '(t)dt, and apparently f f(x)dx= f >[ where (t l ) = a and ;) &r i} ]£/[> (/,.)] *'(*,) M a M = dt, the limits of which are the two integrals above. Now as Ax differs from dx = '(f)dt by an infinitesimal of higher order, so f(.r)Xr will differ from /[<£(£)] <£'(/)(/£ by an infinitesimal of higher order, and with the proper assumptions as to continuity the difference will be uni- form. Hence if the infinitesimals /(.r) A,r be all positive, Duhamel's Theorem may be applied to justify the formula for change of variable. To avoid the restriction to positive infinitesimals it is well to replace Duhamel's Theorem by the new Theorem. Osgood's Theorem. Let a v a,, ■■■, a n be n infinitesimals and let a f differ uniformly by infinitesimals of higher order than A.r from the elements ,/'(•'',-) A.)- t of the integrand of a definite integral Jf(x)dx, where/ is continuous ; then the sum la — a 1 + a + • • • + cc n a approaches the value of the definite integral as a limit when the num- ber n becomes infinite. Let ctj = /(..",) A/', -)-£", Ar,-. where |f,-| '(U)dU will differ uniformly (compare Theorem 18 of § 27 and the above theorem on Ay — dy) by an infinitesimal of higher order, and so will the infinitesimals /(Jf) Ax; and f[4> (<,•)] '(ti) dt t . Hence the change of variable suggested by the hasty substitution is justified. 66 DIFFERENTIAL CALCULUS EXERCISES 1. Show that T Hospital's Rule applies to evaluating the indeterminate form f(x)/4>(x) when x becomes infinite and both /and

' gtana; lo 1 if x is the independent variable. Show that the higher derivatives D 2 y, I>£y, • • • are not the quotients d-y/dx-. d'^y/dx 3 , ■ ■ • if x and y are expressed in terms of a third variable, but that the relations are 2 _ d 2 ydx — d 2 xdy 3 _ dx (dxd 3 y — dyd 3 x) — 3 d 2 x (dxd'-y — dyd 2 x) xV ~ dx 3 ' xl '~ dx- ""' ""' The fact that the quotient d n y/dx n , n > 1, is not the derivative when x and y are expressed parametrically militates against the usefulness of the higher differentials and emphasizes the advantage of working with derivatives. The notation d"y/dx 11 is. however, used for the derivative. Nevertheless, as indicated in Exs. 1 f > 19, higher differentials may be used if proper care is exercised. 11. Compare the conception of higher differentials witli the work of Ex. 5, p. 48. 12. Show that in a circle the difference between an infinitesimal arc and its chord is of the third order relative to either arc or chord. 13. Show that if /3 is of the nth order with respect to a. and 7 is of the first order with respect to laced by one which differs from it by an infinitesimal of higher order than it without affecting the order of the product. 15. Let A and B be two points of a unit circle and let the angle A OB subtended at the center be the primary infinitesimal. Let the tangents at A and B meet at T, and OT cut the chord AB in M and the arc A H in C. Find the trigonometric expression for the infinitesimal difference TC — CM and determine its order. 16. Compute d 2 (x sin x) = (2 cos x — x sin x) dx 2 + (sin x + x cos x) d 2 x by taking the differential of the differential. Thus find the second derivative of x sin x if x is the independent variable and the second derivative with respect to t if x = 1 + t 2 . 17. Compute the first, second, and third differentials, d 2 x ^ 0. (a) x 2 cosx, (£) Vl — x log (1 — x), (7) xe 2x sinx. 18. In Ex. 10 take y as the independent variable and hence express D 2 y, ])J'y in terms of D^x, B 2 x. Cf. Ex. 10, p. 14. 19. Make the changes of variable in Exs. 8. 0. 12. p. 14. by the method of differentials, that is, by replacing the derivatives by the corresponding differential expressions where x is not assumed as independent variable and by replacing these differentials by their values in terms of the new variables where the higher differ- entials of the new independent variable are set equal to 0. 20. Reconsider some of the exercises at the end of Chap. I. say, 17-10. 22. 23. 27, from the point of view of Osgood's Theorem instead of the Theorem of the .Mean, 21. Find the areas of the bounding surfaces of the solids of Ex. 11. p. 18. 68 DIFFERENTIAL CALCULUS 22. Assume the law F—kmm'/r- of attraction between particles. Find the attraction of : (a) a circular wire of radius a and of mass M on a particle m at a distance r from the center of the wire along a perpendicular to its plane ; Ans. kMmr (a- + r-)~ 2 . (/3) a circular disk, etc., as in (a) ; Ans. 2kifma-' 2 (l — r/vV- + a- ). (7) a semicircular wire on a particle at its center ; Ans. 2kMm/7ra-. (5) a finite rod upon a particle not in the line of the rod. The answer should be expressed in terms of the angle the rod subtends at the particle. (e) two parallel equal rods, forming the opposite sides of a rectangle, on each other. 23. Compare the method of derivatives (§ 7). the method of the Theorem of the Mean (§ 17). and the method of infinitesimals above as applied to obtaining the for- mulas for (a) area in polar coordinates, (ft) mass of a rod of variable density. (7) pres- sure on a vertical submerged bulkhead. (0) attraction of a rod on a particle. < )btain the results by each method and state which method seems preferable for each case. 24. Is the substitution dx = 4>'(t)dt in the indefinite integral f f(x) dx to obtain the indefinite integral j f[(t)] 4>'(t)dt justifiable immediately ? 36. Infinitesimal analysis. To work rapidly in the applications of calculus to problems in geometry and physics and to follow readily the books written on those subjects, it is necessary to have some familiarity with working directly with infinitesimals. It is possible by making use of the Theorem of the Mean and allied theorems to retain in every ex- pression its complete exact value ; but if that expression is an infini- tesimal which is ultimately to enter into a quotient or a limit of a sum, any infinitesimal which is of higher order than that which is ultimately kept will not influence the result and may be discarded at any stage of the work if the work may thereby be simplified. A few theorems worked through by the infinitesimal method will serve partly to show how the method is used and partly to establish results which may be of use in further work. The theorems which will be chosen are: 1. The increment A./' and the differential d.r of a variable differ by an infinitesimal of higher order than either. 2. If a tangent is drawn to a curve, the perpendicular from the curve to the tangent is of higher order than the distance from the foot of the perpendicular to the point of tangency. 3. An infinitesimal arc differs from its chord by an infinitesimal of higher order relative to the arc. 4. If one angle of a triangle, none of whose angles are infinitesimal, differs infinitesimally from a right angle and if // is the side opposite and if is another angle of the triangle, then the side opposite is h sin except for an infinitesimal of the second order and the adjacent side is It cos except for an infinitesimal of the first order. TAYLOR'S FORMULA; ALLIED TOPICS 69 The first of these theorems has been proved in § 35. The second follows from it and from the idea of tangency. For take the z-axis coincident with the tangent or parallel to it. Then the perpendicular is Ay and the distance from its foot to the point of tangency is At. The quotient Ay /Ax approaches as its limit because the tangent is horizontal ; and the theorem is proved. The theorem would remain true if the perpendicular were replaced by a line making a constant angle with the tangent and the distance from the point of tangency to the foot of the perpendicular were re- placed by the distance to the foot of the oblique line. For if Z PMN — 9, P/ PM TM TX PJVcsc <>10 PX Tx esc 9 i-^ccte TX and therefore when P approaches T with 9 constant, P M/ TM approaches zero and PM is of higher order than TM. The third theorem follows without difficulty from the assumption or theorem that the arc has a length intermediate between that of the chord and that of the sum of the two tangents at the ends of the chord. Let 9 X and 9., be the angles between the chord and the tangents. Then s - A B A T + Til - All _ A M (sec 9 X - 1) + Mil (sec 0., - 1 ) A M + Mil ~ A M+ MB AM + MB (6) M 11 substitution Now as AB approaches 0, both sec 9 X — 1 and sec 9., — 1 approach and theii coefficients remain necessarily finite. Hence the difference between the arc and the chord is an infinitesimal of higher order than the chord. As the arc and chord are therefore of the same order, the difference is of higher order than the arc. This result enables one to replace the arc by its chord and vice versa in discussing infinitesimals of the first order, and for such purposes to consider an infinitesimal arc as straight. In discussing infinitesimals of the second order, tl would not be permissible except in view of the further theorem given below in § •)'. and even then the substitution will hold only as far as the lengths of arcs are concerned and not in regard to directions. For the fourth theorem let 9 be the angle by which C departs from 00° and with the perpendicular 11M as radius strike an arc cutting JIC. Then by trigonometry AC = A M + MC = h cos + JIM tan 0, BC = h sin.?!. + Ittf(sec 6 - 1). Xow tan 9 is an infinitesimal of the first order with respect to 9 ; for its Maclaurin development begins with 9. And sec 9 — 1 is an infinitesimal of the second order; for its development begins with a term in 9' 1 . The theorem is therefore proved. This theorem is frequently applied to infinitesimal triangles, that is. triangles in which li is to approach 0. 37- As a further discussion of the third theorem it may be recalled that by defi- nition the length of the arc of a curve is the limit of the length of an inscribed polygon, namely, lira (a Arf + A;/f + " A/j + Ays + ■■■ +^ Ar,f + Ay; 70 DIFFERENTIAL CALCULUS / — s 7, /~r~, r^ Ax- + Aw- — dx' 2 — dy 2 Now VAx 2 + Ay 2 - Vdx- + dy 2 = y VAx 2 + Ay 2 + Vdx 2 + dy 2 _(Ar- dx) (Ax + dx) + (Ay - dy) (Ay + dy) ■\ Ac- + Ay 2 + Vdx 2 + dy 2 and VAx- + Ay 2 — Vdx? + dy 2 _ (Ax — dx) Ax + dx VAx- + Ay 2 VAx- + Ay 2 V Ax 2 + Ay 2 + Vdx 2 + dy 2 (Ay - dy ) Ay + dy VAx- + Ay 2 VAx- + Ay 2 + Vdx 2 + dy 2 + Bui Ax — dx and Ay — dy are infinitesimals of higher order than Ax and Ay. Hence the right-hand side must approach zero as its limit and hence a Ax- -h Ay 2 differs from Vdx 2 + dy 2 by an infinitesimal of higher order and may replace it in the sum s = lim X V Ax? + Ay} — lim ^j Vdx: 1 + dy 2 = f ' Vl + y" 2 dx. The length of the arc measured from a fixed point to a variable point is a func- tion of the upper limit and the differential of arc is ds = d ( Vl + y' 2 dx = Vl + y' 2 dx = Vdx 2 + dy 2 . To find the order of the difference between the arc and its chord let the origin be taken at the initial point and the x-axis tangent to the curve at that point. The expansion of the arc by Maclaurin's Formula gives ,s(x) = s(0) + x.s'(O) + \ xV(0) + \xH"'(0x), where s (0) = 0, .s'(0) = Vl + y' 2 = 1, s"(0) = VV \ = 0. ■n l + y" 1 1 o Owing to the choice of axes, the expansion of the curve reduces to y =f(x) = y (0) + x/(0) + J x 2 y"(0x) = J x 2 y"(0x), and hence the chord of the curve is c (x) = Vx 2 + y 2 = x Vl + } x 2 [y" (dx)] 2 = x (1 + x 2 P), where P is a complicated expression arising in the expansion of the radical by Maclaurin's Formula. The difference 8 (x) - c (x) = [x + £ xH"'(0x)-] - [x (1 + x-P)] = x :! (I s"'{6x) - P). This is an infinitesimal of at least the third order relative to x. Now as both ,s(x) and c (x) are of the first order relative to x, it follows that the difference s (x) — c (x) must also be of the third order relative to either .s(x) or c(x). Note that the proof assumes that y" is finite at the point considered. This result, which has been found analytically, follows more simply though perhaps less rigorously from the fact that see 6 X — 1 and sec #., — 1 in (6) are infinitesimals of the second order with 6 X and 6.-,. 38. The theory of contort of plane curves may be treated by means of Taylor's Formula and stated in terms of infinitesimals. Let two curves a = /*(.'') and y = g(x) be tangent at a given point and let the TAYLOR'S FORMULA; ALLIED TOPICS 71 origin be chosen at that point with the sr-axis tangent to the curves. The Maclaurin developments are 2, = /(,•) = \f"(0)x> + ■ ■ ■ + j^rTj- l *"- 1 f (n - 1 XQ) + ^> x " ) f°' ) (0) + ••• If these developments agree up to hut not including the term in x n , the difference between the ordinates of the curves is /(*) - 9 (*) = ^ ■'•" C/ w (0) - !/"\0)] + -.., /(»)(0) * r /«>(0), and is an infinitesimal of the nth order with respect to .r. The curves are then said to have contact of order n — 1 at their point of tangency. In general when two curves are tangent, the derivatives f"(0) and g"(0) are unequal and the curves have simple contact or contort of the first order. The problem may be stated differently. Let PM be a line which makes a constant angle 6 with the ./'-axis. Then, when P approaches T, if RQ be regarded as straight, the proportion lim (PR : PQ) = lim (sin Z PQR : sin Z PRQ) = sin 6 : 1 shows that PR and PQ are of the same order. Clearly also the lines TM and TX are of the same order. Hence if PR . PQ lim =p 0, x, then lim =£ 0, oc . (7-A7< ' ' (TM)" Hence if two curves have contact of the (n — l)st order, the segment of a line intercepted between ~"j£ the two curves is of the wth order with respect to the distance from the point of tangency to its foot. It would also be of the nth order with respect to the perpendicular TF from the point of tangency to the line. In view of these results it is not necessary to assume that the two curves have a special relation to the axis. Let two curves y = f(.r) and y = g (./•) intersect when x = a, and assume that the tangents at that point are not parallel to the //-axis. Then (n— 1)1 n\ ( x — a y-i (? — „y y = !/o + '> - " ) [/'(/') + ■■■ + -, T—f (n - l) (<> ) + , )- — 2 /.' cos r (y — y {) ) = 0, where it remains to determine 7i so that the development of the circle will coincide with that nf the curve as far as written. Differentiate the equation of the circle. and dy 11 sin t + (.r — cos r — (y — ?/,/)]- + [/.' sin r + (x — a)]' 2 W = 1 ds- ' " A' co.- v = y + (•'' - ")/'(") + i (•'' - ")- - 1 i > ..- + ••• TAYLOR'S FORMULA: ALLIED TOPICS is the development of the circle. The equation of the coefficients of (r — «)-, =/"(a), given ic = tfcos 3 T ' v " ° /"(a) B _see 8 r_{l + [/ / (g)]a; This is the well known formula for the radius of curvature and shows that the cir- cle of curvature has contact of at least the second order with the curve. The circle is sometimes called the osculating circle instead of the circle of curvature'. 39. Throe theorems, one in geometry and two in kinematics, will now be proved to illustrate the direct application of the infinitesimal methods to such problems. The choice will be : 1. The tangent to the ellipse is equally inclined to the focal radii drawn to the point of contact. 2. The displacement of any rigid body in a plane may be regarded at any instant as a rotation through an infinitesimal angle about some point unless the body is moving parallel to itself. 3. The motion of a rigid body in a plane may lie regarded as the rolling of one curve upon another. For the first problem consider a secant PP' which may he converted into a tangent TT' by letting the two points approach until they coincide. Draw the focal radii to 7' and P' and strike arcs with F and F' as centers. As F'P + PF - FT' + P'F = 2 a, it follows that XL' = MP'. Now consider the two triangles PP'M and P'PX nearly right-angled at M and X. The sides PP'. PM, PX. P'M, P'X are all infinitesimals of the same order and of the same order as the angles at F and F'. By proposition i of § 36 MP' = PP' cos Z PP'M + e v XP= PP' cos Z P'PX + e. 2 , where e, and c, are infinitesimals relative to MI" and XP or PP'. Therefon lim [cosZ I'P'M- cosZ P'PX] - cosZ TPF- cos Z T PF' = lini '-- L PP 0, ~~J>rB' and the two angles TPF' and T'PF are proved to be equal as desired. To prove the second theorem note first that if a body is rigid, its position is com- pletely determined when the position Ali of any rectilinear segment of the body is known. Let the points .1 and B of the body be de- scribing curves A A' and Lilt' so that, in an infinitesimal interval of time, the line AB takes the neighboring posi- tion A ' IV . Erect the perpendicular bisectors of the lines A A' and BB' and let them intersect at 0. Then the tri- angles AOB and A'OB' have the three sides of the one equal to the three sides of the other and are equal, and The second may be obtained from the first by a mere rotation about through the angle J0.1'= FjOB'. Except for infinitesimals of higher order, the magnitude of the angle is AA'/OA or BB'/OB. Next let the interval of time approach so that A' approaches A and B' approaches B, The perpendicular bisectors will approach 74 DIFFEEEXTIAL CALCULUS the normals to the arcs AA' and Bli' at A and B, and the point will approach the intersection of those normals. The theorem may then be stated that: At any instant of time the motion of a rigid body in a plane may be considered as a rotation through an infinitesimal angle about the intersection of the normals to the paths of any two of its points at that in- stant ; the amount of the rotation will be the distance ds that any point moves divided by the distance of that point from the instantaneous center of rotation ; the angular velocity about Vie instantaneous center will be this amount of rotation divided by the interval of time dt, that is, it will be t/r, where v is the velocity of any point of the body and r is its distance from the instantaneous center of rotation. It is therefore seen that not only is the desired theorem proved, but numerous other details are found. As has been stated, the point about which the body is rotating at a given instant is called the instantaneous center for that instant. As time goes on, the position of the instantaneous center will generally change. If at each instant of time the position of the center is marked on the moving plane or body, there results a locus which is called the moving centrode or body centrode; if at each instant the position of the center is also marked on a fixed plane over which the moving plane may be considered to glide, there results another locus which is called the fixed centrode or the space centrode. From these definitions it follows that at each instant of time the body centrode and the space centrode intersect at the instantaneous center for that instant. Consider a series of positions of the instantaneous center as 7 :> _ : ;/'_ 1 / , P 1 P„ marked in space and Q_oQ_iQQ 1 Q 2 marked in the body. At a given instant two of the points, say P and Q, coincide ; an instant later the body will have moved so as to bring Q x into coin- cidence with Pj ; at an earlier instant Q_i was coincident with P_i. Now as the motion at the instant when P and Q are together is one of rotation through an infinitesimal angle about that point, the angle between PP 1 and QQ X is infinitesimal and the lengths PI\ and QQ 1 are equal ; for it is by the rotation about P and Q that Q x is to be brought into coincidence with P x . Hence it follows 1 3 that the two centrodes are tangent and 2 = that the distances PPj = QQ X which the point of contact moves along the two curves during an infinitesimal inter- val of time are the same, and this means that the two curves roll on one another without slipping — because the very idea of slipping implies that the point of con- tact of the two curves should move by different amounts along the two curves, the difference in the amounts being the amount of the slip. The third theorem is therefore proved. EXERCISES 1. If a finite parallelogram is nearly rectangled, what is the order of infinites- imals neglected by taking the area as the product of the two sides? What if the figure were an isosceles trapezoid '.' What if it were any rectilinear quadrilateral all of whose angles differ from right angles by infinitesimals of the same order '.' 2. On a sphere of radius r the area of the zone between the parallels of latitude X and X -f d\ is taken as 2 7rrcos\- rd\. the perimeter of the base times the slant height. Of what order relative to d\ is the infinitesimal neglected? What if the perimeter of the middle latitude were taken so that 2 -nr- cos (X + \d\)d\ were assumed ? TAYLOR'S FORMULA; ALLIED TOPICS To 3. What is the order of the infinitesimal neglected in taking Awr-dr as the volume of a hollow sphere of interior radius r and thickness dr ? What if the mean radius were taken instead of the interior radius ? Would any particular radius be best? 4. Discuss the length of a space curve y =/(j), z = g (x) analytically as the length of the plane curve was discussed in the text. 5. Discuss proposition 2. p. 68. by Maclaurin's Formula and in particular show that if the second derivative is continuous at the point of tangency. the infinites- imal in question is of the second order at least. How about the case of the tractrix a v — - loir + "\ a- — x 2 , fl + % a- - x- and its tangent at the vertex x = a ? How about s(x) — c(x) of § 37 ? 6. Show that if two curves have contact of order n —1. their derivatives will have contact of order n — 2. What is the order of contact of the kth derivatives k < n - 1 ? 7. State the conditions for maxima, minima, and points of inflection in the neighborhood of a point where /<")( 5f(a), S = 2 s\f{a) + -/"(") + -"-/(*)(£)] , (i 120 J TAYLOR'S FORMULA; ALLIED TOPICS 77 are the areas of the circumscribed trapezoid, the curve, the inscribed trapezoid. Hence infer that to compute the area under the curve from the inscribed or cir- cumscribed trapezoids introduces a relative error of the order 5-, but that to com- pute from the relation S = % (2 /q + !/- 2 + • ■ ■ + 2/2 n] — Vo — V2n] + R by using the work of Ex. 23 and infer that the error R is less than (b — «) 5 4 / (iv) (S)/ 45 - This method of computation is known as Simpson's Rule. It usually gives accu- racy sufficient for work to four or even live figures when 5 = 0.1 and b — a = 1 ; for /( iv )(x) usually is small. 25. Compute these integrals by Simpson's Rule. Take 2?i = 10 equal intervals. Carry numerical work to six figures except where tables must be used to find /(./•) : J™ - (1 1' /* 1 fl T — = log 2 = 0.09315, (fi) I — — = tan-i 1 = - tt = 0.78535, l x Jo 1 + x' 2 4 (7) f sinxr/x = 1.00000, (5) f " log 10 xclx = 21og 10 x- 3/= 0.10776, /»ilo |t (l+x) (& , = r^±^,lx =0.82247. Jo 1 + X- Jo X The answers here given are the true values of the integrals to five places. 26. Show that the quadrant of the ellipse x — 4' /t 78 DIFFERENTIAL CALCULUS 29. If the catenary y = c cosh (x/c) gives the shape of a wire of length L sus- pended between two points at the same level and at a distance I nearly equal to L, find the first approximation connecting L, I. and d, where d is the dip of the wire at its lowest point below the level of support. 30. At its middle point the parabolic cable of a suspension bridge 1000 ft. long between the supports sags 50 ft. below the level of the ends. Find the length of the cable correct to inches. 40. Some differential geometry. Suppose that between the incre- ments of a set of variables all of which depend on a single variable t there exists an equation which is true except for infinitesimals of higher order than At = dt, then the equation will be exactly true for the differ- entials of the variables. Thus if fAx + gAy + hAz + I At + • • • + e x -f e 2 + • • • = is an equation of the sort mentioned and if the coefficients are any func- tions of the variables and if e , e , • ■ ■ are infinitesimals of higher order than dt, the limit of ,Ax A// , A.v At At At At At At At is dx d ii dz dt ' dt dt fdx + gdy + htlz + Mt 0: and the statement is proved. This result is very useful in writing down various differential formulas of geometry where the approximate relation between the increments is obvious and where the true relation between the differentials can therefore be found. For instance in the case of the differential of arc in rectangular coor- dinates, if the increment of are is known to differ from its chord by an infinitesimal of higher order, the Pythagorean theorem shows that the equation A * 2 = A,- 2 + A/ or A.s 2 = A.r 2 + Ay 2 + As 2 (7) is true except for infinitesimals of higher order: and hence ds- = d.r 2 + dif Of ds- = d.r- + dif -f dz~. (7 ') In the ease of plane polar coordinates, the triangle PP'X (see Fig.) has two curvilinear sides PP' and PX and is right- Ar angled at A. The Pythagorean theorem may be J^J -. The two most used systems of coordinates other than rectangular in space are the polar or spherical and the cylindrical. In the first the distance r = OP from the pole or center, the longitude or meridional angle cj>, and the colatitude or polar angle are chosen as coor- dinates ; in the second, ordinary polar coordinates r = OM and <£ in the a'y-plane are combined with the ordinary rectangular z for distance from that plane. The formulas of transformation are z — r cos 6, r sin 6 sin , z P r,, x (8) (9) , x = r cos cf>, r — \j a + I/', 4> = tan" Formulas such as that for the differential of arc may be obtained for these new coordinates by mere transformation of (7') according to the rules for change of variable. In both these cases, however, the value of ds may be found readily by direct inspection of the figure. The small parallelepiped (figure for polar case) of which As is the diagonal has some of its edges and faces curved instead of straight: all the angles, however, are right angles, and as the edges are infinitesimal, the equations certainly suggested as holding except for infinitesimals of higher order are 80 DIFFERENTIAL CALCULUS As 2 = Ay' 2 + r sin'- 6A<£- + r\6 : and A*' 2 = \r + rXtf + A.r (10) or ds 2 = di* + r % sin 2 $d$ 2 + rW and ds* — dr 1 + ihltf -\- dz 2 . (10') To make the proof complete, it would be necessary to show that noth- ing but infinitesimals of higher order have been neglected and it might actually be easier to transform ~\dx 2 -\-dy 2 -\-dz 1 rather than give a rigorous demonstration of this fact. Indeed the infinitesimal method is seldom used rigorously; its great use is to make the facts so clear to the rapid worker that he is willing to take the evidence and omit the proof. In the plane for rectangular coordinates with ridings parallel to the //-axis and for polar coordinates with ladings issuing from the pole the increments of area differ from dA=ydx and dA = I rdcf> (11) respectively by infinitesimals of higher order, and A= C'ydx and A = \ C ' fd$ (IF) are therefore the formulas for the area under a curve and between two ordinates, and for the area between the curve and two radii. If the plane is ruled by lines parallel to both axes or by lines issuing from the pole and by circles concentric with the pole, as is customary for double inte- gration (§§ 131, 134), the increments of area differ respectively by infinitesimals of higher order from dA =d.rd :/ and dA = rdrd. (12) and the formulas for the area in the two cases are A = lim J) A.l =ff'U = fC^di/, (12') A = lim V A. ! = I id. 1 = f Irdrdtft, where the double integrals are extended over the area desired. The elements of volume which are required for triple integration (£§ 133, 131) over a volume in space may readily be written down for the three cases of rectangular, polar, and cylindrical coordinates. In the first case space is supposed to be divided up by planes .r = <<. >/ = />, z — c perpendicular to the axes and spaced at infinitesimal intervals; in the second case the division is made by the spheres r=(t concentric with the pole, the planes = ft through the polar axis, and the cones 8 = c of revolution about the polar axis ; in the third case by the cylin- ders )' = <(, the planes cf> — h, and the planes r: = r. The infinitesimal TAYLOR'S FORMULA; ALLIED TOPICS 81 volumes into which space is divided then differ from do = dxdydz, dv = r sin ddrd^dO, dv = rdrddz (13) respectively by infinitesimals of higher order, and Ixdadz, -sin 0drdd0 > Iff lrddz (13') are the formulas for the volumes. 41. The direction of a line in space is represented by the three angles which the line makes with the positive directions of the axes or by the cosines of those angles, the direction cosines of the line. From the defi- nition and figure it appears that / = cos a = ds m = cos fi ds ' n = cos v = 7 dz ds (14) are the direction cosines of the tangent to the arc at the point; of the tangent and not of the chord for the reason that the increments are replaced by the differ- entials. Hence it is seen that for the direc- tion cosines of' tin' tangent the proportion I : hi : n = ds : dij : dz (14') holds. The equations of a space curve are ■'•=/(0> y = 0(f)> n = h(t) x l in terms of a variable parameter t* At the point (x , >/ (t . £ Q ) where t = t the equations of the tangent lines would then be z P' Ax, y -A/ P Ay Y .'/ - I/O V ~ ?/o As the cosine of the angle 6 between the two directions given by the direction cosines /. w, n and /', ///', ?i' is cos0 = //' + unii' + nn', so IV + w »>' -f- nn' = (16) is the condition for the perpendicularity of the lines. Xow if (.?•, //, z) lies in the plane normal to the curve at ,r , y , z , the lines determined by the ratios .<• — :>• : // — y : z — z Q and (d.r) : ('dy) : (dz) ""'ill be per- pendicular. Hence the equation of the normal pin ne is (■'" - ■'',;,) ( dx )o + (!/— l/o)( rI l/)o + ('- ~ r -^" l: - ►o = ° or /'( f.)( r - .<\) + rj\ fjii/ — i/„ ) + h'{ tjc: — ;:,) = 0. (17) * Fur the sake <>f generality the parametric form in f is assumed : in a particular case a simplification might be made by taking one of the variables as t and one of tbe functions /', ,'/'. /(' would then be 1. Thus in Ex. 8 (e), y should be taken as t. 82 DIFFERENTIAL CALCULUS The tangent plane to the curve is not determinate; any plane through the tangent line will be tangent to the curve. If A be a parameter, the pencil of tangent planes is x + A .'/ - !/o (1+X) 0. There is one particular tangent plane, called the osculating plane ,which is of especial importance. Let * - *o = f%) - + */"(0 - 2 + */'"(*) A r = t-t , f < i < t, with similar expansions for y and «, be the Taylor developments of x, y, z about the point of tangency. When these are substituted in the equation of the plane, the result is 1 2 T ' 'f"(to) +x ff"(Q ./%) /Co) (1+X) h"(t y +-;- aw ' /"'(g ) , A ffM (1 + X) h'"(0 This expression is of course proportional to the distance from any point x, y, z of the curve to the tangent plane and is seen to be in general of the second order with respect to r or ds. It is, however, possible to choose for A that value which makes the first bracket vanish. The tan- gent plane thus selected has the property that the distance of the curve from it in tin- neighborhood of the point of tangency is of the third order and is called the osculating plane. The substitution of the value of A gives /'(O oXQ h '( f o) /"(O o'XQ h "(0 (dyPs - dz ( Py\(x = or \(dx\° (dy\ (dz\\ = (18) K'^'Oo (' 7 V)o (^)ol - .r ) + (dzd-.r - d,-d-r: )t (y - ,,/) + (dxd 2 y - dyd 2 .r )i :: - s Q ) = as the equation of the osculating plane. In ease /"(/„)=//"( f ) =h"(t c ) = 0, this equation of the osculating plane vanishes identically and it is neces- sary to push the development further (Ex. 11). 42. For the case of plane curves the curvature is defined as the rate at which the tangent turns compared with the description of arc, that is, as d/ds if d denotes the differential of the angle through which the tangent turns when the point of tangency advances along the curve by ds. The radius of curvature R is the reciprocal of the curvature, that is. it is ds/dfy. Then ,/<£ = , /tair 1 d i/ dcf> d(f) = 1(1 + dl) + m (m + dm) + n(n + dri). But P + i,P + n* = 1 and (/ + dl) 2 + (m + dm) 2 + (n + efo) 2 = 1. Hence dp + r///r + tfw 2 = 2 — 2 cos (/^> = (2 sin }, dcf>)' 2 , 1_ R- 2 sin \ d(f> ~~ds ~ dl' 1 + dnP + dri 1 ds 2 !> 2 + ,„' 2 + n' 2 , (19') where accents denote differentiation with respect to s. The torsion of a space curve is defined as the rate of turning of the osculating plane compared with the increase of arc (that is, dif//ds, where dip is the differential angle the normal to the osculating plane turns through), and may clearly be calculated by the same formula as the curvature provided the direction cosines L, M, N of the normal to the plane take the places of the direction cosines /, w, n of the tangent line. Hence the torsion is A/A' 2 dL 2 + r/.l/ 2 -f dX 2 ds 2 = L' 2 + M' 2 + X' (20) and the radius of torsion R is defined as the reciprocal of the torsion, where from the equation of the osculating plane .1/ dyd 2 z — dzd 2 ij dziPs — d.nPn dxiPij — d ijd 2 x v sum of squares The actual computation of these quantities is somewhat tedious. (20') The vectorial discussion of curvature and torsion (§ 77) gives a better insight into the principal directions connected with a space curve. These are the direction of the taut/cut. that of the normal in the osculating plane and directed towards the concave side of the curve and called the principal normal, and that of the normal to the osculating plane drawn upon that side which makes the three direc- tions form a right-handed system and called the binormal. In the notations there given, combined with those above, r = xi + yi + zk, t = /i + »dc)t+(n.cZc)n. But as t.c = n.c = 0, t.dc = — c«cZt and n.cZc = — ocZn. Hence dc = - (odt)t - (ccZn)n = - Ctds + Tnds = cZs + - cZ.s. Hence £=-£ + £. £=- = + £• T =-£+?■ <*> ds 7i R ds R R tZ.s L R Formulas (22) are known as Frenet's Formulas ; they are usually written with — R in the place of R because a left-handed system of axes is used and the torsion, being an odd function, changes its sign when all the axes are reversed. If accents denote differentiation by s, \x' y' z x" y" z above formulas, -— -—- — — — - — , usual formulas, — '-— — y — — "—• (23) risdit-handed R **-+V"'- + *'- left-handed R xf'- + y"- + z"- EXERCISES 1. Show that in polar coordinates in the plane, the tangent of the inclination of the curve to the radius vector is rd

. 6) = and F(r. 0) = in polar coordinates in space are respectively cones and surfaces of revolution about the pcilar axis. What sort of surface would the equation F(r, ' w - wr + {h'f"-m- 2 + (fV'-(iTT ]i p- Vj; +y - + z — — -3 ' where in the first case accents denote differentiation by s, in the second by t. 14. Show that the radius of curvature of a space curve is the radius of curva- ture of its projection on the osculating plane at the point in question. 15. From Frenet's Formulas show that the successive derivatives of x are , _ „ _ _ x „, _ y x/r _ 1 ir l ~r' x ~ T:~ Iv ~ ~ ni~ ' ui + rr' where accents denote differentiation by s. Show that the results for y and z are the same except that m. fx. .\f or n. v. X take the places of l. X, L. Hence infer that for the nth derivatives the results are X U0 = ip^ + \p 2 + LI'.,. //(") = ml\ + 11I', + .VP 3 , z("> = nP 1 + vP, 2 + XI'.. , where P., /'.,. J'., are rational functions of li and R and their derivatives by s. 16. Apply the foregoing to the expansion of Ex. 10 to show that ] .. _ n- 1!' ., ^ _ s 3 where R and R are the values at the origin where ,s = 0, I = n = N = 1, and the other six direction cosines m, ?/. X. v. L. M vanish. Find s and write the expan- sion of the curve of Ex. 8 (7) in this form. 17. Note that the distance of a point on the curve as expanded in Ex. 16 from the sphere through the origin and with center at the point (0, R, R'R) is Vx- + (y- /.')'- + (z - R'Rf-VR- + R'*R* (./•- + if- - 2 Ry + z- - 2 R'Rz) a ./•- + (;/ - R) 2 + (2- R'R)Hv^ + /."-R- and consequently is of the fourth order. The curve therefore has contact of the third order with this sphere. Can the equation of this sphere be derived by a limiting process like that of Ex. 18 as applied to the osculating plane 80 DIFFERENTIAL CALCULUS 18. The osculating plane may be regarded as the plane passed through three consecutive points of the curve ; in fact it is easily shown that lim &r, 5?/, Sz Aa\ Ay, Az approach X y Z 1 *0 Vq z o J x + 8x y + Sy z + Sz 1 x + Ax y + Ay z + Az 1 z - *o v - y z ~ z o (dx) (dy) (dz) (d' 2 x) (d?y) (d?z) = 0. 19. Express the radius of torsion in terms of the derivatives of x, y, z by t (Ex. 10, p. 07). 20. Find the direction, curvature, osculating plane, torsion, and osculating sphere (Ex. 17) of the conical helix x = t cos t. y = t sin t, z = kt at t = 2 tt. 21. Upon a plane diagram which shows As, Ax. Ay, exhibit the lines which represent ds, dx, dy under the different hypotheses that x, y, or s is the independ- ent variable. CHAPTER IV PARTIAL DIFFERENTIATION; EXPLICIT FUNCTIONS 43. Functions of two or more variables. The definitions and theo- rems about functions of more than one independent variable are to a large extent similar to those given in Chap. II for functions of a single variable, and the changes and difficulties which occur are for the most part amply illustrated by the case of two variables. The work in the text will therefore be confined largely to this case and the generaliza- tions to functions involving more than two variables may be left as exercises. If the value of a variable z is uniquely determined when the values (x, if) of two variables are known, z is said to be a function z = f(x, if) of the two variables. The set of values [(.r, ?/)] or of points P(x, if) of the »'i/-plane for which z is defined may be any set, but usually consists of all the points in a certain area or region of the plane bounded by a curve which may or may not belong to the region, just as the end points of an interval may or may not belong to it. Thus the function 1/ vl — y 1 — if is defined for all points within the circle .r' 1 -\- if = 1, but not for points on the perimeter of the circle. For most purposes it is sufficient to think of the boundary of the region of definition as a polygon whose sides are straight lines or such curves as the geometric intuition naturally suggests. The first way of representing the function z =/(■>', y) geometrically is by the surface z =f(x, if), just as y =f(x) was represented by a curve. This method is not available for u =f(r, //, z), a function of three vari- ables, or for functions of a greater number of variables ; for space has only three dimensions. A second method of representing the function z =/(./•, f) is by its contour lines in the .ry-plane, that is, the curves f (■''■> y) — const, are plotted and to each curve is attached the value of the constant. This is the method employed on maps in marking heights above sea level or depths of the ocean below sea level. It is evident that these contour lines are nothing but the projections on the .ry-plane of the curves in which the surface z = f(oc, y) is cut by the planes z — const. This method is applicable to functions u=f(x, y, z) of three variables. The contour surfaces a = const, which are thus obtained 87 DIFFERENTIAL CALCULUS are frequently called equlpotentlal surfaces. If the function is single valued, the contour lines or surfaces cannot intersect one another. The function z =f(x, y) is continuous for (a, b) when either of the following equivalent conditions is satisfied : 1°. lim/(.r, y) =f(a, b) or lim/(a5, y) =/(lim.r, lim y), no mutter how the variable point P(x, y) approaches {a, b). 2°. If for any assigned e, a number 8 may be found so that I/O''? V) — /(«> ] ') i < e lc]l en \x — a\j), (a„, 6.,), • ■ • . (a n , b H ) but are not tangent (the polynomials have common solutions which are not mul- tiple roots). Show that the value of the function will change by 2 kir if (s, y) describes a circuit which includes k of the points. Illustrate by taking for P/Q the fractions in Ex. 2. 10. Consider regions or volumes in space. Show that there are regions in which some circuits cannot be shrunk away to nothing ; also regions in which all circuits may be shrunk away but not all closed surfaces. 46. First partial derivatives. Let z=f(x,y) be a single valued function, or one branch of a multiple valued function, denned for (<•>, //) and for all points in the neighborhood. If y be given the value A, then z becomes a function f(j', l>) of x alone, and if that function has a derivative for x = >>, that derivative is called the partial derivative of z =f(x, //) with respect to x at (". li). Similarly, if .'• is held fast and equal to a and if /(>>. //) has a derivative when y = l>. that derivative is called the partial derivative of z with respect to y at (", //). To obtain these derivatives formally in the case of a given function f(x, y) it is merely necessary to differentiate the function by the ordinary rules, treating y as a constant when finding the derivative with respect to x and x as a constant for the derivative with respect to y. Notations are cf __£•"_.,_>_ / _ . _ _ l']±_ ex ex ' x ' '' "'' '''"" \dx 94 DIFFERENTIAL CALCULUS for the ^-derivative with similar ones for the ^-derivative. The partial derivatives are the limits of the quotients .. /(„ + /,./,)-/(„,?,) /(„,/, + ;,■) _/y „, /,) hm : - j Inn > (2) provided those limits exist. The application of the Theorem of the Mean to the functions f(x, b) and f(a, //) gives f(a + hj h) _/(„, h) = hf >( a + e x h, I), < 9 X < 1, /(a, & + /.•) -/(«, A) = A;/; (a, & + 0/), < 2 < 1, under the proper but evident restrictions (see § 26). Two comments may be made. First, some writers denote the partial derivatives by the same symbols dz/dx and dz/dy as if z were a function of only one variable and were differentiated with respect to that variable ; and if they desire especially to call attention to the other variables which are held constant, they affix them as subscripts as shown in the last symbol given (p. 93). This notation is particularly prevalent in thermodynamics. As a matter of fact, it would probably be impos- sible to devise a simple notation for partial derivatives which should be satisfac- tory for all purposes. The only safe rule to adopt is to use a notation which is sufficiently explicit for the purposes in hand, and at all times to pay careful atten- tion to what the derivative actually means in each case. Second, it should be noted that for points on the boundary of the region of definition of f(x, y) there may he merely right-hand or left-hand partial derivatives or perhaps none at all. For it, is necessary that the lines y = b and x = a cut into the region on one side or the other in the neighborhood of (a. b) if there is to be a derivative even one-sided ; and at a corner of the boundary it may happen that neither of these lines cuts into the region. Theorem. If f(x, //) and its derivatives f' x and f' y are continuous func- tions of (x, //) in the neighborhood of («., b), the increment Af may be written in any of the three forms \f = .f(" + h, h + /.■) -/(", h) = kf' x (a + 0J>, h) + kf y {a + h. h + 6Jc) = />/;.(" + Oh. l> + 6k) + kf'iQi + OIi, h + Ok) = hfX«, h) + /.■/;(", h) + zji + i 2 k, where the #\s are proper fractions, the £'s infinitesimals. To prove the first form, add and subtract /(a + h, b) ; then A/=[/(« + h. b)-f(a, b)] + [/(a + h. b + 1c) -/(« + h. &)] = kC(« + Ojh, b) + kfj(a + h, b + 0,k) by the application of the Theorem of the Mean for functions of a single variable (§§ 7, 26). The application may be made because the function is continuous and the indicated derivatives exist. Now if the derivatives are also continuous, they may be expressed as f',(n + e x h. h) =/;(«, b) + r,. /,>' + '<■ & + 0J) =//(«, &) + f 2 PARTIAL DIFFERENTIATION; EXPLICIT 95 where f x , f, may be made as small as desired by taking h and k sufficiently small. Hence the third form follows from the first. The second form, which is symmetric in the increments h, k, may be obtained by writing x = a + th and y = b + tk. Then/(.r. y) = <3>(t). As/ is continuous in (x, y), the function 4> is continuous in t and its increment is A* = f(a + t + Ath, b + t + Atk) -f(a + th, b + tk). This may be regarded as the increment of / taken from the point (x, y) with At ■ h and A( • k as increments in x and y. Hence A may be written as A* = At- ///; (a + th, b + tk) + At ■ kf,' f (a + th, b + tk) + ^At ■ h + f 2 Ai • k. Now if A<{> be divided by At and At be allowed to approach zero, it is seen that li„, _J! = k f i a + th, b + tk) + kf' (a + th, b + tk) = — . At J dt The Theorem of the Mean may now be applied to to give (1) —

dij cy where the indices x and y introduced in d x f and d y f indicate that x and y respectively are alone allowed to vary in forming the corresponding partial differentials. The total differential df=dJ'+d, J f= C £dx + f^d !/ , (6) which is the sum of the partial differentials, may be defined as that sum ; but it is better defined as that part of the increment A/ = f- Ax + 4- Ay + LAx + CA'J (7) ex cy l - which is obtained by neglecting the terms £ x Ax -f- CA/A which are of higher order than Ax and Ay. The total differential may therefore be computed by finding the partial derivatives, multiplying them respec- tively by dx and dy, and adding. The total differential of z = f(x, y) may be formed for (cr , y) as -*-©.<*-^ + (D.<"-*>' (8) where the values x — x Q and y — y are given to the independent differ- entials dx and dy, and df= dz is written as z — z . This, however, is 96 DIFFERENTIAL CALCULUS the equation of a plane since x and y are independent. The difference Af — df which measures the distance from the plane to the surface along a parallel to the s-axis is of higher order than Va./: 2 + Ay' 2 ; f or Af-Jf Va.x- 2 + Ay 2 LAx + LAy Va.x' 2 + A// 2 < 1/ I + \n = o. Hence the plane (8) will be defined as the tangent plane at (x Q , y , z ) to the surface z = f(x, y). The normal to the plane is f) c !J/ n -1 (9) which will be defined as the normal to the surface at (x , ?/ , 2 Q ). The tangent plane will cut the planes y = y and x = x in lines of which the slope is /^ and /^ . The surface will cut these planes in curves which are tangent to the lines. In the figure, PQSR is a portion of the surface z=f(x, y) and PT'TT" is a cor- responding portion of its tangent plane at P(x Q , y , z). Xow the various values may be read off. P'Q = A r ,f, P'T' = dj, 1'"^ = \f, P"T" = d y f, x's = a/; PP' = Ax, p't'/pp' =/;., J']'" = Ay, p"T"/pp"=f;„ P'T' + P"T" = -VT, x'r = df= 48. If the variables ,r and y are expressed as ,r = <£(£) and y = ^(f) so that /(.'', y) becomes a function of £, the derivative 1 of /'with respect to t is found from the expression for the increment of/'. At ex At dy At il At - A* lim A/' _ 'If = cf ilx cf dy a< = o At dt Ox dt cy dt (10) The conclusion requires that x and y should have finite derivatives with respect to t. The differential of /'as a function of t is c/f , df d.r cf da , df , of , rf£ c.r e/7 cy c/£ cr cy ' (11) and hence it appears that /7h> differential has the same form as the total differential. This result will be generalized later. PARTIAL DIFFERENTIATION; EXPLICIT 97 As a particular case of (10) suppose that x and y are so related that the point (x, y) moves along a line inclined at an angle t to the .r-axis. If s denote distance along the line, then x — x + s cos t, y = y + s sin r, dx = cos rds, dy = sin rds (12) and -f : =^-T : -f T--f- f' x cos t + /^ sin r. (13) '^f_^fdxdfdy ds ex ds cy ds The derivative (13) is called the directional derivative of f in the direc- tion of the line. The partial derivatives f' x , f' y are the particular direc- tional derivatives along the directions of the ./'-axis and ?/-axis. The directional derivative of f in any direction is the rate of increase of f along that direction ; if z = f(x, //) be inter- preted as a surface, the directional derivative is the slope of the curve in which a plane through the line (12) and perpendicular to the cry-plane cuts the surface. If f(x, y) be represented by its contour lines, the derivative at a point (x, y) in any direction is the limit of the ratio A// As = AC'/A.s of the increase of/, from one contour line to a neigh- boring one, to the distance between the lines in that direction. It is therefore evident that the derivative along any contour line is zero and that the derivative along the normal to the contour line is greater than in any other direction because the element dn of the normal is less than ds in any other direction. In fact, apart from infinitesimals of higher order, A?t As — = cos ^, A/ A.s A/ COS lj/, if df -— = — COS f ds /, z, • • •) is a function of any number of variables. The reasoning of the foregoing paragraphs may be repeated without change except for the additional number of variables. The increment of /will take any of the forms A/WO* + M + k, c + I, • ■ •) -f(a, b, c, ■ ■ •) = hf;(a + eji, b, c, • • •) + kf w (a + h, b + 6Jc, e, + if z (a + h, b + k, c + 0J, •••) + ••• = V'.C + kfy + Ifz H ]a + Ph. b + e*. r + fll, . . . = AC + kfy + {f~ + --- + ZJi + Lj* + £/ + • • ■, •) 98 DIFFERENTIAL CALCULUS and the total differential will naturally be defined as and finally if x, y, z, • • • be functions of t, it follows that df_ _of_dx df_dy df_dz dt c.r dt cy dt cz dt ^ > and the differential of /'as a function of t is still (16). If the variables x, y } z, ■ • • were expressed in terms of several new variables r, s, ■ • ■ , the function / would become a function of those vari- ables. To find the partial derivative of / with respect to one of those variables, say /■, the remaining ones, s, •••, would be held constant and / would for the moment become a function of /• alone, and so would x, y, z, ••■. Hence (17) may be applied to obtain the partial derivatives df = cfdx ,c£dy_ ,dfdz cr c.r or cy or cz or Cf Cf C.r Cf Ci/ Of CZ and t- = t^-— + -^--^- + 7^— + •••. etc. OS C.r Cs Cy CS CZ Cs These are the formulas for change of variable analogous to (4) of § 2. If these equations be multiplied by A/-, An, ■ • • and added, Cf K , %f A , Cf (C.r ^ | C.r t \ cf/ci/ T~ r- A/' Ar + f- As + ■ ■ ■ = ih — A>> + — As + • • • + 7^ ~ A;- + • • • + • • •> cr cs c.r\cr cs / cy\cr ) or df= d J. dx + d l du + d l d , + ... C.r 01/ CZ for when r, s, ••• are the independent variables, the parentheses above are dx, dy, dz, ■ ■ ■ and the expression on the left is df. Theorem. The expression of the total differential of a function of ./•, >/, z, •■• as i/f = f'//.r -\- f'ydij -\-fjIz + ••■ is the same whether x, y. z, ••• are the independent variables or functions of other independent variables r, s, • • • ; it being assumed that all the derivatives which occur, whether of / by x, y, z, ■ • • or of x, y, z, ■ ■ ■ by r, s, • ■ •, are continuous functions. By the same reasoning or by virtue of this theorem the rules d(cu) = edit, d (it -f" v — "•) = il a -\- dr — tln\ /„\ rd,/ — ,i,h- (19) d(uv) = udo + vdu, d(-)= - , of the differential calculus will apply to calculate the total differential of combinations or functions of several variables. If by this means, or any other, there is obtained an expression PAETIAL DIFFERENTIATION; EXPLICIT 99 df = R (r, s, t,...)dr + S(r,s,t,...) ds + T (r, s, t,...)dt + ... (20) for the total differential in which r, s, t, • ■ ■ are Independent variables, the coefficients R, S, T, ■ ■ ■ are the derivatives R = ^, S = d l, T= d l,.... (21) Cf cs ct For in the equation df= Rdr + Sds + Tdt -\ =f r dr+f s ds +f t dt -\ , the variables r, s, t, ■ ■ •, being independent, may be assigned increments absolutely at pleasure and if the particular choice dr = 1, ds = dt = • • • = 0, be made, it follows that R =/,': and so on. The single equation (20) is thus equivalent to the equations (21) in number equal to the number of the independent variables. As an example, consider the case of the function tan- 1 (y/x). By the rules (19), d tan _ i v _ a (v/ x ) _ ,l y/x - y(x) and normal to the curve. 9. If df '/tin is defined by the work of Fx. 7 (a), prove (14) as a consequence. 10. Apply the formulas for the change of variable to the following cases : («) r = Vx 2 +lr, * = tan-^. Find ^ *L, J(^Y + Wf. x ex by \\cx! \ajj (/S) x = rcostf>, y = rsin0. Find '--, (f -, (—)"+ \(—\- cr c \cr/ r' 2 \d

are polar ex ty cy ex dr r c

+ g sin 0, 67 = — /sin

= LL f » = EL f» = d T Jxx ex 2 ' Jxy cydx Jyx cxiif J ' JU otf where the derivative of cf/cy with respect to x is written cf/excy with the variables in the same order as required in Z/^D,,/' and opposite to the order of the subscripts in f" x . This matter of order is usually of no importance owing to the theorem: If the derivatives J"., f' y //are derU'titices f', f' which ore continuous in (./.', y) in the neighborhood of any point (./• , ?/ ), the derivatives f xy and f' y ' x ore equal, that is, f*y( x v !/ ) =L'A-' , v U Q )- The theorem may be proved by repeated application of the Theorem of the Mean. For [/K + h - Vo + k)-f(x , y + k)] - [/(x + h, y )-f(x w y )] = [4>(y + k)-(y) stands for /(x + h, y)-f(x , y) and 0(x) for /(x, y Q + k) -/(/, y ). Now (y ) = k + *. Vo + M)-f' y (x , y + Ok)] = W£(x + vh, y + 6k), Hfx( x o + e ' h - Vo + k )~fx( x o + e ' h - 2/o)l = W£{z + &% y + ifk). Hence fyx( x o + ^ Vo + 0k ) -fxy ( x o + &% Vq + v' k )- As the derivatives f" f'f y are supposed to exist and be continuous in the variables (x. y) at and in the neighborhood of (x , y ), the limit of each side of the equation exists as h = 0, k = and the equation is true in the limit. Hence J yx \ X V{)) = Jxy v^oi Vq) • The differentiation of the three derivatives/^.,/,^ = f' vx ,fy,, will give six derivatives of the' third order. Consider /"'.,, and f' xyx . These may be written as {f' x )'Jy and (f x ) yx and are equal by the theorem just proved (provided the restrictions as to continuity and existence are satisfied). A similar conclusion holds for f yxy and f yyx ; the number of distinct derivatives of the third order reduces from six to four, just as the number of the second order reduces from four to three. In like manner for derivatives of any order, the value of the derivative depend* not on t/te order in which the individual differentiations with respeet to x and 1/ are performed, but only on the total number of differentiation* with respeet to each, and the result may be written with the differentiations collected as - m + n • D?D* f = C ^L = fOn + ») t / 22 ) - 1 vJ C.r'"C;/" l •""■"" ' V ' Analogous results hold for functions of any number of variables. If several derivatives are to be found and added together, a symbolic form of writing is frequently advantageous. Fur example, cH c'f (n?.n y D? + jy;if= ^V-73 + ^ • " JJ c.i-cycz 6 cif or (d j: + D y ff= (/>; + 2 i>j) y + n*)f=f£ + 2f;; +/;;. 51. It is sometimes necessary to change the variable in higher deriv- atives, particularly in those of the second order. This is done by a repeated application of (18). Thus f' r ' r would be found by differentiat- ing the first equation with respect to r, and f' s by differentiating the first by s or the second by /•, and so on. Compare p. 12. The exercise below illustrates the method. It may be remarked that the use of higher different ills is often of advantage, although these differentials, like the higher differentials of functions of a single variable (Exs. 10, 16-19, p. 67), have the disadvantage that their form depends on what the independent variables are. This is also illustrated below. It should be particularly borne in mind that the great value of the first differential 104 DIFFERENTIAL CALCULUS lies in the facts that it may be treated like a finite quantity and that its form is independent of the variables. To change the variable hw^ + v" to polar coordinates and show Z-v Z-v _ c 2 v 1 cv 1 Z-v ( x = rcos, y = r sin 0. £x' 2 Zy' 2 cr- rcr r- c cv cv Zr cv ccp Then — = V — — > — = H - Zx cr Zx ccp Zx cy cr cy Z

'- c-v c cv c cv cr c cv Zcp Next —, = — — = — — ■ — + — — --r- Zx- Zx Zx cr Zx Zx Zcp ex Zx [c-vx cv c x c-v — y Zv Z — ylx cr* r cr cr r creep r- e/ ~— - + — — - + — , — r + — — - r\—r- ceper r creep r ccp- r- ccp ccp r- J r- The differentiations of x/r and — ?//;•- may be performed as indicated with respect to r, 0, remembering that, as r,

= ~ ( - x + . V cr ZxZr cy cr ex cy r\Zx cy Z I Zt\ ic-v c-v . \ I c-i c-v . \ cv cv . — [r — ) = I — - cos0 + sin x + I cos

dx + sinepdy, rd

— sin 2 c 2 f cosd>sinrf>~| + 2 — cos sin + - — ------ ., \_cr- \crc

" x>" are the three brackets which are the coefficients of dx 2 , 2dxdy, dy 2 . The value of v^ + v" y is as found before. 52. The condition f' i ' IJ =f' ll ' e which subsists in accordance with the fundamental theorem of § 50 gives the condition that M(X, y)dx + X(.r, y)(ly = f- d.r + % d )/ = df cx c jj IOC DIFFERENTIAL CALCULUS be the total differential of some function f(x, y). In fact o cf c.M cX c cf Cij ex cy ex ex cy CM cX /dM\ /dX\ and ■— = -j- or (—-) = (_). (2G) cy ex \dy J x \dxj y The second form, where the variables which are constant during the differentiation are explicitly indicated as subscripts, is more common in works on thermodynamics. It will be proved later that conversely if this relation (26) holds, the expression Mdx + Xdy is the total differ- ential of some function, and the method of finding' the function will also be given (§§ 92, 124). In case Mdx + Xdy is the differential of some function fix. //) it is usually called an exact differential. The application of the condition for an exact differential may be made in connection with a problem in thermodynamics. Let S and U be the entropy and energy of a gas or vapor inclosed in a receptacle of volume v and subjected to the pressure p at the temperature 7'. The fundamental equation of thermodynamics, connecting the differentials of energy, entropy, and volume, is iV-TdS-pd,; and (f ) = - (g)_ (27) is the condition that dU be a total differential. Now, any two of the five quantities U, S, r. T. j> may be taken as independent variables. In (27) the choice is S. r ; if the equation were solved for dS, the choice would be U, r : and U, S if solved for '//'. In each case the cross differ- entiation to express the condition (26) would give rise to a relation between the derivatives. If p. T were desired as independent variables, the change of variable should be made. The expression of the condition is then f d U f}+ T 4 -"4' = T ^r -(%)-*£*• dpi t cl (ji (l(ii (pel W/ ',, (pel where the differentiation on the left is made with p constant and that on the right with T constant and where the subscripts have been dropped from the second derivatives and the usual notation adopted. Everything cancels except two terms which srive PARTIAL DIFFERENTIATION; EXPLICIT 107 -)=-(-) or im\=-m. (28 ) dp/r \dTj P T\dp/r \dT/ P K ' The importance of the test for an exact differential lies not only in the relations obtained between the derivatives as above, but also in the fact that in applied mathematics a great many expressions are written as differentials which are not the total differentials of any functions and which must be distinguished from exact differentials. For instance if dll denote the infinitesimal portion of heat added to the gas or vapor above considered, the fundamental equation is expressed as dll = dU + pdv. That is to say, the amount of heat added is equal to the increase in the energy plus the work done by the gas in expanding. Now dll is not the dif- ferential of any function II ([', v) ; it is dS = dll/T which is the differential, and this is one reason fur introducing the entropy S. Again if the forces X, Y act on a particle, the work done during the displacement through the arc dx = Vrix- + dy' 2 is written dW = Xdx -f Ydy. It may happen that this is the total differential of some function ; indeed, if dW=-dV(x, y), Xdx + Ydy = - dV, X=- — , Y = -—, ex dy where the negative sign is introduced in accordance with custom, the function V is called the potential energy of the particle. In general, however, there is no poten- tial energy function 1'. and d II" is not an exact differential ; this is always true when part of the work is due to forces of friction. A notation which should dis- tinguish between exact differentials and those which are not exact is much more needed than a notation to distinguish between partial and ordinary derivatives; but there appears to be none. Many of the physical magnitudes of thermodynamics are expressed as deriva- tives and such relations as (26) establish relations between the magnitudes. Some definitions : (dii\ i as specific heat at constant volume is C v = ( -^ ) = T specific heat at constant pressure \dT/v \dT 1 \dTJ P \dTj P (dll\ ^/dS latent heat ot expansion is L r = — 1 = I coefficient of cubic expansion h )t \dv/T 1 / dr v [dfj , • y /dp modulus of elasticity (isothermal) is L r = — vl - Iv/t dp\ modulus of elasticity (adiabatic) is E s \dv 53. A polynomial is said to be homogeneous when each of its terms is of the same order when all the variables are considered. A defini- tion of homogeneity which includes this ease and is applicable to more general eases is : A function f (or, y, z, • ■ ■) of any number of variables is called homogeneous if the function is multiplied by some power of A when all the variables are mult'qAied by A: and the power of A which factors 108 DIFFEBEXTIAL CALCULUS out is called the order of homogeneity of the function. In symbols the condition for homogeneity of order n is f(\x, Xy, \z, ■■■) = k»f(x, y, z, • • •). (29) Thus xe* + — , -4 + tan" 1 - 5 -^= (29') x z- z VW- + ?/ 2 V J are homogeneous functions of order 1, 0, — 1 respectively. To test a function for homogeneity it is merely necessary to replace all the vari- ables by A times the variables and see if A factors out completely. The homogeneity may usually be seen without the test. If the identity (29) be differentiated with respect to A, with x'=Xx, etc., Id d d \ , [ X dx~' + 1/ d^' + Z dz~' + " jf^' X V> Xz > ' • = n\ n ~ x f{x, y,z,-- ■). A second differentiation with respect to A would give Now if A be set equal to 1 in these equations, then x' = x and x% + y% + z% + --- = nf(x,y,z,---), (30) OX ( If v.* 2 &f , Q ' C 'f , 2 ^'f I O &f J / 1 N Jf, N C.r" ^ C.rCy C if CXCZ ' v y In words, these equations state that the sum of the partial derivatives each multiplied by the variable with respect to which the differentia- tion is performed is n times the function if the function is homogeneous of order n ; and that the sum of the second derivatives each multiplied by the variables involved and by 1 or 2, according as the variable is repeated or not, is v(ji — 1) times the function. The general formula obtained by differentiating any number of times with respect to A may be expressed symbolically in the convenient form (■'■/>, + ///>, + zD t + • • •)*/= n (n - 1). • • (n - k + 1)/. (31) This is known as Eider's Formula on homogeneous functions. It is worth while noting that in a certain sense every equation which represents a geometric or physical relation is homogeneous. For instance, in geometry the magnitudes that arise may he lengths, areas, volumes, or angles. These magni- tudes are expressed as a number times a unit ; thus, v2 ft., 3 sq. yd., it eu. ft. PARTIAL DIFFERENTIATION; EXPLICIT 109 In adding and subtracting, the terms must be like quantities; lengths added to lengths, areas to areas, etc. The fundamental unit is taken as length. The units of area, volume, and angle are derived therefrom. Thus the area of a rectangle or the volume of a rectangular parallelepiped is A = a ft. x b ft. = ab ft. 2 = ah sq ft., V = a ft. x b ft. x c ft. = abc ft. 3 = abc cu. ft., and the units sq. ft., cu. ft. are denoted as ft.-, ft. 3 just as if the simple unit ft. had been treated as a literal quantity and included in the multiplication. An area or volume is therefore considered as a compound quantity consisting of a number which gives its magnitude and a unit which gives its quality or dimensions. If L denote length and [X] denote "of the dimensions of length," and if similar nota- tions be introduced for area and volume, the equations [A] — [LJ 2 and [F] = [L] 3 state that the dimensions of area are squares of length, and of volumes, cubes of lengths. If it be recalled that for purposes of analysis an angle is measured by the ratio of the arc subtended to the radius of the circle, the dimensions of angle are seen to be nil, as the definition involves the ratio of like magnitudes and must therefore be a pure number. When geometric facts are represented analytically, either of two alternatives is open : 1°, the equations may be regarded as existing between mere numbers ; or 2°, as between actual magnitudes. Sometimes one method is preferable, sometimes the other. Thus the equation x 2 + y' z = r 2 of a circle may be interpreted as 1°, the sum of the squares of the coordinates (numbers) is constant ; or 2°, the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (Pythagorean Theorem). The second interpretation better sets forth the true inwardness of the equation. Consider in like manner the parabola y' 2 = Apx. Gen- erally y and x are regarded as mere numbers, but they may equally be looked upon as lengths and then the statement is that the square upon the ordinate equals the rectangle upon the abscissa and the constant length 4p ; this may be inter- preted into an actual construction for the parabola, because a square equivalent to a rectangle may be constructed. In the last interpretation the constant p was assigned the, dimensions of length so as to render the equation homogeneous in dimensions, with each term of the dimensions of area or [L]-. It will be recalled, however, that in the definition of the parabola, the quantity p actually has the dimensions of length, being half the distance from the fixed point to the fixed line (focus and directrix). This is merely another corroboration of the initial statement that the equations which actually arise in considering geometric problems are homogeneous in their dimensions, and must be so for the reason that in stating the first equation like magnitudes must be compared with like magnitudes. The question of dimensions may be carried along through such processes as differentiation and integration. For let y have the dimensions [y] and x the dimen- sions [x]. Then A?/, the difference of two ?/'s, must still have the dimensions [y] and Ax the dimensions [x]. The quotient Ay/Ax then has the dimensions [?/]/[.'']. For example the relations for area and for volume of revolution, dA dv ., . rcLn [A] nivi [V] rri2 — = V, — = "ty ! 3' lv e - = - — - = \L1, = - — - = \L\-, dx dx ° Idxj [i] L J Idxj [i] L J and the dimensions of the left-hand side check with those of the right-hand side. As integration is the limit of a sum, the dimensions of an integral are the product 110 DIFFERENTIAL CALCULUS of the dimensions of the function to be integrated and of the differential dx. Thus if r x dx 1 , x = I = - tan- 1 - + c Jo a- + x- a a V - + were an integral arising in actual practice, the very fact that a- and x' 2 are added would show that they must have the same dimensions. If the dimensions of x be Til, then r ,._cfc_i = r_i_-| j_ = i_ and this checks with the dimensions on the right which are [£] -1 , since angle has no dimensions. As a rule, the theory of dimensions is neglected in pure mathe- matics ; but it can nevertheless be made exceedingly useful and instructive. In mechanics the fundamental units are length, mass, and time ; and are denoted by [i], [.!/]. ["/']. The following table contains some derived units: velocity — --, acceleration— — -, force - — , [T] [Tf [7"f , i • [i]- , • LV] r-vi rz-i areal velocity - — — > density - — -, momentum — — — -, - [T] [L]8 [T] , . l rjn rz.]2 WW 2 angular velocity , moment - — — — — , energy - — — — — . [T] [Tf - y [iy With the aid of a table like this it is easy to convert, magnitudes in one set of units as ft., lb., sec, to another system, say cm., gm., sec. All that is necessary is to substitute for each individual unit its value in the new system. Thus g = 321 ^^ , 1 ft, = 30.48 cm., g = 32J x 30.48 — — = 980| -^1. sec- sec- " sec. 2 EXERCISES 1. Obtain the derivatives/,'.',., f" /" f" and verify/" = /,/,.. V , os , .'-•'- +J/ 2 xy 2. Compute l-v/cy- in polar coordinates by the straightforward method. {a) sin-i y - , 08) log 5l±T , ( 7 ) Jl) + f {xy) x xy \x/ 3. Show that a- — = - - if u =/(x + at) + (x — at). ex' 1 ct 2 4. Show that this equation is unchanged in form by the transformation : ~ + 2 xy- '; - + 2 (y - y") ;- + x-y-f = ; u = xy, v = 1/y. cX- c x c >j 5. In polar coordinates z = r cos#. ./■ = r sin # cos./). // = r sin sin

'\ cr/ sin'- 6* r"c//- sin # c# V (v'_\ The work of transformation may be shortened by substituting successively x = i\ cos = ij'/ir. (Of importance for the Hertz oscillator.) Take if / i

/') dy. (6) x-y 2 (dx + dy). 8. Express the conditions that P(x. y. z)dx + Q(x, y. z)dy + R(x. y. z)dz be an exact differential dF(x, y. z). Apply these conditions to the differentials : (a) 3 x-y-zdx + 2 xhjzdy + x s y 2 dz, (/3) (y + z) dx + (x + z) dy + (x + y) dz. 9. Obtain | ) = (— -] and ( — ] = [ — -] from (27) with proper variables. \dTj v \dv) T \dSjp \dp) s 10. If three functions (called thermodynamic potentials) be defined as ^ = U - TS, x = U + Pi: t=U- TS + pv, show df - — Sd T — pdv, d x = TdS + vdp, d{ = — Sd T + vdp, and express the conditions that df. dx, d£ be exact. Compare with Ex. 9. 11. State in words the definitions corresponding to the defining formulas, p. 107. 12. If the sum (Mdx -f Ndy) + (Pdx + Qdy) of two differentials is exact and one of the differentials is exact, the other is. Prove this. 13. Apply Euler's Formula (31), for the simple case k = 1, to the three func- tions (20') and verify the formula. Apply it for k = 2 to the iirst function. 14. Verify the homogeneity of these functions and determine their order : (a) y-/x + x (log x - log y), (13) X '"- / " , (7) \ x 2 + y 2 "•'- + ty + '-z - r ,v ^ - Vy ( 5 ) xy&fi + z\ (e) yfx cot- ^ , (f) __ -1 • z VX + v y 15. State the dimensions of moment of inertia and convert a unit of moment of inertia in ft. -lb. into its equivalent in cm.-gm. 16. Discuss for dimensions Peirce's formulas Nos. 93, 124-12-"). 220. 300. 1 n ,, ■ ,, , _ .,„, , d ix cv , d cT . ix cT 17. ( oiitinue hx. 1 1. p. 101. to show - -= — and = mv 1 dt i'i; i'i^p x + '}.,<'.,. If T' denote T' = T, where T' is considered as a function of p v p„ while T is con- sidered as a function of q v q 2 , prove from I" = q x p x + q 2 p 2 — T that cT _ . iT _ _iT cpi ' cq t cqt 112 DIFFERENTIAL CALCULUS 19. If (x v ?/,) and (x 2 , y„) are the coordinates of two moving particles and d' 2 x. „ d 2 y, x , d 2 x„ ,, d 2 w„ m, — ^ = A,, to, - --* = F,, m„ —^ = X„, to„ -- 2 - = F 2 1 di 2 l l at 2 v 2 dt 2 2 - dt 2 2 are the equations of motion, and if x v y v x 2 , y 2 are expressible as x i=fi( 7.3) ;ill( ^ ' s homogeneous of the second degree in q v q 2 , q 3 . Tlie work may be carried on as a generalization of Ex. 17, p. 101, and Ex. 17 above. It may be further extended to any number of particles whose positions in space depend on a number of variables q. 20. In Ex. 19 if pi = — , generalize Ex. 18 to obtain cqi — = - — . Q, = -i + — • cpi cqi (' = andr'(K, cf c(f> x =/(«, v), ?/ = («, u), — = -/ , -~ = - ^ • c« cu el) cm 25. For an orthogonal transformation (Ex. 10 (f). p. 100) c 2 V ( 2 v crV c-V ( 2 v F 2 r r- + - -, + . , = . - + - + — • (X- c y- a- ex - cy- cz - 54. Taylor's Formula and applications. The development of /(.r, y) is found, as was the Theorem of the .Mean, from the relation (p. 95) PARTIAL DIFFERENTIATION; EXPLICIT 113 A/ = * (1) - * (0) if (t) = f(a + th, b + tk). If &({) be expanded by Maclaurin's Formula to n terms, *(9 - *(0) = «>'(<)) + yi *"(°) + • • • + (/t -"i)! ^"- 1 \°) + 5 * (,, W- The expressions for '(f) and 3>'(0) may be found as follows by (10) : $'(0 = KC + % *'(0) = V'f* + *£]«=«> then *"(*) = A (AC + ftQ + ft (ft/;; + ftQ = ft 2 /;; + 2 //ft/;;; + ft;/;;; = (ftD, + *d„) 2 /, ¥'\t) = (hD x + hD y yf, ¥\0) = l(hD x + /.•/>„)'/],._. And f(a + ft, & + ft) -/(«, &) = A/= $(1) -$(0) = (ft/),. + ftA,)/(", &) + |j (AD, + *W(«, &) + ••• + Tjy^y, (AD, + ^)- 1 /(«, 6) + ^ (AD, + ft/>,)"/(" + #', * + **)• (32) In this expansion, the increments ft and ft may be replaced, if de- sired, by x — a and // — h and then /(./-, y) will be expressed in terms of its value and the values of its derivatives at (a, b) in a manner entirely analogous to the case of a single variable. In particular if the point (it, b) about which the development takes place be (0, 0) the development becomes Maclaurin's Formula for /(./■, y). /(*, V) =/(0, 0) + (*D, + yD,)f(0, 0) + A (.,■/>, + 2/Dj 2 /(0, 0) + • • • +^zi)T C^+^-Wi °) + ii (* D *+ &*>,)" f(fo, Oy)- (32') Whether in Maclaurin's or Taylor's Formula, the successive terms are homogeneous polynomials of the 1st, 2d, • • •, (n — l)st order in ./•, // or in jc — a, y — b. The formulas are unique as in § IY2. Suppose Vl — x- — y' 2 is to be developed about (0, 0). The successive deriva- tives are fx = ~ x — , /,; = - - — — . /; (o, o) = o, /; (o, o> = o, Vl — x- — //'- V 1 — x' 2 — y 2 „ _ - 1 + if f „ xy „ _ - 1 + X' (1 - x- - if) 5 (1 - x- - //-) 2 (1 - x 2 - y-)z „, _ \{\ — y-)x ,„■ _ ?/ :i — 2 xy 2 — y (1 - x- - y 2 )t (1 - x 2 - if) - and V l - x- - if = 1 + (0 x + y) + l{- x- + xy - if) + J (0 x 3 + ...) + ..., or V 1 — x- — ?/ 2 = 1 — I (x- + y' 2 ) + terms of fourth order + • • • . In this case the expansion may be found by treating x- + if as a single term and expanding by the binomial theorem. The result would be 114 DIFFERENTIAL CALCULUS [1 - (x 2 + y 8 )]* = 1 - 1 (x 2 + !/-) - I (x* + 2 x-y- + y*) - f 6 (* 2 + 2/ 2 ) 3 • That the development thus obtained is identical with the Maclaurin development that might be had by the method above, follows from the uniqueness of the devel- opment. Some such short cut is usually available. 55. The condition that a function z =f(x, //) have a minimum or maximum at («, b) is that A/> or A/*< for all values of // = Ar and k = A?/ which are sufficiently small. From cither geometrical or analytic considerations it is seen that if the surface z = /('•'', >/) has a minimum or maximum at (ft, />), the curves in which the planes y = h and .'■ = a cut the surface have minima or maxima at ./■ = / — I b. The corresponding z is I c ami the volume V is therefore abc/9 or | of the volume cut off from the first octant by the plane. It is evident that this solution is a maximum. There are other solut ions of Y' x = Y" — which have been discarded because they give 1' = 0. The conditions f' x =fl = may be established analytically. For A/=(/; + QA^-r-(/; + t)A>/. Xow ;ts £ £ o are infinitesimals, the signs of the parentheses are deter- mined by the signs of f' x , f' unless these derivatives vanish; and hence unless f x = 0, the sign of \f for \.r sufficiently small and positive ami A// = would be opposite to the sign of A/ for A./' sufficiently small and negative and A// = 0. Therefore for a winiiHitni or uio.riitni/i) /',' = 0; uml hi like manner _/',' = 0. Considerations like these will serve to establish a criterion for distinguishing between maxima and minima PARTIAL DIFFERENTIATION; EXPLICIT 115 analogous to the criterion furnished by /"(•'') in the eu.se of one vari- able. Tor if f' x =fy = 0, then A / = i ( ] '~f*X + 2 hkf£ + k 2 fyy) x=a + h ,,J= b + 9k, by Taylor's Formula to two terms. Xow if the second derivatives are continuous functions of (.<•, y) in the neighborhood of (a, I>), each deriv- ative at (a -+- 0/t, b + 6k) may be written as its value at (a, b) plus an infinitesimal. Hence \f = \ W~ + 2 hkf xy + 7r/;;) ( „, h) + \ (h% + 2 /,/.■£, + k%). Xow the sign of A/' for sufficiently small values of //, /,■ must be the same as the sign of the first parenthesis provided that parenthesis does not vanish. Hence if the quantity > for every (Jt, k), a minimum '' ; < for every (It, k), a maximum. (Jt'fxx + 2 Jtkf" y + k-f wi , ( As the derivatives are taken at the point (a, b), they have certain constant values, say A, B, C. The question of distinguishing between minima and maxima therefore reduces to the discussion of the possible signs of a quadratic form All 2 + 2 Bltk + Ck 2 for different values of h and k. The examples /,-2 + £2, _ /(2 _ fc 2 5 ft 2 _ J. 2j ± (/ t _ fc)2 show that a quadratic form may be : either 1°, positive for every (It. k) except (0. 0) ; or 2°, negative for every (It. k) except (0. 0); or 3°, positive for some values (It, k) and negative for others and zero for others ; or finally 4°, zero for values other than (0. 0), but either never negative or never positive. Moreover, the four possibilities here mentioned are the only cases conceivable except 5 D , that A = B = C — anil the form always is 0. In the first case the form is called a definite positive form, in the second a definite negative form, in the third an indefinite form, and in the fourth and fifth a singular form. The first case assures a minimum, the second a maxi- mum, the third neither a minimum nor a maximum (sometimes called a minimax) ; but the case of a singular form leaves the question entirely undecided just as the condition /" (x) = did. The conditions which distinguish between the different possibilities may be ex- pressed in terms of the coefficients A, B. C. Ppos. def., B 2 ; 3° indef ., B 2 > AC ; 2°neg. def., B- < AC. A,C<0; -P sing., B- = AC. The conditions for distinguishing between maxima and minima are : f~ = "I „ , f f".. f".. > minimum ; f = j J I /xj-i fiji/ < maximum : fry > fxxfJm minima* ; J*? = f'xxfw < '■) ■ It may be noted that in applying these conditions to the case of a definite form it is sufficient to show that either/", or/" is positive or negative because they neces- sarily have the same sign. 116 DIFFERENTIAL CALCULUS EXERCISES 1. Write at length, without symbolic shortening, the expansion of /(x, y) by Taylor's Formula to and including the terms of the third order in x — a, y — b. "Write the formula also with the terms of the third order as the remainder. 2. Write by analogy the proper form of Taylor's Formula for/(x, y, z) and prove it. Indicate the result for any number of variables. 3. Obtain the quadratic and lower terms in the development (a) of xy- + sin xy at (1, I tt) and (/3) of tan- 1 (ij/x) at (1. 1). 4. A rectangular parallelepiped with one vertex at the origin and three faces in the coordinate planes has the opposite vertex upon the ellipsoid X 2/ a 2 + y2 /b 2 + Z 2/ C 2 _ 1m Find the maximum volume. 5. Find the point within a triangle such that«the sum of the squares of its distances to the vertices shall be a minimum. Note that the point is the intersec- tion of the medians. Is it obvious that a minimum and not a maximum is present ? 6. A floating anchorage is to be made with a cylindrical body and equal coni- cal ends. Find the dimensions that make the surface least for a given volume. 7. A cylindrical tent has a conical roof. Find the best dimensions. 8. Apply the test by second derivatives to the problem in the text and to any of Exs. 4-7. Discuss for maxima or minima the following functions : (a) x-y + xy- — x, (p) x 8 + y 3 - x-y- - I (x 2 + y-), (y) x- + ir + x + y, (S) \ ir - xy- + x-y - x, ( e ) x? + ,/' - 9x?/ + 27, ( f ) x* + y i - 2 x- + 4 xy - 2 y*. 9. State the conditions on the first derivatives for a maximum or minimum of function of three or any number of variables. Prove in the case of three variables. 10. A wall tent with rectangular body and gable roof is to be so constructed as to use the least amount of tenting for a given volume. Find the dimensions. 11. Given any number of masses m v in.,. ■ • •, m n situated at (x 1? y^). (x.„ y ), ■ • •, (x n , y„). Show that the point about which their moment of inertia is least is their center of gravity. If the points were {x v y v z x ). • ■ ■ in space, what point would make 2m>' 2 a minimum ? 12. A test for maximum or minimum analogous to that of Ex. 27, p. 10, may be given for a function f{x, y) of two variables, namely : If a function is positive all over a region and vanishes upon the contour of the region, it must have a max- imum within the region at the point for which ./',' =/„' = 0. If a function is finite all over a region and becomes infinite over the contour of the region, it must have a minimum within the region at the point for which f' x =f = 0. These tests are subject to the proviso that/,' =/ ;/ ' = has only a single solution. Comment on the test and apply it to exercises above. 13. If a. b. c, /• are the sides of a given triangle and the radius of the inscribed circle, the pyramid of altitude // constructed on the triangle as base will have its maximum surface when the surface is \ (a + b + c)\'r- + k-. CHAPTER V PARTIAL DIFFERENTIATION; IMPLICIT FUNCTIONS 56. The simplest case ; F(x, y) = 0. The total differential dF = F' x dx + F' y dy = dO = • v dy F' r dx F' ... indicates ~ = ,- ■> — - = — ', (V) dx F y dy F x as the derivative of y by x, or of x by y, where y is defined as a function of x, or x as a function of y, by the relation F(x, y) = ; and this method of obtaining a derivative of an implicit function without solving expli- citly for the function has probably been familiar long before the notion of a partial derivative was obtained. The relation F(x, y) = is pictured as a curve, and the function y= <£(#), which would be obtained by solu- tion, is considered as multiple valued or as restricted to some definite portion or branch of the curve F(x, y) = 0. If the results (1) are to be applied to find the derivative at some point (x Q , y ) of the curve F(x, y) = 0, it is necessary that at that point the denominator F' y or F' x should not vanish. These pictorial and somewhat vague notions may be stated precisely as a theorem susceptible of proof, namely : Let x be any real value of x such that 1°, the equation F(x , y) = has a real solution y Q ; and 2°, the function F(x, y) regarded as a function of two independent variables (./•, y) is continuous and has continuous first partial derivatives F' x , F' it in the neighborhood of (,/- , y) ; and 3°, the derivative F y (x , y (j )=h0 does not vanish for (x , y ) ; then F(x, //) = may be solved (theoretically) as y=cf}(x~) in the vicinity of x = x. and in such a manner that y o = (f>(.r ), that <£(./") is continuous in x, and that (x) has a derivative <£'(>) = — F' x /F' y ; and the solution is unique. This is the fundamental theorem on implicit functions for the simple case, and the proof follows. By the conditions on F x , F' the Theorem of the Mean is applicable. Hence F (x, y)-F (/„ . y ) =F{x,y) = (h F' x + kF' y ) Xo + eh , Vn + ek . (2) Furthermore, in any square \h\<8, |A'j<5 suiTOimding (x , y ) and sufficiently small, the continuity of F' x insures |-F^|< M and the continuity of F' y taken with 117 118 DIFFERENTIAL CALCULUS Y X i/o+ 5 5 i i 5 i %-* ! 1 25 m M X the fact tliat F'(x , y ) ■£ insures \F y \>m. Consider the range of x as furtliei restricted to values such that \x — x \m8 but Ux — x Q )Fx\(x) such that F(x, y) = 0, equation (2) gives at once fc _ y-y _ Ay _ _ F; : {x + Oh, y + 6k) dy _ _ K (x , y„) it ~ x - x ~ Ax ~~ F y (x + 6h. y + Ok) ' dx ~ F y (x , y ) ' As F' x , F' are continuous and F' y ^ 0, the fraction k/h approaches a limit and the derivative (p'(x ) exists and is given by (1). The same reasoning would apply to any point x in the interval. The theorem is completely proved. It may be added that the expression for '{x) itself is continuous. The values of higher derivatives of implicit functions are obtainable by successive total differentiation as f;. + f'J = o, f;;. + 2 y;j + F y y 2 + f;,/ 1 = o, (3) etc. It is noteworthy that these successive equations may be solved for the derivative of highest order by dividing by F' rl which has been assumed not to vanish. The question of whether the function ij = (,.'') defined implicitly by F(.r, if) = has derivatives of order higher than the first may be seen by these equations to depend on whether F(.r, if) has higher partial derivatives which are continuous in (./■, //). 57. To find the maxima and minima of y = (.»•), that is. to find the points where the tangent to F(x, if) = is parallel to the .c-axis, observe that at such points //' = 0. Equations (3) give /•-;. = o, /■;;;, + f,V' = o. (i) Hence always under the assumption that F' — 0, there arc maxima at the, intersections of F= ami /■",'= if /•',',' end F'„ hare the same si;/n, and minima at the intersections for vliich F" r and F' tl hare oj/jjosiie aif/nsj the cast' F" = still remains undecided. PARTIAL DIFFERENTIATION; IMPLICIT 119 For example if F(x, y) = x 3 + y 3 — 3 axy = 0, the derivatives are dy x 2 — ay 3 (x 2 - ay) + 3 (y 2 - ax) y' = 0, 6 x — 6 ay' + yy' 2 + 3 (y 2 - ax) y" - 0, dx y 2 — ax d 2 y 2 a 3 xy dx 2 (y 2 — ax) 3 To find the maxima or minima of j/asa function of x, solve F x = = x 2 - ay, F = = x 3 + y z - 3 axy, F^ ^ 0. The real solutions of F' x — and F = are (0. 0) and (j/2a, "v^a) of which the first must be discarded because F^(0, 0) = 0. At (V2a, Via) the derivatives F' and F^, are positive; and the point is a maximum. The curve F=0 is the folium of Descartes. The role of the variables x and y may be interchanged if F' x =£ and the equation F(x, y) = may be solved for .v = "/'(.y), the functions <£ and \\t being inverse. In this way the vertical tangents to the curve F = may be discussed. For the points of F — at which both F' x = and Fy = 0, the equation cannot be solved in the sense here defined. Such points are called singular points of the curve. The questions of the singular points of F = and of maxima, minima, or minimax (§ oo) of the surface z = F(x, y) are related. For if F x = F' y = 0, the surface has a tangent plane parallel to z = 0, and if the condition z = F = is also satisfied, the surface is tangent to the .ry-plane. Now if -: = F(.r, //) has a maximum or minimum at its point of tangency with z = 0, the surface lies entirely on one side of the plane and the point of tangency is an isolated point of F(.c, y) = ; whereas if the surface has a mini- max it cuts through the plane z = and the point of tangency is not an isolated point of F(x, y) = 0. The shape of the curve F = in the neighborhood of a singular point is discussed by developing F(.r, //) about that point by Taylor's Formula. For example, consider the curve F(x. y) = x 3 + y ?> — x 2 y 2 — I (x 2 + y 2 ) = and the surface z = F(x, y). The common real solutions of f; = 3 x 2 - 2 xy 2 - x = 0, F' y = 3 y 2 - 2 x 2 y - y = 0, F (x, y) = are the singular points. The real solutions of F x = 0. F' — are (0, 0), (1, 1), (J, i) and of these the first two satisfy F(x. y) = but the last does not. The singular points of the curve are therefore (0. 0) and (1. 1). The test (34) of § 55 shows that (0. 0) is a maximum for z = F(x, y) and hence an isolated point of F(x, y) = 0. The test also shows that (1, 1; is a minimax. To discuss the curve F(x, y) = near (1, 1) apply Taylor's Formula. = F(x, y) = | (3 h 2 - 8 M- + 3 A: 2 ) + J (6 h 3 - 12 /i 2 & - 12 M 2 + A: 3 ) + remainder = | (3 cos 2 — 8 sin cos + 3 sin 2 0) + r (cos 3 — 2 cos 2 sin 0—2 cos sin 2 + sin 3 0) + • • • . 120 DIFFERENTIAL CALCULUS if polar coordinates h = rcos0, k = rsiu0 be introduced at (1, 1) and r- be can- celed. Now for very small values of r, the equation can be satisfied only when the first parenthesis is very small. Hence the solutions of 3-4 sin 2^ = 0, sin 2 = f , or = 24° 17*', 05° 42*', and + 7r, are the directions of the tangents to F(x, y)= 0. The equation F= is = (13, — 2 sin 2 0) + r(cos0 + sin0)(1 — 1* sin 2 0) if only the first two terms are kept, and this will serve to sketch F(x, y) = for very sm-.ll value.', of r, that is, for very near to the tangent directions. 58. It is important to obtain conditions for the maximum or minimum of a function z = f(x, y) where the variables x, y are connected by a relation F(x, y) = so that z really becomes a function of ./• alone or y alone. For it is not always possible, and frequently it is inconvenient, to solve F(x, y) = for either variable and thus eliminate that variable from z = f(x, y) by substitution. When the variables x, y in z = f(x, y) are thus connected, the minimum or maximum is called a constrained minimum or maximum ; when there is no equation F(x, y) = between them the minimum or maximum is called free if any designation is needed.* The conditions are obtained by differentiating .-.' = /'(./■, y\ and F(x,y)= totally with respect to x. Thus dz _ of of dy dO _ dF cF dy _ dx ex cy dx ax ex dy ; =/; + xf; = o, ; =,/;; + xf; = o. (7) These two equations taken with F=0 constitute a set of three from which the three values x, y, X may be obtained by solution. Note that * The adjective "relative" is sometimes used for constrained, and "absolute!" for free: but the term "absolute" is best kept Cor the greatest of the maxima or least of the minima, and the term " relative " for the other maxima and minima. PARTIAL DIFFERENTIATION; IMPLICIT 121 A cannot be obtained from (7) if both F' x and F' y vanish ; and hence this method also rejects the singular points. That this method really deter- mines the constrained maxima and minima of f(x, y) subject to the constraint F(x, //) = is seen from the fact that if A. be eliminated from (7) the condition f' x F' y — f' u F' x = of (5) is obtained. The new method is therefore identical with the former, and its introduction is more a matter of convenience than necessity. It is possible to show directly that the new method gives the constrained maxima and minima. For the conditions (7) lire those of a free extreme for the function 4>(.r, //) which depends on two independent variables (;r, //). Now if the equa- tions (7) be solved for (,r, y\ it appears that the position of the maximum or minimum will be expressed in terms of A as a parameter and that consequently the point (.''(A.), //(A)) cannot in general lie on the curve F(.r, //) = 0; but if A be so determined that the point shall lie on this curve, the function <£(.r, //) has a free extreme at a point for which F= and hence in particular must have a constrained extreme for the particular values for which F(x, y) = 0. In speaking of (7) as the con- ditions for an extreme, the conditions which should be imposed on the second derivative have been disregarded. For example, suppose the maximum radius vector from the origin to the folium of Descartes were desired. The problem is to render /(a;, y) = x' 2 + y- maximum subject to the condition F(x, y) = x' A + tf' — 3 axy = 0. Hence 2 x + 3 X (/- — ay) = 0, 2 y + 3 X (//- — ax) = 0, x 3 + y 3 — 3 axy = or 2 x • 3 (i/~ — ax) — 2 y ■ 3 (x' 2 — ay) — 0, ,r ! -f y" — 3 axy = are the conditions in the two cases. These equations may be solved for (0, 0), (1 .', a, 1.1 a), and some imaginary values. The value (0, 0) is singular and X cannot be determined, but the point is evidently a minimum of x' 1 + y' 1 by inspection. The point (1 1 a, 11 «) gives X = — ] I a. That the point is a (relative constrained) maxi- mum of /- + y' 1 is also seen by inspection. There is no need to examine d' 2 f. In most practical problems the examination of the conditions of the second order may be waived. This example is one which may be treated in polar coordinates by the ordinary methods ; but it is noteworthy that if it could not be treated that way. the method of solution by eliminating one of the variables by solving the cubic, F(x, y) = would be unavailable and the methods of constrained maxima would be required. EXERCISES 1. By total differentiation and division obtain dy/dx in these cases. Do not substitute in (1). but. use the method by which it was derived. (<() ax- + 2 bxy + a/ 2 —1 = 0, (jj) x* + ?/ 4 = 4 a 2 xy, (y) (cos/)" — (sin y) x = 0, (5) (x- + y-)' 2 = a 2 (x' 2 — y' 2 ), (e) e x + & — 2 xy, (f) x~ ' 2 y~ 2 = tan- 1 xy. 2. obtain the second derivative d' 2 y/dx 2 in Ex. 1 (a), (/3), (e), (f) by differen- tiating the value of dy/dx obtained above. Compare with use of (3). 122 DIFFERENTIAL CALCULUS , p d 2 y F?F^ - 2 Jffi F^ + F?F £ 6. Prove — = = — — ■ dx 1 F'* v 4. Find the radius of curvature of these curves : ( + 2 xy- = x 2 + v/ 4 . 12. Make these functions maxima or minima subject to the given conditions. Discuss the work both with and without a multiplier: , . a b , . sinx u (a) 1 — , a tan x + b tan y = c. Ans. = - • wcosx vcosy amy v ((3) x 2 + y 2 , ax 2 + 2bxy + ry 2 = f. Find axes of conic. (7) Find the shortest distance from a point to a line (in a plane). 13. Write the second and third total differentials of F(x, y) = and compare with (3) and Ex. 5. Try this method of calculating in Ex. 2. 14. Show that F' r are obtained by ordinary division after setting dy = and dx = re- spectively. If this division is to be legitimate, F' z must not vanish at the point considered. The immediate suggestion is the theorem : If, when real values (x Q , y ) are chosen and a real value z Q is obtained from F(z, a* , y Q ) = by solution, the function F(x, >/. z) regarded as a function of three independent variables (./■. y, z) is continuous at and near (x Q , y Q , z ) and has continuous first partial derivatives and •FV( ,r o> y , z) 4=- 0, then F(x, //. z) = may be solved uniquely for z = 4>( J ') y) an d $(■''■> !/) will be continuous and have partial derivatives (9) for values of (./•, //) sufficiently near to (x y\. The theorem is again proved by the Law of the Mean, and in a similar manner. F(x, y, z) - F(x , y , z ) = F(x, y, z) = (hF' x + kF' y + ^).r„ + 0A,.v o + ^,.- o + ^. As F' r . F', F.. are continuous and F'{x () , y (r z ) ^ 0. it is possible to take 5 so small that, when \h ]< 5. \k\ < 5. \l\< 5, the derivative |K|> »' and |^| '»• Hence if (x. y) be held fixed, there is one and only one value of z for which the parenthesis vanishes between z„ + 8 and z — 5. Thus z is defined as a single valued function of (x. y) for sufficiently small values of h = x — x , k = y — //,,. use. / = F;(x + ffh. y (t + ek.z (l + di) i_ r;(...) h ~ F^{x o + 0h,y o + 8k,z o + 0l)' k~ F 2 '(.--) when k and /; respectively are assigned the values 0. The limits exist when h = or k = 0. But in the first case / = Az = A,z is the increment of z when x alone varies, and in the second case I = Az =A u z. The limits are therefore the desired partial derivatives of z by x and y. The proof for any number of variables would be similar. 124 DIFFEBEXTIAL CALCULUS If none of the derivatives F' x , F' y , F' z vanish, the equation F(x, y, z) = may be solved for any one of the variables, and formulas like (9) will express the partial derivatives. It then appears that •lz\ (dx\ ex c.r F\. F: 1, (10) dxl y \dz/ y c.r cz F' z F' x in like manner. The first equation is in this case identical with ( i) of § 2 because if // is constant the relation F(x, y, z) = reduces to G(x, z) = 0. The second equation is new. By virtue of (10) and simi- lar relations, the derivatives in (11) may be inverted and transformed to the right side of the equation. As it is assumed in thermodynamics that the pressure, volume, and temperature of a given simple substance are connected by an equation F(p, r, T) = 0, called the characteristic equation of the substance, a relation between different thermodynamic magnitudes is furnished by (11). 60. In the next place suppose there are two equations F(x,y,v, r) = 0, G(x,y,v, /■) = (12) between four variables. Let each equation be differentiated. dF = = F x dx + F' y dy + F' u da + F' v dv, dG = = G' x dx + G' y dy + G' H ihi + G/h: (13) If it l)e desired to consider h, r as the dependent variables and ./'. // as independent, it would be natural to solve these equations for the differ- entials du and dv in terms of d.v and dy ; for example. (f;.g;. - f;/q d* + ( K/;;. - fv;,; , dy dll = — —- , ; — — ■ (lo F U G V - l\.d u v ' The differential dv would have a different numerator but the same de- nominator. The solution requires F' U G[, — F' V G' U =£ 0. This suggests the desired theorem : If (/' n . r f ) are solutions of F = 0. G = corresponding to (x Q , y ( ) and if F' U G[, — F' r G' u does not vanish for the values (r , y n , i/ f) . r ), the equations F — 0, G = may be solved for u = (x, y), v = \p(x, //) and the solution is unique and valid for (./', //) sufficiently near <.'■ . // ) — it being assumed that jPand G regarded as functions in four variables are continuous and have continuous first partial derivatives at and near (./• , y , u , v ) ; moreover, the total differentials da, dc are given by (13') and a similar equation. PARTIAL DIFFERENTIATION; IMPLICIT 125 The proof of this theorem may be deferred (§ G4). Some observations should be made. The equations (13) may be solved for any two vari- ables in terms of the other two. The partial derivatives 8u (x, y) r !'(■>•, r) dx (u, v) c.r(u, -if) ex (14) of u by x or of x by u will naturally depend on whether the solution for u is in terms of (x, //) or of (x, r), and the solution for ./• is in (m, /■) or (if, //). Moreover, it must not lie assumed that Cu /dx and cx/cn are reciprocals no matter which meaning is attached to each. In obtaining relations between the derivatives analogous to (10), (11), the values of the derivatives in terms of the derivatives of F and G may be found or the equations (12) may hrst be considered as solved. Thus if Then M : = /lx + 4>,(hj, v ■ = i (•<•• y), dv - - \p',,dx + fydy. (IX : 'x¥y - xf'y - tij^'x dx K (X cv - -*'» C cu

yf' x 4>xfy — yfx cu ex ex cu cv dx (X cv - 1, and — = , — = : . etc. du $y v - y x cv (x, //) litis at that point a tangent plane parallel to z = and there is a maximum, minimum, or minimax. To distinguish between the possibilities further investigation must be made if necessary ; the details of such an investi- gation will not be outlined for the reason that special methods are usually available. The conditions for an extreme of u its a function of (./•, jj) defined implicitly by the equations (13') are seen to be KG'v - K&* = 0, K°v - K r *' v = 0, F = 0, G = 0. (17) The four equations may be solved for .;•, y, a, v or merely for .'', //. 120 DIFFERENTIAL CALCULUS Suppose that the maxima, minima, and minimax of u =f(x, ?/, z) sub- ject either to one equation F(x, y, z) = or two equations /•'(./-, //. z) = 0, G(x, //, z) = of constraint are desired. Xote that if only one equation of constraint is imposed, the function u = /'(•'•. //, z) becomes a function of two variables; whereas if two equations are imposed, the function u really contains only one variable and the question of a minimax does not arise. The method of multipliers is again employed. Consider *(*, y, z)=f+kF or $> = f+\F+iiG (18) as the case may be. The conditions for a free extreme of are $; = o, 4>; = o, /. z which will then be expressed as functions of A or of A and fi according to the case. If then A or A and /x be determined so that (x, >/. z) satisfy F = or F = and G = 0, the constrained extremes of u =/{.r. //, z) will be found except for the examination of the conditions of higher order. As a problem in constrained maxima and minima let the axes of the section of an ellipsoid by a plane through the origin be determined. Form the function * = x- + y- + z 2 + X ("^ + ? f, + ~ - 1 ) + ix (Ix + my + nz) \a- b- c- 1 by adding to x 2 + y- + z". which is to be made extreme, the equations of the ellipsoid and plane, which are the equations of constraint. Then apply (19). Hence X + X— + /x ? = 0. y -p X — + - »? = 0. z + \- + -n = a- 2 b- 2 c- 2 taken with the equations of ellipsoid and plane will determine x. y. z. X. fj.. If the equations are multiplied by x. y. z and reduced by the equations of plane and ellipsoid, the solution for X is X =— r- =— (x- + y- -\- z 2 ). The three equations then become 1 /xlil- 1 fj.)hh- 1 fine' 2 . , , x = - — , y — — , z — - — , with Ix + my + nz — 0. 2 r- — a 2 2 r 2 — b 2 2 r- — c- Hence — -\ 1 — — = determines r". (20) r- — a- r- — b 2 r 2 — c 2 The two roots for r are the major and minor axes of the ellipse in which the plane cuts the ellipsoid. The substitution of x. y. z above in the ellipsoid determines H 2 ( ill \- / bm \- , / en \- . x 2 y 2 z 2 since — + - + — = 1. (21) 4 \ /■- — a-/ \r 2 — b-J \r- — c 2 / a 2 b 2 Xow when (20) is solved for any particular root r and the value of fx is found by (21 ). th" actual coordinates x, y. z of the extremities of the axes mav he found. PARTIAL DIFFERENTIATION ; IMPLICIT 127 EXERCISES 1. Obtain the partial derivatives of z by x and y directly from (8) and not by substitution in (9). Where does the solution fail ? («) ~, + )~ + ~ = 1, (p) x + y + z= , a- b- c- xyz (j) (*r 2 + V 1 + z 2 ) 2 = a 2 x 2 + b 2 y 2 + c 2 z 2 , (5) xyz = c. 2. Find the second derivatives in Ex. 1 (a), (/3), (5) by repeated differentiation. 3. State and prove the theorem on the solution of F(x, y, z, u) = 0. 4. Show that the product a p E T of the coefficient of expansion by the modulus of elasticity (§ 52) is equal to the rate of rise of pressure with the temperature if the volume is constant. 5. Establish the proportion E s : E T = C p : C v (see § 52). „ . cucxcycz cucx 6. If i- (x, y, z, u) — 0, show - = 1, - — = 1. ex cy cz cu ex cu 7. ^Yrite the equations of tangent plane and normal line to F(x, y, z) = and find the tangent planes and normal lines to Ex. 1 (/S), (5) at x — 1. y = 1. 8. Find, by using (13), the indicated derivatives on the assumption that either x, y or u, v are dependent and the other pair independent : (a) u 5 + r' + x:' — ?> y = 0, ir + v" + y z + ?>x = 0, //,'. u' u " v'/ r (13) x + y + u +v = a, x 1 + y 2 + u 2 + r- = 6, x' u . u' x , v' y , v'.' y (y) Find dy in both cases if x, v are independent variables. 9. Prove ^ ~ + ~ -? = if F(./\ y, u, v) = 0, G (x, y, u, v) = 0. ex cu tx CV 10. Find du and the derivatives u' r . u' u'. in case X 2 + y 2 + z 1 — uv, xy = u' 1 + v 2 + w 2 , xyz = mi-?/;. 11. If F(x, y, z) — 0, G(x, y, z) = define a curve, show that x - Jq V-Vq 2 - z„ ( F ;o; - f;g;) - (f:g' x - f; : g^ ~ (Kg; - f;gq o is the tangent line to the curve at (x , y , z ). Write the normal plane. 12. Formulate the problem of implicit functions occurring in Kx. 11. 13. Find the perpendicular distance from a point to a plane. 14. The sum of three positive numbers is x + y + z = N, where N is given. Determine x. y. z so that the product xJ'y'iz r shall be maximum if p. q, r are given. Ans. X : y : Z : X — p : q : r : (p + q + >'). 15. The sum of three positive numbers and the sum of their squares are both given. Make the product a maximum or minimum. 16. The surface (x 2 + y 2 + z 2 )' 2 = ax 2 + by 2 +cz 2 is cut by the plane lx+my + nz = 0. Find the maximum or minimum radius of the section. Ans. > — = 0. A 1 /■- - a 128 DIFFERENTIAL CALCULUS 17. In ca.se F(x, y, u, v) = 0, G{x, y, u, v) = consider the differentials dv , dv , , dx dx , , ?y . ?// , au = — ax + — ay, ax = — du -\ dv, dy = — du + ' dv. dx dy du dv du dv Substitute in the first from the last two and obtain relations like (15) and Ex. f). 18. If f(x. y, z) is to be maximum or minimum subject to the constraint F(x, y, z) = 0, show that the conditions are that dx : dy : dz = : : are indeter- minate when their solution is attempted from f x dx + f y dy + f'Az = and F' x dx + F' y dy + F' z dz = 0. From what geometrical considerations should this be obvious '.' Discuss in connec- tion with the problem of inscribing- the maximum rectangular parallelepiped in the ellipsoid. These equations, dx : dy ■. dz =/;f: -f z F' y -fzK-fLK ■■f',K -f' t ,K = ° : ° : ^ may sometimes be used to advantage for such problems. 19. Given the curve F(x, y, z) = 0, G(x, y, z) = 0. Discuss the conditions for the highest or lowest points, or more generally the points where the tangent is parallel to z = 0, by treating « =f(x, y, z) = z as a maximum or minimum sub- ject to the two constraining equations F — 0, G = 0. Show that the condition F' X G' — F'G' X which is thus obtained is equivalent to setting dz = in F'jlx + F' y dy + F'Az = and Gjlx + G'yly + Gjlz = 0. 20. Find the highest and lowest points of these curves : (a) x" + V 1 = 2 3 + 1, x + y + 2z = 0, (p) X - + \ r - + -"- = 1, Ix + my + nz = 0. a- b" f- 21. Show that F' x dx + F'/ly + F'Az = 0, with dx = £ - x, dy = v — //, i"„ ---- ><','■ PARTIAL DIFFERENTIATION; IMPLICIT 129 62. Functional determinants or Jacobians. Let two functions u = 4>(x,y), v = if/(x,y) (22) of two independent variables be given. The continuity of the functions and of their first derivatives is assumed throughout this discussion and will not be mentioned again. Suppose that there were a relation F(u, v) = or F((f>, if/) = between the functions. Then F(,ij/) = 0, F&+FM = 0, F^; + F r >;=0. (23) The last two equations arise on differentiating the first with respect to x and y. The elimination of F' u and F' v from these gives 4>&'v - 4vV* = 0; f; d(u, v) T ( u i i %, y) (£)-* ™ The determinant is merely another way of writing the first expression ; the next form is" the customary short way of writing the determinant and denotes that the elements of the determinant are the first deriva- tives of u and v with respect to x and y. This determinant is called the functional determinant or Jam/nan of the functions u, r or cf>, if/ with respect to the variables x, y and is denoted by J. It is seen that : //' there is a functional relation F(, if/") = between tiro functions, the Jacobian of the functions vanishes identically, that is, vanishes for all values of the variables (x, //) under consideration. Conversely, if the Jacobian vanishes identically over a tiro-dimensional region for (x, y\ the functions are connected by a functional /■elation. For, the functions u, v may be assumed not to reduce to mere constants and hence there may be assumed to be points for which at least one of the partial derivatives eft'., { J} if/'., ip' f does not vanish. Let x be the derivative which does not vanish at some particular point of the region. Then u = (x, y) may be solved as x = x("> V) m the vicinity of that point and the result may be substituted in v. cr , c\ , , ex v = t (x> y)> — = x by (11) and substitution. Thus cv/cy = J/ x ; and if ./ = 0, then cv/cy = 0. This relation holds at least throughout the region for which x =f= 0, and for points in this region cv /cy vanishes identically. Hence v does not depend on y but becomes a function of u alone. This es- tablishes the fact that v and u are functionally connected. 130 DIFFERENTIAL CALCULUS These considerations may be extended to other eases. Let u = (x, y, z), v = ij/(x, y, z), w = x (r, V, *)• If there is a functional relation F(u, v, w) = 0, differentiate it. F' u $' x + F$' x + F'„. x ' r = 0, f u >; + /•-;X tyx X.r 'y ¥y Xy <$>': ¥z Xz = 0, (25) (26) c (, «A, x) c("> '"> "') , n or —. 7 = —, ^ = ./ = U. The result is obtained by eliminating F' u , F'.. F' u , from the three equations. The assumption is made, here as above, that F' u , F' v , F' w do not all vanish ; for if they did, the three equations would not imply .1 = 0. On the other hand their vanishing would imply that F did not contain //. r. v, — as it must if there is really a relation between them. And now con- versely it may be shown that if ./ vanishes identically, there is a func- tional relation between u, r. a-. Hence again the necessary and sufficient conditions that tin' three functions (25) be functionally connected is that their Jacobin n vanish. Tht! proof of the converse part is about as before. It may be assumed that at least one of the derivatives of u. v. v or 0. \p, ^ by X, y. z does not vanish. Let (x. y. z) may be solved as x = w(w, y. z) and the result may be substituted in v and w as v = f (x. y. z) = \p (co. y. z). w = x (■'■■ II- ~-) = X (*>, V- %)• Next the Jacobian of v and w relative to y and z may be written as Its— + x f / y X.r- + X„ (y ({i ([/ cy ( r ( a- cz CZ , ( X , , (X f,-— + f z Xr — + Xz ty X'„ +z : i + --p., x / + x.,. Xz -~ IK + d x ': 1 X; . As J" vanishes identically, the Jacobian of r and v expressed as functions of y. ?, also vanishes. Hence by the case previously discussed there is a functional rela- tion F(v. w) = f » independent of y. z : and as r. w now contain u, this relation may be considered as a functional relation between u, r. w. 63. If in (22) the variables u, v be assigned constant values, the equations define two curves, and if n. r be assigned a series of such values, the equations (22) define a network of curves in some part of the PARTIAL DIFFERENTIATION: IMPLICIT 131 rry-plane. If there is a functional relation u = F(v), that is, if the Jacobian vanishes identically, a constant value of v implies a constant value of u and hence the locus for which v is constant is also a locus for which u is constant ; the set of y-curves coincides with the set of «-curves and no true network is formed. This ? . case is uninteresting. Let it be assumed that the Jacobian does not vanish identically and even that it does not vanish for any point (x, y) of a certain region of the ./'//-plane. The indi- cations of § GO are that the equations (22) may then be solved for ./•, y in terms of u, v at any point of the region and that there is a pair of the curves through each point. It is then proper to consider (a, r) as the coordinates of the points in the region. To any point there corre- spond not only the rectangular coordinates (x, y) but also the curvi- linear coordinates (u, v). The equations connecting the rectangular and curvilinear coordinates may be taken in either of the two forms u = (x, y), v = (./■, y) or x =f(u, r), y = a(u, r), (22') each of which are the solutions of the other. The Jacobians Y l\ 7^ X = 1 (27) are reciprocal each to each ; and this rela- tion may be regarded as the analogy of the relation (4) of § 2 for the case of. the function y — (.r) and the solution ./• =zf(y) = ~ l (y) in the case of a single variable. The differential of arc is (x+d v x, y+d v y) (a, v+dv) (x + dx, y+dy) {u + du, v+dv) v+dv (x+d u x, y+d u y) (u + du, v) ds 2 = 0, the transformation is called direct ; but if J < 0, the transformation is called perverted. The significance of the distinction can be made clear only when the ques- tion of the signs of areas has been treated. The transformation is called conformed when elements of are in the neighborhood of a point in the cry-plane are proportional to the elements of arc, in the neighborhood of the corresponding point in the wr-plane, that is, when dx 2 + dif = I- (dtr + dr-) = Mo*. (30) PARTIAL DIFFERENTIATION : IMPLICIT 133 For in this case any little triangle will be transformed into a little tri- angle similar to it, and hence angles will be unchanged by the transfor- mation. That the transformation be conformal requires that F = and K = G. It is not necessary that E = G = Jc be constants ; the ratio of similitude may be different for different points. 64. There remains outstanding the proof that equations may be solved in the neighborhood of a point at which the Jacobian does not vanish. The fact was indicated in § 60 and used in § 63. Theorem. Let p equations in n + p variables be given, say, Fi(*v ;*.+p) = 0, r 0. (31) Let the i> functions be soluble for x, , .r, , • • •, x„ when a particular set of the other n variables are given. Let the functions '(P+l)o> } X (n+p) and their first derivatives be continuous in all the n -f- p variables in the neighborhood of (x lg , a*^, • • •, ."'' ( , ( + /;)o )- Let the Jacobian of the functions with respect to x , x .,, • • • , x p , **„ d ll. d I± cx } dx x dl \. d J± cx„ ex, *o, (32) > -''Oi+p)o fail to vanish for the particular set mentioned. Then the p equations may be solved for the y variables x , ./•„, • • •, x p , and the solutions will be continuous, unique, and differentiable with continuous first partial derivatives for all values of x p+1 , ■••, x n+p sufficiently near to the values i'\ p+1)ii , •••, .''(/i + /,)o- Theorem. The necessary and sufficient condition that a functional relation exist between j> functions of /> variables is that the Jacobian of the functions with respect to the variables shall vanish identically, that is, for all values of the variables. The proof s of these theorems will naturally be given by mathematical induction. Each of the theorems lias been proved in the simplest cases and it remains only to show that the theorems are time for p functions in ease they are for p — i . Expand the. determinant J. ex. " ex., J l , • • ■ , J p , mini For the first theorem J ^ and hence at least one of the minors J x , ■ • •. J p must fail to vanish. Let that one lie J v which is the Jacobian of J' 1 .,, • • • . F t , with respect to x. 2 , • • •, x p . By the assumption that the theorem holds for the case p — 1, these £> — 1 equations may be solved for x.,. • • •, x p in terms of the n + 1 variables jj, 134 DIFFERENTIAL CALCULUS Xp+i, • • • , x n+p , and the results may be substituted in F x . It remains to show that F x = is soluble for x r Now dF, cF, cF. cx„ cF, cx„ r T — - i = ^r- L + ^- L H + "- + — i - J? = , • • • , x v . Hence the equations have been solved for x x , x.,, • ■ •, x p in terms of x p +1 , • • • , x n+p and the theorem is proved. For the second theorem the procedure is analogous to that previously followed. If there is a relation F(u x , ■ ■, u p ) = between the p functions u i = i(*i, ■ • •, x p ), ■ • •, Up = p (x x , ■■■, x p ), differentiation with respect to x x , ■•-, x p gives p equations from which the deriva- tives of F by u x , ■ ■ -, u p may be eliminated and jt — — - — - ') = becomes the con- dition desired. If conversely this Jacobian vanishes identically and it be assumed that one of the derivatives of w; by Xj, say cu x /cx x , does not vanish, then the solution x x = ui(i(,, x„, • • •. x p ) may be effected and the result may be substituted in ».,, • • •, Up. The Jacobian of u 2 , • • •. u p with respect to x.,. • • •. x p will then turn out to be J -f- tu x /cx x and will vanish because ./ vanishes. Now, however, only p — 1 functions are involved, and hence if the theorem is true for p — 1 functions it must be true for p functions. EXERCISES 1. If u = ax. + by + c and v = a'x -f b'y + c' are functionally dependent, the lines u = and v = are parallel ; and conversely. 2. Prove x + y + z. xy + yz + zx, x- + y- + z- functionally dependent. 3. If u = ax + by + cz + d, v = a'x + b'y + c'z + d' , w = a"x + b"y + e"z + d" are functionally dependent, the planes u = 0, v = 0, w = are parallel to a line. 4. In what senses are — and \p' of (24') and - — - and — } - of (32') partial or total cy dx x cx x derivatives ? Are not the two sets completely analogous '? \1/ y 5. Given (25), suppose y r y f \^£ 0. Solve v — \p and ?o = x for ?/ and z, substi- tute in u = , y = a'u + b'v + c'w, z = a"u + b"v + c"w is conformal, is it orthogonal ? See Ex. 10 (f), p. 100. 12. Show that the areas of the triangles whose vertices are (u, v). (u + du, t). («, i' + t7r) and (u + du, v + (a), y — ij/((t) with F(, if/, a) = 0, (34) where the first equations express the dependence of the points on the envelope upon the parameter a and the last equation states that each point of the envelope lies also on some curve of the family F(.r, y, a) = 0. Differentiate (34) with respect to a. Then F&'(a) + Ftf(a) + F' a = 0. (35) Now if the point of contact of the envelope with the curve F = is an ordinary point of that curve, the tangent to the curve is F' x (x - x ) + F' y (y - y) = ; and Ftf + Ftf = 0, since the tangent direction dy : dx = if/' : ' along the envelope is by definition identical witli that along the enveloping curve; and if the point of contact is a singular point for the enveloping curve, F' x = F' y = 0. Hence in either case F' a = 0. Thus for points on the envelope the t/r<> equations Fix, y, a) = 0, F' a (x, y, a) = (30) are satisfied end tin' equation of the envelope of the family F =-- may In- found liy solving (36) 1(a\ y=.\p(a\ of tli e envelope or by eliminating a between (36) to Jind the equation of the, envelope in the form <£(./'. //) = 0. It should be remarked that the locus found by this process may contain other curves than the envelope, for instance if the curves of the family F= have singular points and if ./• = (a), y= i]/(a) be the locus of the singular points as + dx > //o + d !/> < x o + lla ) ~ F ( r o> 1/o> a o) = F'jlx + F' y d,j + F' a da = (37) holds except for infinitesimals of higher order. The distance from the point on a 4- da to the tangent to a Q at (.r , // o ) is /.•;,/.,. -fj-',;,/// _ _± ffrto r//i (38) except for infinitesimals of higher order. Tins distance is of the first order with da, and the normal derivative daldn of § 48 is finite except when F' a = 0. The distance is of higher order than da, and daldn is infinite or dn/da is zero when F' a = 0. It appears therefore that f//e envelope is the locus of points of which the distance between two neigh- boring curves is of higher order than da. This is also apparent geomet- rically from the fact that the distance from a point on a curve to the 138 DIFFERENTIAL CALCULUS tangent to the curve at a neighboring point is of higher order (§ 36). Singular points have been ruled out because (38) becomes indetermi- nate. In general the locus of singular points is not tangent to the curves of the family and is not an envelope but an extraneous factor ; in exceptional cases this locus is an envelope. If two neighboring curves F(x, y, a) = 0, F(x, y, a + Aa) = inter- sect, their point of intersection satisfies both of the equations, and hence also the equation — [F(x, y, a + \a) - F(x, y, «)] = F' a (x, y, a + 0\a) = 0. If the limit be taken for A« == 0, the limiting position of the intersec- tion satisfies F' a = and hence may lie on the envelope, and will lie on the envelope if the common point of intersection is remote from singular points of the curves F(x, y, a) = 0. This idea of an envelope as the limit of points in which neighboring curves of the family intersect is valuable. It is sometimes taken as the definition of the envelope. But, unless imaginary points of intersection are considered, it is an inade- quate definition ; for otherwise y = (x- — af would have no envelope according to the definition (whereas y = is obviously an envelope) and a curve could not be regarded as the envelope of its osculating circles. Care must be used in applying the rule for finding an envelope. Otherwise not only may extraneous solutions be mistaken for the envelope, but the envelope may be missed entirely. Consider y — sin ax — or a — x~ l sin -1 y = 0. (39) where the second form is obtained by solution and contains a multiple valued function. These two families of curves are identical, and it is geometrically clear that they have an envelope, namely y — i 1. This is precisely what would lie found on applying the rule to the first of (39) ; but if the rule be applied to the second of (39), it is seen that F^ = l, which does not vanish and hence indicates no envelope. The whole matter should be examined carefully in the light of implicit functions. Hence let F(x. y. a) — be a continuous single valued function of the three variables (x. y. a) and let its derivatives V' r . F' F' a exist and he continuous. Con- sider the behavior of the curves of the family near a point (x () . y ) of the curve for a — (x tt provided that (,r ir y ) is an ordinary (nonsingular) point of the curve and that the derivative F^(x , y , a ) does not vanish. As F' a ?± and either F' x ^ or F ^ for (x , y . a lt ), it is possible to surround (x , y„) with a region so small that F(x. y. a) = may be solved for a =f(x. y) which will he single valued and differentiable ; and the region may further he taken so small that F' x or F' remains different from throughout the region. Then through every point of the region there is one and only one curve a =/(j, y) and the curves have no singular points within the region. In particular no two curves of the family can be tangent to each other within the re»ion. PARTIAL DIFFERENTIATION; IMPLICIT 139 Furthermore, in such a region there is no envelope. For let any curve which traverses the region be x = (t), y = \p (t). Then a (t) = f(

(t), y = \f/(t) be tangent to this curve, dy = dx = \j/' : ', y, z, a, ft) = depending on two parameters. The envelope may be defined by the property of tangency as in § 65; and the conditions for an envelope would be F(x, y, z, a, ft) = 0, F' a = 0, F, = 0. (40) These three equations may be solved to express the envelope as x = cf> (a, ft), ij = $ (a, ft), :: = X («, ft) parametrically in terms of a, ft; or the two parameters may be elimi- nated and the envelope may be found as <£(.'•, //, z) = 0. In any case extraneous loci may be introduced and the results of the work should therefore be tested, which generally may be done at sight. It is also possible to determine the distance from the tangent plane of one surface to the neighboring surfaces as ,.y, r + F ; //f/ + /.•;,/, _ ,. y ft + j.^/p _ V/-: 2 + f; + f:; ^f: 1 + f; 2 + F' z 2 and to define the envelope as the locus of points such that this distance is of higher order than \da\ + \*'ft\- The equations (40) would then also follow. This definition would apply only to ordinary points of the sur- faces of the family, that is, to points for which not all the derivatives F' r , F' F' z vanish. But as the elimination of a, ft from (40) would give an equation which included the loci of these singular points, there would be no danger of losing such loci in the rare instances where they, too, happened to be tangent to the surfaces of the family. 140 DIFFERENTIAL CALCULUS The application of implicit functions as in § 66 could also be made in this case and would show that no envelope could exist in regions where no singular points occurred and where either F' a or F^ failed to vanish. This work could be based either on the first definition involving tangency directly or on the second definition which involves tangency indirectly in the statements concerning infinitesimals of higher order. It may be added that if F(x, y, z, a, (3) = were not single valued, the surfaces over which two values of the function become inseparable should be added as possible envelopes. A family of surfaces F(x, y, z, a) = depending on a single param- eter may have an envelope, and the envelope Is found from F(x, y, z, a) = 0, F&x, y, z, a) = (42) by the elimination of the single parameter. The details of the deduction of the rule will be omitted. If two neighboring surfaces intersect, the limiting position of the curve of intersection lies on the envelope and the envelope is the surface generated by this curve as a varies. The surfaces of the family touch the envelope not at a point merely but along these curves. The curves are called characteristics of the family. In the case where consecutive surfaces of the family do not intersect in a real curve it is necessary to fall back on the conception of imagi- naries or on the definition of an envelope in terms of tangency or infinitesimals ; the characteristic curves are still the curves along which the surfaces of the family are in contact with the envelope and along which two consecutive surfaces of the family are distant from each other by an infinitesimal of higher order than da. A particular case of importance is the envelope of a plane which depends on one parameter. The equations (42) are then Ax + By + Cz + ]) --= 0, A'x -f- B'y + C'z + D' = 0, (43) where A, B, C, D are functions of the parameter and differentiation with respect to it is denoted by accents. The case where the plane moves parallel to itself or turns about a line may lie excluded as trivial. As the intersection of two planes is a line, the characteristics of the system are straight lines, the envelope is a ruled surface, and a plane tangent to the surface at one, point of the fines is tangent, to the surface throughout the whole extent of the line. Cones and cylinders are exam- ples of this sort of surface. Another example is the surface enveloped by the osculating planes of a curve in space; for the osculating plane depends on only one parameter. As the osculating plane (§ 41) may be regarded as passing through three consecutive points of the curve, 1 wo consecutive osculating planes may be considered as having two consecu- tive points of the curve in common and hence the characteristics are PARTIAL DIFFERENTIATION ; IMPLICIT 141 the tangent lines to the curve. Surfaces which are the envelopes of a plane which depends on a single parameter are called developable surfaces. A family of curves dependent on two parameters as F(x, y, s, a, (3) = 0, G (x, y, ::, a, /3) = (44) is called a congruence of curves. The curves may have an envelope, that is, there may be a surface to which the curves are tangent and which may be regarded as the locus of their points of tangency. The envelope is obtained by eliminating a, /3 from the equations f = o, g = o, f' u g; - f;g; = o. (45) To see this, suppose that the third condition is not fulfilled. The equa- tions (44) may then be solved as a = f(x, y, z), f3 = ;/(■'■. >/. z). Reason- ing like that of § 66 now shows that there cannot possibly be an envelope in the region for which the solution is valid. It may therefore be inferred that the only possibilities for an envelope are contained in die equations (45). As various extraneous loci might be introduced in the elimination of a, (3 from (45) and as the solutions should therefore be tested individually, it is hardly necessary to examine the general question further. The envelope of a congruence of curves is called the focal surface of the congruence and the points of contact of the curves with the envelope are called the focal 'points on the curves. EXERCISES 1. Find the envelopes of these families of curves. In each ease test the answer or its individual factors and check the results by a sketch : (a) y = 2 ax + « 4 , (3) y- = a{x - or), (-,) y = ax + k/a, (5) a(y + «) 2 = x s , (e) y = a(x + a)-, (f) .//- = a(x - erf. 2. Find the envelope of the ellipses x~/a- + y-/h- = 1 under the condition that (a) the sum of the axes is constant or (8) the area is constant. 3. Find the envelope of the circles whose center is on a given parabola and which pass through the vertex of the parabola. 4. Circles pass through the origin and have their centers on x- — y- = c 2 . Find their envelope. Ans. A lemniscate. 5. Find the envelopes in these cases : (a) x + xya = sin- J xy, (B) x + a = v ers- l y + V2 y — y-, {y) y + a = Vl— 1/x. 6. Find the envelopes in these cases : (a) ax + 8y + aBz = 1. (8) - + \ + — — - = 1, o 2 2 iX P *■ ~ (X _ P (l) — + — + — = 1 with aBy = k z . a- 3- y- 7. Find the envelopes in Ex. (a). (B) if a = 8 or il a = — 3. 142 DIFFERENTIAL CALCULUS 8. Prove that the envelope of F(x, y, z, a) = is tangent to the surface along the whole characteristic by showing that the normal to F(x, y. z. a) = and to the eliminant of F — 0, F' a = are the same, namely F X :F :F 2 and 7^ + 7^ v - -: 7 + 1 a — : F z + F a — , " ex oy cz where a{x, y, z) is the function obtained by solving F' a = 0. Consider the problem also from the point of view of infinitesimals and the normal derivative. 9. If there is a curve x = (u, r), y = f(u, r), z = x (", r) of a surface may be taken in the unsolved, the solved, or the parametric form. The parametric form is equivalent to the solved form provided u, v he taken as ./•. y. The notation cz cz c 2 z c 2 r: _ d 2 z c.r cy ex 1 excy cy 1 is adopted for the derivatives of z with respect tor and //. The applica- tion of Taylor's Formula to the solved form gives A.v = r h + fjk + I (/•//- + 2 shh + fir) + • • • (47) with h = A./\ /.■ = A//. The linear terms pli + '//.' constitute the differ- ential dz and represent that part of the increment of z which would he obtained by replacing the surface by its tangent plane. Apart from infinitesimals of the third order, the distance from the tangent plane up or down to the surface along a parallel to the .-.--axis is given by the quadratic terms \(rlr A- 2 shk + tlr). Hence if the quadratic terms at any point are a positive definite form (§ 55), the surface lies above its tangent plane and is concave up; but if the form is negative definite, the surface lies below its tangent plane and is concave down or convex up. If the form is indefinite but not singular, the surface lies partly above and partly below its tangent plane and may be called concavo-convex, that is, it is saddle-shaped. If the form is singular nothing can be definitely stated. These statements 144 DIFFERENTIAL CALCULUS are merely generalizations of those of § 55 made for the case where the tangent plane is parallel to the .r//-plane. It will be assumed in the further work of these articles that at least one of the derivatives r, s, t is not 0. To examine more closely the behavior of a surface in the vicinity of a particular point upon it, let the ,r //-plane be taken in coincidence with the tangent plane at the point and let the point be taken as origin. Then Maclaurin's Formula is available. z = i(nc 2 + 2 sxy + tif) + terms of higher order — h P 2 ( r cos2 -j- 2 s sin 9 cos 9 + t sin 2 9) + higher terms, where (p, 9) are polar coordinates in the x //-plane. Then — = /• cos 2 9 + 2 s sin 9 cos 9 -f- t sin' 2 9 = -~ R dp" ■«s (48) (49) is the curvature of a normal section of the surface. The sum of the curvatures in two normal sections which are in perpendicular planes may be obtained by giving 9 the values 9 and 9 + \ ir. This sum reduces to r -f t and is therefore independent of 9. As the sum of the curvatures in two perpendicular normal planes is constant, the maximum and minimum values of the curvature will be found in perpendicular planes. These values of the curvature are called the ftrincipal values and their reciprocals are the principal radii of curvature and the sections in which they lie are the principal sections. If s = 0, the principal sections are 6 = and 9 = \ it ; and conversely if the axes of x and // had been chosen in the tangent plane so as to be tangent to the principal sections, the derivative s would have vanished. The equation of the surface would then have taken the simple form z = ^(rx 2 + tif) + higher terms. (50) The principal curvatures would be merely r and t, and the curvature. in any normal section would have had the form 1 cos 2 9 sin 2 9 ., „ . , „ — = — 1 — = r cos 2 9 + t sin'- 9. 1 2 If the two principal curvatures have opposite signs, that is, if the signs of r and t in (50) are opposite, the surface is saddle-shaped. There are then two directions for which the curvature of a normal sec- tion vanishes, namely the directions of the lines 9 = ± tan- 1 V— /.', /ll x or Vp| x =± V| t \ >/. These are called the asymptotic directions. Along these directions the surface departs from its tangent plane by infinitesimals of the third PARTIAL DIFFERENTIATION; IMPLICIT Ho order, or higher order. If a curve is drawn on a surface so that at each point of the curve the tangent to the curve is along one of the asymp- totic directions, the curve is called an asymptotic curve or line of the surface. As the surface departs from its tangent plane by infinitesimals of higher order than the second along an asymptotic line, the tangent plane to a surface at any point of an asymptotic line must be the oscu- lating plane of the asymptotic line. The character of a point upon a surface is indicated by the Du-pln indicatr'ix of the point. The indicatrix is the conic y + ^ = 1, of. * = *('■*" + '*"), (51) 1 2 which has the principal directions as the directions of its axes and the square roots of the absolute values of the principal radii of curvature as the magnitudes of its axes. The conic may be regarded as similar to the conic in which a plane infinitely near the tangent plane cuts the surface when infinitesimals of order higher than the second are neg- lected. In case the surface is concavo-convex the indicatrix is a hyper- bola and should be considered as either or both of the two conjugate hyperbolas that would arise from giving z positive or negative values in (51). The point on the surface is called elliptic, hyperbolic, or parabolic according as the indicatrix is an ellipse, a hyperbola, or a pair of lines, as happens when one of the principal curvatures vanishes. These classes of points correspond to the distinctions definite, indefinite, and singular applied to the quadratic form r/r + 2shk + tic' 1 . Two further results are noteworthy. Any curve drawn on the surface differs from the section of its osculating plane with the surface by infinitesimals of higher order than the second. For as the osculating plane passes through three consecutive points of the curve, its inter- section with the surface passes through the same three consecutive points and the two curves have contact of the second order. It follows that the radius of curvature of any curve on the surface is identical with that of the curve in which its osculating plane cuts the surface. The other result is Meusniei J s Theorem : The radius of curvature of an oblique section of the surface at any point is the projection upon the plane of that section of the radius of curvature of the normal section which passes through the same tangent line. In other words, if the radius of curvature of a normal section is known, that of the oblique sections through the same tangent line may be obtained by multiplying it by the cosine of the angle between the plane normal to the surface and the plane of the oblique section. 146 DIFFERENTIAL CALCULUS The proof of Meusnier's Theorem may be given by reference to (48). Let the r-axis in the tangent plane be taken along the intersection with the oblique plane. Neglect infinitesimals of higher order than the second. Then y = (x) = \ ax 2 , z = \ (rx 2 + 2 sxy + ty-) = \ rx 2 (48') will be the equations of the curve. The plane of the section is az — ry = 0, as may be seen by inspection. The radius of curvature of the curve in this plane may be found at once. For if u denote distance in the plane and perpendicular to the x-axis and if v be the angle between the normal plane and the oblique plane az — ry = 0, u = z sec v — y esc v — \ r sec v ■ x- = -| a esc v ■ x 2 . The form u = \ rsec v ■ x 2 gives the curvature as rsec v. But the curvature in the normal section is r by (48'). As the curvature in the oblique section is secv times that in the normal section, the radius of curvature in the oblique section is cos v times that of the normal section. Meusnier's Theorem is thus proved. 69. These investigations with a special choice of axes give geometric proper- ties of the surface, but do not express those properties in a convenient analytic form ; for if a surface z =f{x. y) is given, the transformation to the special axes is difficult. The idea of the indicatrix or its similar conic as the section of the surface by a plane near the tangent plane and parallel to it will, however, deter- mine the general conditions readily. If in the expansion Az - dz = I (rh 2 + 2 shk + tk 2 ) = const, (52) the quadratic terms be set equal to a constant, the conic obtained is the projection of the indicatrix on the xy-plane, or if (52) be regarded as a cylinder upon the xy-plane, the indicatrix (or similar conic) is the intersection of the cylinder with the tangent plane. As the character of the conic is unchanged by the x>rojection, the point on the surface is elliptic if .s' 2 < rt. hyperbolic if s 2 > rt, and parabolic if s 2 = rt. Moreover if the indicatrix is hyperbolic, its asymptotes must project into the asymptotes of the conic (52), and hence if dx and dy replace h and k, the equation rdx- + 2 ndxdy + tdy 2 = (53) may be regarded as the differential equation of the projection of the asymptotic lines on the xy-plane. If r. s. t be expressed as functions f^.f'^.f',,,, of (x, y) and (53) be factored, the integration of the two equations -V(.r. y)dx + X(x. y)dy thus found will give the finite equations of the projections of the asymptotic lines and, taken with the equation z =f(x. y). will give the curves on the surface. To find the lines of curvature is not quite so simple ; for it is necessary to deter- mine the directions which are the projections of the axes of the indicatrix. and these are not the axes of the projected conic. Any radius of the indicatrix may be regarded as the intersection of the tangent plane and a plane perpendicular to the xy-plane through the radius of the projected conic. Hence z~ z =p (x - x ) + q(y- y ), (x - x ) k = (y - y ) h are the two planes which intersect in the radius that projects along the direction determined by /*, k. The direction cosines V/V- + k 2 + (ph + qk) 2 PARTIAL DIFFERENTIATION; IMPLICIT 147 are therefore those of the radius in the indicatrix and of its projection and they determine the cosine of the angle between the radius and its projection. The square of the radius in (52) is h 2 + k 2 , and (h 2 + k 2 )sec 2 = h 2 + k 2 + (ph + qk) 2 is therefore the square of the corresponding radius in the indicatrix. To deter- mine the axes of the indicatrix, this radius is to be made a maximum or minimum subject to (52). With a multiplier X, h + ph + qk + X (rh + sk) = 0, k + ph + qk + X (sh + tk) = are the conditions required, and the elimination of X gives ifi [, (i + pi) _ Mr] + hk [t ( i + pi) _ r (1 + 7' 2 )] - fc' 2 [t (1 + T) - pqt] = as the equation that determines the projection of the axes. Or (1 + p 2 ) dx + pqdy pqdx + (1 + q 2 ) dy — (oo) rdx + sdy sdx + tdy is the differential equation of the projected lines of curvature. In addition to the asymptotic lines and lines of curvature the geodesic or shortest lines on the surface are important. These, however, are better left for the methods of the calculus of variations (§ 159). The attention may therefore be turned to finding the value of the radius of curvature in any normal section of the surface. A reference to (48) and (49) shows that the curvature is 1 _ 2 z _ rh 2 + 2 shk + tk 2 _ rh 2 + 2 shk + tk 2 Tl~ p 2 ~ p 2 h 2 + k 2 in the special case. But in the general case the normal distance to the surface is (Az — dz) cos 7, with sec y = VI + p 2 + q 2 , instead of the 2 z of the special case, and the radius p 2 of the special case becomes p 2 sec' 2 = h 2 + k 2 + (ph + qk) 2 in the tangent plane. Hence 1 2(Az - dz) cos 7 rl 2 + 2 slm + tin 2 — — = — — , (Oil) R h- + k- + (ph + qk)- A i + pi + qi where the direction cosines I, m of a radius in the tangent plane have been intro- duced from (54), is the general expression for the curvature of a normal section. The form 1 rh 2 + 2.shk + tk 2 1 t , an — = — , (5b ) R h- + k- + (ph + qk)- Vl + p 2 + q 2 where the direction h. k of the projected radius remains, is frequently more con- venient than (56) which contains the direction cosines /. m of the original direction in the tangent plane. Meusnier's Theorem may now be written in the form cos v rP 4- 2 slm + tin 2 R VI + p 1 + q 1 where v is the angle between an oblique section and the tangent plane and where l. m are the direction cosines of the intersection of the planes. The work here given has depended for its relative simplicity of statement upon the assumption of the surface (4/ gives the square root: if u represents a range of values of a variable x, the expression f(x) or fx denotes a function of x; if u be a function of x, the operation of dif- ferentiation may be symbolized by 7> and the result ]>u is the deriva- tive ; the symbol of definite integration J (#)<7# converts a function nix) into a number; and so on in great variety. The reason for making a short study of operators is that a consider- able number of the concepts and rules of arithmetic and algebra may be so defined for operators themselves as to lead to a calculus of opera- tion* which is of frequent use in mathematics; the single application to the integration of certain differential equations (§ 95) is in itself highly valuable. The fundamental concept is that of a product : If u is oper- ated upon by f to yirefu = r and if r is operated upon by is J . Some oper- ations have no inverse; multiplication by is a case, and so is the square when applied to a negative number if only real numbers are considered. Other operations have more than one inverse; integra- tion, the inverse of J), involves an arbitrary additive constant, and the inverse sine is a multiple valued function. It is therefore not always true that/ _1 /= 1, but it is customary to mean by / -1 that particular inverse of / for which f~ l f = ff~ l = 1. Higher negative powers are defined by the equation f~ n = (/ -1 )", :UK ^ ^ readily follows that f'f~ n = 1, as may be seen by the example /»/-» =//•(/./- y- 1 /- 1 =f(f-f- 1 )f- 1 =ff- 1 = i. The l"ie of indices /'"/"= /"' + * also holds for negative indices, except in so far as f~\f may not be equal to 1 and may be required in the reduction of /'"'/'" to f" l+ ". COMPLEX NUMBERS AND VECTORS 151 If u, v, and u + v are operands for the operator f and if f(it + v)=fu+fo, (5) so that the operator applied to the sum gives the same result as the sum of the results of operating on each operand, then the operator f is called linear or distributive. If f denotes a function such that f(x + y) =/(./•) + /(//), it has been seen (Ex. 9, p. 45) that / must be equivalent to multiplication by a constant and fx = Ox. For a less specialized interpretation this is not so ; for D (a + r) = Du + Dv and J (u + ?•) = / a + I v are two of the fundamental formulas of calculus and show operators which are distributive and not equivalent to multiplication by a constant. Nevertheless it does follow by the same reasoning as used before (Ex. 9, p. 45), that /Vim = nftt. if f is distributive and if n is a rational number. Some operators have also the property of addition. Suppose that u is an operand and/, g are operators such thaty*« and gu are things that may be added together as fit + gu, then the sum of the operators, / + g, is defined by the equation (f + g)u =fu + gu. If furthermore the operators f g, h are distributive, then * (/ + a) = ] >f + ] ".t and (/ + y) h = / 7 < + r/ 7 ' > ( 6 ) and the multiplication of the operators becomes itself distributive. To prove this fact, it is merely necessary to consider that h [(/+ if) "] = h (/» + .'/") = 7 '/« + 7 ',V" and (/ + y ) (A a) = fit u + gh a. Operators which are associative, commutative, distributive, and which admit addition may be treated algebraically, in so far as polynomials are concerned, by the ordinary algorisms of algebra • for it is by means of the associative, commutative, and distributive laws, and the law of indices that ordinary algebraic polynomials are rearranged, multiplied out, and factored. ~S<.nv the operations of multiplication by constants and of differentiation or partial differentiation as applied to a function of one or more variables x, y, z, ■ ■ ■ do satisfy these laws. For instance r(Du) = D(CU), I>J>„I< = 1 >„!{,", ( I>. r + ]) u ) D z U = D x Tj z U + /'A':: 11 - (") Hence, for example, if y be a function of x, the expression J)"y + ajr-^y + • • • + "„-i^>/ + "„//, where the coefficients a are constants, may be written as (/y + ^/z-i + ... + t , n _ 1 /, + l , n)i/ (8) 152 DIFFERENTIAL CALCULUS and may then be factored into the form [(/> - a l} (JJ -a J--- (D - a n _ l} {D - a n y\y, (8') where a v a , ■ ■ •, a n are the roots of the algebraic polynomial .'•" + v'" _1 + ■■• + "»-!■'• + "„ = °- EXERCISES 1. Show that (fgh)-' 1 = h-^g- 1 /- 1 , that is, that the reciprocal of a product of operations is the product of the reciprocals in inverse order. 2. By definition the operator gfg -1 is called the transform of/by (/. Show that (cr) the transform of a product is the product of the transforms of the factors taken in the same order, and (0) the transform of the inverse is the inverse of the transform. 3. If .s- 5^ 1 but .s' 2 = 1, the operator ,s is by definition said to be involutory. Show that («) an involutory operator is equal to its own inverse; and conversely (j3) if an operator and its inverse are equal, the operator is involutory; and (7) if the product of two involutory operators is commutative, the product is itself involu- tory : and conversely (5) if the product of two involutory operators is involutory. the operators are commutative. 4. If /and g are both distributive, so are the products/;'/ and gf. 5. If /is distributive and n rational, show /mm = nfu. 6. Expand the following operators first by ordinary formal multiplication and second by applying the operators successively as indicated, and show the results are identical by translating both into familiar forms. (a) (D-l)(B-2)y, Ans. pt 3^ + 2y, (/3) (I)-1)D(D + 1) y, (7) l> (1> - 2) [D + 1) (I) + 3) y. 7. Show that {!)— ") <"•' I i—" r A'ilx = A', where .V is a function of x, and hence infer that t" r j c~" J '(*)dx is the inverse of the operator (I) — /)(*). 8. Show that l){L a - r ij) = c ar (D + a) y and hence' generalize to show that if l'{b) denote any polynomial in I) with constant coefficients, then P(U) ■ («•'// = e«*P(D+ n)y. Apply this to the following and check the results. (a) (IP -3IJ+ 2),->!, = e---(Tfi + B)y = e--'('p / - + ^Y \dx- ax/ ( (3 1 , //- _ 3 ]) _ 9) « ■'•(/, (■) ) < If - 8 I) + 2) < *y. 9. If /j is a function of x and x = <' show that l) J .y = e-'J),y, 1/j.y = e~ - <1), (l) t - l)y. • ... I) 1 ;,/ = e-i>'l) t (l),- l)---(D t -p+ l)y. 10. Is the expression (ltl),,. -f 1:J>„)". which occurs in Taylor's Formula ($ 'A), the nth power of the operator ///>,. -f kD,, or is ii merely a conventional .symbol '.' The same question relative to {sl) x + yJl,) 1 ' occurring in Euler's Formula i ,' ->Jj '.' COMPLEX NUMBERS AND VECTORS 153 71. Complex numbers. In the formal solution of the equation >i,c -f- l/x + c = 0, where b 1 < 4ac, numbers of the form m + n V— 1, where m and n are real, arise. Such numbers are called complex or imaginary ; the part w is called the real part and » V- 1 the inire imaginary part of the number. It is customary to write V— 1 = i and to treat i as a literal quantity subject to the relation ir = — 1. The defini- tions for the equality, addition, and multiplication of complex num- bers are , . . . . „ _ . . „ a + bi = c + «* it and only it « = c, /> = e/, [a + o't] + [c + rfi] = (« + c) + (J + 'O i, . (9) [« + it] [c + fZt] = (W — itf) + (arf + //'•) /. It readily follows that the commutative, associative, and distributive hues hold in the domain of complex numbers, namely, a 4 (3 = /? + «, (a + j3) + y = a + (£ + y), a/3 = pa, ( a P)y = a (Py)> (1°) « (/3 + y) = a/3 + ay, (a + /?) y = ay + /3y, where Greek letters have been used to denote complex numbers. Division is accomplished by the method of rationalization. a + bi _ a + it r — di _ (rttf + /;<■/) -f (be — ml) i C -f- ) may be considered as representing the number a + bi. If OP and OQ are two directed lines representing the two numbers a 4- bi and c 4 di, a reference to the figure shows that the line which 154 DIFFERENTIAL CALCULUS <«,6) ! represents the sum of the numbers is OR, the diagonal of the parallelo- gram of which OP and OQ are sides. Thus the geometric Inn- fur adding complex numbers is the same as the law for compounding forces and is known as the parallelogram law. A segment AB of a line possesses magnitude, the length AB, and direction, the direction of the line AB from A to B. A v n.^4?rS a+c i h + d) quantity which has magnitude and direction is cidlcd a vector; and the parallelogram, law is called the late of rector addition. < 'omplex no in- jjf hers may therefore he regarded as rectors. From the figure it also appears that 0(1 and PR have the same mag- nitude and direction, so that as vectors they are equal although they start from different points. As OP -f- PR will be regarded as equal to OP + OQ, the definition of addition may he given as the triangle law instead of as the parallelogram law: namely, from the terminal end ]' of the first vector lay off the second vector PR and (dose the triangle by joining the initial end of the first vector to the terminal end R of the second. The absolute cubic of a complex number a + hi is the magnitude of its vector OP and is equal to V« 2 -+- b' 1 , the square root of the sum of the squares of its real part and of the coefficient of its pure imaginary part. The absolute value is denoted by \a + bi\as in the case of reals. If a: and /3 are two complex numbers, the rule ' a I + J3 === \a + fi\ is a consequence of the fact that one side of a triangle is less than the sum of the other two. If the absolute value is given and the initial end of the vector is fixed, the terminal end is thereby constrained to lie upon a circle concentric with the initial end. 72. When the complex numbers are laid off from the origin, polar coordinates may be used in place of Cartesian. Then (13) ;• = V a' 1 + b' z , = tan~V^V/*, a = ,■ cos <£. b = r sin and a -f- ib = /'(cos c£ + < s h' »• The absolute value /' is often called the modal as or magnitude of the complex number; the angle 4> ' s failed the angle or argument of the number and suffers a certain indetermination in that 2 mr, where n is a positive or negative integer, may be added to without affecting the numher. This polar representation is particularly useful in discussing products and quotients. For if a = /'jfcos .,), then aB = ;y„[ci)S (tj> 1 -f <£./) -f / sin < ^ + <£.,")], * As both cos'/" and >\w!> are known, the quadrant of this angle is deterniineil. (14 COMPLEX NUMBERS AXI) VECTORS 155 as may be seen by multiplication according to the rule. Hence the magnitude of a product is the product of the magnitudes of the factors, and the angle of a product is the sum, of the angles of the factors y the general rule being proved by induction. The interpretation of multiplication by a complex numher as an oper- ation is illuminating. Let /3 be the multiplicand and a the multiplier. As the product a(3 has a magnitude equal to the product of the magni- tudes and an angle equal to the sum of the angles, the factor a used as a multiplier may be interpreted as effecting the rotation of (3 through the angle of a and the stretching of yS in the ratio laj : 1. From the geometric viewpoint, therefore, multiplication lnj a complex numher is an operation of rotation and stretching in the plane. In the case of a = cos -f- i sin ?i) ; (15) and (cos 4> + <• sm <£)" = ( ' os n 4> + i sin //<£, O-^') which is a special cast 1 , is known as J)e Moirre's Theorem and is of use in evaluating the functions of ?i and sin ?i. Hence n (n — 1) cos n = cos"

n ( n -j)(n-2)(» -3) . H — ■ cos" 4 <£ sm 4 <£ — • • • (16) _ , n (n — 1) (n — 2) „ . „ sin n

sin ~ cos" "9 sin + • ■ •• o . As the nth root Va of a must be a number which when raised to the nth power gives a, the ?;th root may be written as Va = VWcos (f>/n + i sin /n). (17) The angle , however, may have any of the set of values . + 2 7T, <£ -f 4 7T, • • • . £ + 2 (n - 1) 7T, 156 DIFFERENTIAL CALCULUS and the nth parts of these give the n different angles tf> 2 IT 4 7T 2 (n - 1) IT n 11 n n ii 11 11 7 Hence there may be found just n different rath roots of any given com- plex number (including, of course, the reals). The roots of unity deserve mention. The equation x" = 1 lias in the real domain one or two roots according as n is odd or even. But if 1 be regarded as a complex number of which the pure imaginary part is zero, it may be represented by a point at a unit distance from the origin upon the axis of reals; the magnitude of 1 is 1 and the angle of 1 is 0, 2tt, • • •, 2(n — 1)tt. The nth roots of 1 will therefore have the magnitude land one of the angles 0, 2ir/n, •••, 2 (n — \)ir/n. Then nth roots are therefore 2-rr . . 2ir , 47r . . 4tt I. 1 . VI* \ ] + (i+ o- V-2 - i Va 1 + V^-s ..,' (0 - <)- 6. Plot and find the modulus and angle in the following cases: (a) - 2, ((3) - 2 V- 1, ( 7 ) 3 + ii, (5) 1 - i v'^3. (y) V- 3/ V2 . 2 + 2 V3 i, tt) (0 + ■1 Vs i, COMPLEX NUMBERS AND VECTORS 157 7. Show that the modulus of a quotient of two numbers is the quotient of the moduli and that the angle is the angle of the numerator leas that of the denominator. 8. Carry out the indicated operations trigonometrically and plot: (a) The examples of Ex. 5, (8) Vl + i \'l — i. (5) ( VT+1 + Vl^K (e) Vv"2 + V^2, (77) v'lG (cos 200° + i sin 200°), (0) V^T, 9. Find the equations of analytic geometry which represent the transforma- tion equivalent to multiplication by it = — 1 + v — 3. 10. Show that \z — ct\ = r, where z is a variable and a a fixed complex number, is the equation of the circle (x — a) 2 + (// — b)' 2 = r". 11. Find cos ox and cos8x in terms of cosx, and sin 6 x and sin 7 x in terms of sin x. 12. Obtain to four decimal places the five roots VI. 13. If z = x + iy and z' = x' + '//'• show that 2' = (cos<£ — isin0)z — a is the formula for shifting the axes through the vector distance it = a + ib to the new origin (a, b) and turning them through the angle + i sin *) (19) will be completely determined in value if z is given. Such a function is called a com p)l ex function (and not a function of the complex variable, for reasons that will appear later). The magnitude and angle of the function are determined by X . Y R = Va' 2 + Y\ cos = — , sin <1> = — . (20) 1L It, 158 DIFFERENTIAL CALCULUS The function F is continuous by definition when and only when both X and Y are continuous functions of (x, y) ; It is then continuous in (x, y) and F can vanish only when 11 = ; the angle <£ regarded as a function of (x, y) is also continuous and determinate (except for the additive 2 mi) unless R = 0, in which case X and Y also vanish and the expression for $ involves an indeterminate form in two variables and is generally neither determinate nor continuous (§ 44). If the derivative of F with respect to z were sought for the value z = a + ib, the procedure would be entirely analogous to that in the case of a real function of a real variable. The increment Az = Ax -f iAy would be assumed for z and AF would be computed and the quotient AF/Az would be formed. Thus by the Theorem of the Mean (§ 46), af = at + /Ar = (.v; + ;Y: r) \r + (x; + /)-,;•) a.v , . Az Ax + iAy Ax + iAy ^ ^" ' where the derivatives are formed for (a, b) and where £ is an infinitesi- mal complex number. When Az approaches 0, both Ax and Ay must approach without any implied relation between them. In general the limit of AF/Az is a double limit (§ 44) and may therefore depend on the way in which Ax and Ay approach their limit 0. Xow if first Ay = and then subsequently A./- = 0, the value of the limit of AF/Az is X x + lY' x taken at the point (a, l>) : whereas if first Ax ~ and then Ay = 0, the value is — iX y -|- Y' r Hence if the limit of AF/Az is to be independent of the way in which Az approaches 0, it is surely necessary that cX ,cY cx cx .cX cY ■ i- h w~> cX cY cX dY cx cy cy cx or — = — and — = — - — (22) cx cy cy cx v J And conversely if these relations are satisfied, then AF (cX , . CY\ „ (cY .cX\ Az \cx cx J " \cy cy) and the limit is A'' + iY' x = Y' tl — iX' y taken at the point (", b), and is independent of the way in which As approaches zero. The desirability of having at least the ordinary functions differentiable suggests the definition: A complex function F(x, y) = X(x, y) -f- iY(x, y) is con- sidered as a function of the complex variable z = x -f- iff v ' nrn antJ f>ni ll vlicnX and Y are in general differential >le and satisfy the relations (22). In this case the derivative is COMPLEX NUMBERS AND VECTORS 159 ,, s dF cX .cY cY .cX , nn F \ z ) = -r = — + 1^ = ^-1^- • (23) v J dz ex ex cij cy These conditions may also be expressed in polar coordinates (Ex. 2). A few words about the function <£>(x. y). This is a multiple valued function of the variables (x, y). and the difference between two neighboring branches is the con- stant 2tt. The application of the discussion of § 45 to this case shows at once that, in any simply connected region of the complex plane which contains no point (a, b) such that E (a, b) = 0, the different branches of 4>(.r, y) may be entirely separated so that the value of <£ must return to its initial value when any closed curve is de- scribed by the point (./•. ?/). If. however, the region is multiply connected or contains points for which 11 = (which makes the region multiply connected because these points must be cut out), it may happen that there will lie circuits for which , although changing continuously, will not return to its initial value. Indeed if it can be shown that does not return to its initial value when changing continuously as (,r, y) describes the boundary of a region simply connected except for the excised points, it may be inferred that there must be points in the region for which 11 = 0. An application of these results may be made to give a very simple demonstration of the fundamental theorem of algebra tliat every equation of the nth degree has at least one root. Consider the function F(z) = z» + «!«»-! + ■ • ■ + a H -iz + a„ = X(x. y) + iY(x. y). where X and Y are found by writing z as x -f iy and expanding and rearranging. The functions X and 1" will be polynomials in (x, y) and will therefore be every- where finite and continuous in (x. y). Consider the angle <$> of F. Then $ = ang. of F= ang. of z" 1 1 + — + ••• + "" — + ' : '-'- ) = aim. of z» + ang. of (1 + • • •). \ z z"- 1 z"J Xext draw about the origin a circle of radius r so large that "i I yn—\\ ' ?n n A K'n-l \ a n\ 11 + ••• +UL_LI + l_!Ll must change by 2 mr and does not return to its initial value. Hence there is within the circle at least one point (a. b) for which U (a, b) = and consequently for which A' (a, b) = and Y{a. b) = and F(a, b)-0. Thus if a = a + ib. then F(a) = and the equation F(z) — is seen to have at least the one root a. It follows that z — a is a factor of F(z) ; and hence by induc- tion it may be seen that F{z) = has just n roots. 74. The discussion of the algebra of complex numbers showed how the sum, difference, product, quotient, real powers, and real roots of such numbers could be found, and hence made it possible to compute the value of any given algebraic expression or function of z for a given value of z. It remains to show that any algebraic expression in z is 160 DIFFERENTIAL CALCULUS really a function of z in the sense that it has a derivative with respect to z, and to find the derivative. Now the differentiation of an algebraic function of the variable x was made to depend upon the formulas of dif- ferentiation, (6) and (7) of § 2. A glance at the methods of derivation of these formulas shows that they were proved by ordinary algebraic manipulations such as have been seen to be equally possible with imagi- naries as with reals. It therefore may be concluded that an ahjebraic expression in z litis a derivative with respect to z and that derivative may be found just as if z were a real variable. The case of the elementary functions e*, logs, sins, cos z, ••• other than algebraic is different; for these functions have not been defined for complex variables. Now in seeking to define these functions when z is complex, an effort should be made to define in such a way that: 1° when z is real, the new and the old definitions become identical : and 2° the rules of operation with the function shall be as nearly as possi- ble the same for the complex domain as for the real. Thus it would be desirable that ])<' z = r z and e z + w = e z e v ', when z and >'■ are complex. With these ideas in mind one may proceed to define the elementary functions for complex arguments. Let e z = /.' (>', //) [cos (:r, y ) + i sin $ (./•, //)]. (24) The derivative of this function is, by the first rule of (23), c . c De? = — (I! cos ) C.r - CX. = ( l!' r cos — 11 sin

,') + i ( A',', sin <$ + R cos 4> • <&,'.), and if this is to be identical with e~ above, the equations I!' r cos 4> — R<$>' r sin = R cos /,'.'. = /.' or R' x sin $> + A','. cos = R sin ' = 1. It therefore appears that if the derivative of <■'-. however computed, is to be i' : . then /?; = r, R' y = o, ; = 1 are four conditions imposed upon R and $. These conditions will be satisfied if R = r r and 4> = //.* Hence define ,,~ _ ,,.<• I- i'J _ ,.<-( ( .( )S y _j_ I S i U y)_ ( 'J,') I * The use of tin- more general solutions It >i + < ' would lead to expressions which would not reduce to <■■'' when >/ and z .c or would nor satisfy < ~ + "' = c~c"'. COMPLEX NUMBERS AND VECTORS 161 With this definition Dc z is surely e z , and it is readily shown that the exponential law e z + w = eV holds. For the special values \ iri, iri, 2 iri of z the value of e z is Hence it appears that if 2 mri be added to z, e 2 is unchanged ; e 2 + 2nnt_^ period 2 iri. (26) Thus m the complex domain e z has the period 2ttI, just as cos,/' and since have the real period 2 tt. This relation is inherent; for e- ,/f = cos y + I sin >/, e~ ! " = cos y — ! sin //, and cos y = > sin y = — • (27) The trigonometric functions of a real variable y may be expressed in terms of the exponentials of yl and — yi. As the exponential has been defined for all complex values of z, it is natural to use (27) to define the trigonometric functions for complex values as e" -f >'-•"-' r'-' — e~ zl cos z = ■ — > sm z = — (2 1 ) With these definitions the ordinary formulas for cos (z + "'), 1 } s i n ■-', • ■ ■ may be obtained and be seen to hold for complex arguments, just as the corresponding formulas were derived for the hyperbolic functions (§ 5). As in the case of reals, the logarithm logs will be defined for com- plex numbers as the inverse of the exponential. Thus if t z = w, then log v: — z + 2 mri, (28) where the periodicity of the function c z shows that the logarithm is not uniquely determined hut admits the addition, of 2 mri to any on*' of its values, just as tan -1 ,r admits the addition of mr. If v is written as a complex number u + iv with modulus /• = VV 2 -\- v' 2 and with the angle . it follows that u- = u + ir = »-(cos 4- i sin cf>) = re*' = e losr + '* t ; (29) and log v — log r + i = log ^ i. 4. From the definitions given above prove the formulas (a) sin (x + iy) = sin x cosh y + i cos x sinh ?/, (/3) cos (x + iy) = cos x cosh y — i sin x sinh y, sin2.r + /sinh 2 y coszx + cosh 2 a 5. Find to three decimals the complex numbers which express the values of: (a) ei 771 ', 08) e\ (y) ei + i^ (5) e-i-', (e) sinlTTt, (f) cos/. (r?) sin (i + I V- 3 ). (4) tan (- 1 - /), (0 log(-l), (k) log/, (X) log(i + ^ V-li). (a) lf.g(-l- /). 6. Owing to the fact that log a is multiple valued. a b is multiple valued in such a manner that any one value may be multiplied by c ln ~ bi . Find one value of each of the following and several values of one of them: (a) 2>, (/3) i\ (y) < r L (5)^2, (OG + l^)'' COMPLEX NUMBERS AND VECTORS 7. Show that Da z = a 2 log a when a and z are complex. 163 8. Show that (a b ) c = a'" : ; and fill in such other steps as may be suggested by the work in the text, which for the most part has merely been sketched in a broad way. 9. Show that if f(z) and g(z) are two functions of a complex variable, then f(z)±y(z), ccf(z) with a a complex constant, f(z)g (z), f(z)/g(z) are also func- tions of z. 10. Obtain logarithmic expressions for the inverse trigonometric functions. Find sin- 1 i. 75. Vector sums and products. As stated in § 71, a vector is a quan- tity which lias magnitude and direction. If the magnitudes of two vectors are equal ami the directions of the two vectors are the same, the vectors are said to he equal irrespective of the position which they occupy in space. The vector — a is by definition a vector which has the same <^i magnitude as a hut the opposite direction. The ^ vector ma is a vector which lias the same direction as a (or the opposite) and is /// (or — ///) times as long. The law of vector or geometric addition is the parallelogram or triangle law (§ 71) and is still applicable when the vectors do not lie in a plane hut have any directions in space; for any two vec- tors brought end to end determine a plane in which the construction may he carried out. Vectors will he designated by Greek small letters or by letters in heavy type. The relations of equality or similarity between triangles establish the rules a + /? = /3 + a, a -f- (/? + y) = (a + (3 ) -f- y. m (a + (3 ) = ma + m(3 (30) as true for vectors as well as for numbers whether real or complex. A vector is said to be zero when its magnitude is zero, and it is writ- ten 0. From the definition of addition it follows that a + = a. In fact as fur as addition, subtraction, and multiplication bij numbers arc concerned, rectors obey the same formal lairs as numbers. A vector p may be resolved into components par- allel to any three given vectors a, (3. y which are not parallel to any one plane. For let a parallelepiped be constructed with its edges parallel to the three given vectors and with its diagonal equal to the vector whose compo- nents are desired. The edges of the parallelepiped are then certain 164 DIFFERENTIAL CALCULUS multiples xa, yfi, zy of a, /3. y ; and these are the desired components of p. The vector p may be written as p = xa + y/3 + zy* (31) It is clear that two equal vectors would necessarily have the same components along three given directions and that the components of a zero vector would all be zero. Just as the equality of two complex numbers involved the two equalities of the respective real and imagi- nary parts, so the equality of two vectors as p = xa + ///? + ::y = x'a + y'fi + z'y = p' (31') involves the three equations x = x', y = //', z = z'. As a problem in the use of vectors let there be given the three vectors a. £. y from an assumed origin to three vertices of a parallelogram ; required the vector to the other vertex, the vector expressions for the sides and diagonals of the paral- lelogram, and the proof of the fact that the diagonals bisect each other. Consider the figure. The side AB is, by the triangle law, that vector which when added to OA = a gives OB = j3. and hence it must be that AB = j3 — a. In like manner AC = y— a. Now 01) is the sum of 00 and CD, and CD = AB; hence OD = y + )3 — a. The diag- onal AD is the difference of the vectors OD and OA, and is therefore y + /3 — 2 a. The diagonal BC is y — (3. Now the vector from to the middle point of BC may be found by adding to OB one half of BC. Hence this vector is j8 + | (7 — /3) or \ (/3 + 7). In like manner the vector to the middle point of A 1) is seen to be a + I (7 + ,3 — '1 a) or \ (7 4- /3). which is identical with the former. The two middle points therefore coincide and the diagonals Insect each other. Let a and /3 be any two vectors, \a\ and |/?| their respective lengths, and Z. (a, ft) the angle between them. For convenience the vectors may be considered to be laid off from the same origin. The product of the lengths of the vectors by the cosine of the tingle between the vectors is called the scalar product, scalar product = a./3 = |a|| j8| cos Z (a. /3), (32) of the two vectors and is denoted by placing a dot between the letters. This combination, called the scalar product, is a number, not a vector. As ft I'usZ (a. f3) is the projection of fi upon the direction of a, the scalar product may be stilted to be equal to the product of the length of either vector by the length of the projection of the other upon it. In particular if either vector were of unit length, the scalar product would be the projection of the other upon it. with proper regard for * The numbers ./•. >i. z are the oblique coordinates of the terminal end of p (if the initial end be at the orijjin) referred to a set of axes which are parallel to a, (3, 7 and upon which the unit lengths are taken as the lengths of a, 13, 7 respectively. COMPLEX NUMBERS AND VECTORS 165 the sign ; and if both vectors are unit vectors, the product is the cosine of the angle between them. The scalar product, from its definition, is commutative so that «•/?=/?•«:. Moreover (jna)»fi = /?) y means that the vector y is multiplied by the number oc»(S, whereas a (/8»y) means that a is multiplied by (/3«y). a very different matter. The laws of cancellation cannot hold : for if a.jl = 0, then \a\\p\ cos Z (7r, /8) = 0, (34) and the vanishing of the scalar product a -ft implies either that one of the factors is or that" the two vectors are perpendicular. In fact effi = is called the condition of perpendicularity. It should be noted, however, that if a vector p satisfies p.ct = 0, p.p = 0, p.y = 0, (35) three conditions of perpendicularity with three vectors a, /3, y not parallel to the same plane, the inference is that p = 0. 76. Another product of two vectors is the vector product, vector product = axfi = v\a\\(3\ sin Z (a, ft), (36) where v represents a vector of unit length normal to the plane of a and ft upon that side on which rotation from a to ft through an angle of less than 180° appears posi- ax/3' tive or counterclockwise. Thus the vector product is itself a vector of which the direction is perpen- dicular to each factor, and of which the magni- aX <_ tude is the product of the magnitudes into the sine of the included angle. The magnitude is therefore equal to the area of the parallelogram of which the vectors a and ft are the sides. 1GG DIFFERENTIAL CALCULUS The vector product will be represented by a cross inserted between the letters. As rotation from ft to a is the opposite of that from a to ft, it follows from the definition of the vector product that ftxa = - axft, not axft = ftxa, (37) and the product is not commutatlre, the order of the factors must be carefully observed. Furthermore the equation ax (3 = v (i\\ft\smZ(a, ft) = (38) implies either that one of the factors vanishes or that the vectors a ami ft are parallel. Indeed the condition axft = is called the condition of parallelism. The laws of cancellation do not hold. The associative law also does not hold; for (axft)xy is a vector perpendicular to axft and y. and since ax ft is perpendicular to the plane of a and ft. the vector (ax/3)xy perpendicular to it must lie in the plane of a and ft: whereas the vec- tor ax(ftxy), by similar reasoning, must lie in the plane id' ft and y : and hence the two vectors cannot be equal except in the very special case where each was parallel to ft which is common to the two planes. But the operation (nta)xft = ax(mft) = m(axft), which consists in allowing the transference of a numerical factor to any position in the product, does hold; and so does the dlstrlhutlre lair ax( ft -f y ) = ax ft + axy and (a -f- ft)xy = axy -f- ftxy, i .">'.> } the proof of which will be given below. In expanding according to the distributive law care must be exercised to keep the order of tin- factors in each vector product the same on both sides of the equation, owing to the failure of the commutative law: an interchange of the order of the factors changes the sign. It might seem as if any algebraic operations where so many of the laws of elementary algebra fail as in the case of vector products would be too restricted to be very useful; that this is not so is due to the astonishingly great number of problems in which the analysis can be carried on with only the laws of addition and the distributive law of multiplication combined with the possibility of transferring a numerical factor from one position to another in a product; in addition to these laws, the scalar product a-ft is commuta- tive and the vector product ax ft is commutative except for change of sign. In addition to segments of lines, plum* ureas ma ij he regarded as rector quantities ; for a plane area has magnitude (the amount of the area ) and direction (the direction of the normal to its plane). To specify on which side of the plane the normal lies, some convention must be made. If the area is part of a surface inclosing a portion of space, the COMPLEX NUMBERS AND VECTORS 16' Ab normal is taken as the exterior normal. If the area lies in an isolated plane, its positive side is determined only in connection with some assigned direction of description of its hounding curve ; the rule is : If a person is assumed to walk along the boundary of an area in an assigned direction and upon that side of the plane which causes the inclosed area to lie upon his left, he is said to be upon the positive side (for the assigned direction of description of the boundary), and the vector which represents the area is the normal to that side. It has been mentioned that the vector product represented an area. That the projection of a plane area upon a given plane gives an area which is the original area multiplied by the cosine of the angle between the two planes is a fundamental fact of projection, following from the simple fact that lines parallel to the intersection of the two planes are unchanged in length whereas lines perpendicular to the intersection are multiplied by the cosine of the angle between the planes. As the angle between the normals is the same as that between the planes, the projection of on area upon a plane and flu- projection of the vector rep- resenting the area upon the normal to the plane are equivalent. The projection of a closed area upon a plane is zero: for the area in the projection is covered twice (or an even number of times) with opposite signs and the total algebraic sum is therefore 0. To prove the law ax(j8 + y) = orx/3 + axy and illustrate the use of the vector interpretation of areas, construct a triangular prism with the triangle on /3, y, and (3 + y as base and a as lateral edge. The total vector expression for the surface of this prism is fixa + yx« + ax((3 + y) + i(/3xy) - \ £xy = 0, and vanishes because the surface is closed. A cancel- lation of the equal and opposite terms (the two bases) and a simple transposition combined with the lade fixa = — ax^3 gives the result ax((3 -f- y) = — (3*a — yx + &J + 2» g k, then /? connecting the vertex to the middle point, of the opposite side CD is trisected by the diagonal AD and trisects it. 4. Show that the medians of a triangle meet in a point and are trisected. 5. If m. and in., are two masses situated at 7', and P„, the coder of gravity or center of mass of »(, and »i., is defined as that point 67 on the line PjP which divides P 1 P, inversely as the masses. Moreover if 67, is the center of mass of a number of masses of winch the total mass is .1/, and if 67., is the center of mass of a number of other masses whose total mass is .1/".,, the same rule applied to M 1 and 3f„ and 67, and 67., gives the center of gravity 67 of the total number of masses. Show that m.r. + m.,r„ , m.x, 4- m.,r., 4- • . • + m„r„ zZmr r= — " "- and r — — '- ■'- -- ' - — , m t + m 2 m, 4- in., 4- • • ■ 4- '",, S?n where r denotes the vector to the center of gravity. Resolve into components to show «, „., ., 2.m.r 2,iii// zZmz x- , y = , z 2Zl)l ZiHl 2.111 6. If a and /3 are two fixed vectors and p a variable vector, all being laid off from the same origin, show that (p — p)»a = is the equation of a plane through the end of rf perpendicular to or. 7. Let a, /3, 7 be the vectors to the vertices A, J'>, C of a triangle. Write the three equations of the planes through the vertices perpendicular to the opposite sides. Show that the third of these can be derived as a combination of the other two; and hence infer that the three planes have a line in common and that the perpendiculars from the vertices of a triangle meet, in a point. COMPLEX NUMBERS AND VECTORS 109 8. Solve the problem analogous to Ex. 7 for the perpendicular bisectors of the sides. 9. Note that the length of a vector is V (pxy) and (nrx^).y = (j3xy)>a ; and hence infer that in a product of three vectors with cross and dot, the position of the cross and dot may be interchanged and the order of the factors may be permuted cyc- lically without altering the value. Show that the vanishing of (dxp).y or any of its equivalent expressions denotes that or, p, 7 are parallel to the same plane ; the condition axp»y = is called the condition of complanarity. 14. Assuming a = a x i + <<.,) + a 3 k, p = c.,i + !>.,] + b 3 k, 7 = (\i + <-.,] + <\,k, expand /3, and ax(pxy) in terms of the coefficients to show ax(pxy) — (ct>y)p — (a»p)y; and hence (cixp)xy = (a»y) p — (y-p) a. 15. The formulas of Ex. 14 for expanding a product with two crosses and the rule of Ex. 13 that a dot and a cross may be interchanged may be applied to expand («x/3)x( 7 x8) = (a.y*8)p-(p.yx8)a = (a*p.5)y - (a*p.y)8 and (axp).(yxS) = (a.y)(p.S) - (p.y)(«.8). 16. If or and p are two unit vectors in the ./'//-plane inclined at angles and to the x-axis, show that a = icos# + j sin 0, p = ieos + jsin ; and from the fact that «./3 = cos(0— 0) and — 6) and eos( — 0). 17. If /. hi, x are direction cosines, the vector l\ + »i] + )/k is a vector of unit length in the direction for which l,.m, n are direction cosines. Show that the condition for perpendicularity of two directions (/, to, )/) and (/', ///'. n") is II' + mm' + hh' = 0. 18. With the same notations as in Ex. 14 show that I i j k I I //, //., Co or.or = a'l +■ «| + «| and axp = rf, //., , /., /< . ■ ''1 &- ''3 I 1 ''1 ''• '':: 170 DIFFERENTIAL CALCULUS 19. Compute the scalar and vector products of these pairs of vectors ("H f 6i + 0.3 j — 5 k 1^0.1 i- 4.2 j + 2.5k, (/*) 1 f i + 2 j + 3 k L - 3 i - 2 j + k, (7) Ji + k ti + i. 20. Find the areas of the parallelograms defined by the pairs of vectors in Ex. 19. Find also the sine and cosine of the angles between the vectors. 21. Prove ax[px(yx8)] = (cx-y^S) (i — cx-fiyxd = /3«5 axy — fi-y ax8. 22. What is the area of the triangle (1, 1, 1), (0. 2. 3). (0. 0, - 1) ? 77. Vector differentiation. As the fundamental rules of differentia- tion depend on the laws of subtraction, multiplication by a number. the distributive law, and the rules permitting rearrangement, it follows that the rules must lie applicable to expressions containing vectors without any changes except those implied by the fact that «x/3 =£ fixa. As an illustration consider the application of the definition of differen- tiation to the vector product Uxv of two vectors which are supposed to be functions of a numerical variable, say x. Then A(uxv) = (U + AU)x(V + Av) — UxV = UxAv -f AUxy + AUxAv, A (Uxv) A.r rf(UxV) dx Av Au Ux 1 xV = lim A.'' = - A./' Ar A (UxV ) A./- AUxAv r ^A^~' the rurrr : in particular if the variable .#■ were the arc s, the derivative would have the magnitude unity and would be a unit vector tangent to the curve. The derivative or differential of a vector of constant length is per- pendicular to the vector. Tins follows from the fact thai the vector dx v Ar dx T Al * = Inn - = = Jim - d.r A./' ds An COMPLEX NUMBEES AND VECTORS 171 then describes a circle concentric with the origin. It may also he seen analytically from the equation d(r.r) = tfr.r + r.rfr = 2 r.r/r = d const, = 0. (43) If the vector of constant length is of length unit}-, the increment Ar is the chord in a unit circle and, apart from infinitesimals of higher order, it is equal in magnitude to the angle subtended at the center. Consider then the derivative of the unit tangent t to a curve with respect to the arc s. The magnitude of <^ the normal to the surface F = (' is more rapid than along any other direction ; for the change in F be- tween the two surfaces is dF = dC and is constant, whereas the distance dn between the two surfaces is least (apart from infinitesimals of higher order) along the normal. In fact if dr denote the distance along any other direction, the relations shown by the figure are dr = sec Odti and — r- = —— cos 0. (46) dr dn If now n denote a vector of unit length normal to the surface, the product RdF/dn will be a rector quantity which has both the magnitude and, the direction of most rapid increase of F. Let n — - = VF = grad F (47) dn be the symbolic; expressions for this vector, where VF is read as "del /•"' and grad F is read as "the gradient of /<'." If dT be the vector of which dr is the length, the scalar product n^/r is precisely cos Qdr, and hence it follows that ( /r.VF = dF and r^V !■' = '-—- , (48) where r } is a unit vector in the direction dr. The second of the equa- tions shows that the directional deriratire in any direction is the com- ponent or projection of the gradient in that direction. From this fact the expression of the gradient may be found in terms of its components along the axes. For the derivatives of F along the axes are dF/dx, cF/dy, cF/dz, and as these are the components of VF along the directions i, j, k, the result is . OF . cF , OF VF = grad F = l . - + i — + k . - c.r Cy cz Hence V = i — + j - + k — C.r J cy cr: (49) may be regarded as a symbolic vector-differentiating operator which when applied to /•' gives the gradient of /•'. The product dt-VF = ( my, F = -(my). (It at /KiN (51) T7 ' /V ' /f f "P * ' /V ' /r Hence F = m — = m — r = nil it I = — = — -. • dt df dt dt 1 From the equation.-; it appears that the force F is the product of the mass m by a vector f which is the rate of change of the velocity regarded * In applications, it is usual to denote vectors by heavy type and to denote the magni- tudes of those vectors by corresponding italic letters. 174 DIFFERENTIAL CALCULUS as a vector. The vector f is called the acceleration; it must not be con- fused with the rate of change dv/dt or d 2 s/df 2 of the speed or magnitude of the velocity. The components /,., f ;n f z of the acceleration along the axes are the projections of f along the directions i, j, k and may be written as f«i, f «j, f «k. Then by the laws of differentiation it follows that /* = f-i = dv . Tt' 1== d (vi) dv x dt " dt /, = w = A" = /// f =11) — dt- It It is noteworthy that the force must lie in the osculating plane. If r and r -f- Ar are two positions of the radius vector, the area of the sector included by them is 1 except for infinitesimals of higher order) COMPLEX NUMBERS AND VECTORS 175 AA = i rx(r + Ar) = -i- rxAr, and is a vector quantity of which the direction is normal to the plane of r and r -+- Ar, that is, to the plane through the origin tangent to the curve. The rate of description of area, or the ureal velocity, is therefore 2\?Tt- x it)> 2\ x &-**)• (o4) If the force F acting on the mass m passes through the origin, then r and F lie along the same direction and rxF = 0. The equation of motion may then be integrated at sight. m — = F, ?«rx— = rxF = 0, dt ' dt ' dy d rx— - = — (rxv) = 0, rxv = const. dt dt v J It is seen that in this case the rate of description of area is a constant vector, which means that the rate is not only constant in magnitude but is constant in direction, that is, the path of the particle m must lie in a plane through the origin. When the force passes through a fixed point, as in this case, the force is said to be central. Therefore when a particle moves under the action of a central force, the motion takes place in a plane passing through the center and the rate of description of areas, or the areal velocity, is constant. 80. If there are several particles, say ?;. in motion, each has its own equation of motion. These equations may lie combined by addition and subsequent reduction. d 2 r„ „ m n — - = F„, dt- and m, - + m„ -%-\" + • ■ • + m„ Vr = F, + F„ + • • • + F„. But dt- d' 2 r, _ d-r., ?«, ' = F, . m — - = :F 2 , d-r, c7-r„ - } + m„ — - H + m, dt- - dt- d-r„ '~d¥ d-r., d-r n - m„ — = + • • • + m„ : - dt- dt- _ d 2 "dt 1 Let m l t 1 + m„r 2 + • • • + m n r n = (nij + m. 2 + • • • + m„) r = M r _ m 1 r 1 + m 2 r„ + ■ ■ ■ + m„r„ __ Zmr _ Smr or m 1 + m 2 + • • • + m n 1m M jl-r Then 31 — = F. + F., + • • . + F„ = Vf. (55) dt? l - "A- 170 DIFFERENTIAL CALCULUS Now the vector r which has been here introduced is the vector of the center of muss or center of gravity of the particles (Ex. 5, p. 168). The result (55) states, on comparison with (51), that the center of gravity of the n masses moves as if all the mass M were concentrated at it and all the forces applied there. The force F,- acting on the ith mass may be wholly or partly due to attractions, repulsions, pressures, or other actions exerted on that mass by one or more of the other masses of the system of n particles, in fact let F,- be written as F t - = F l0 + F/, + Fa + ■ ■ ■ + F„„ where Fy is the force exerted on »/,• by m, and F,-,, is the force due to some agency external to the n masses which form the system. Now by Newton's Third Law. when one particle acts upon a second, the second reacts upon the first with a force which is equal in magnitude and opposite in direction. Hence to Fy above there will correspond a force F/,- =— Fy exerted by ni{ on mj. In the sum ZF; all these equal and opposite actions and reactions will drop out and ZF,- may be re- placed bySFfo, the sum of the external forces. Hence the important theorem that : The motion of the center of mass of a set of particles is as if all the mass were concen- trated there and all the external forces vjere applied there (the internal forces, that is, the forces of mutual action and reaction between the particles being entirely neglected). The moment of a force about a given point is defined as the product of the force by the perpendicular distance of the force from the point. If r is the vector from the point as origin to any point in the line of the force, the moment is therefore rxF when considered as a vector quantity, and is perpendicular to the plane of the line of the force and the origin. The equations of n moving masses may now be combined in a different way and reduced. Multiply the equations by r n r„, • • •, r„ and add. Then ''V, dv., dv„ „ _ _ m.r.x ' + )H.,r.,x = _u . . . j. Hi.,r„x — r.xF, + r.,xF„ + • • • + r„xF„ 11 ill - - dt dt l x 2 2 d d d or m, r.xv, + vi., r.,xv„ + . . . -f m„ - r,,xv„ = r.xF. -f r.,xF„ + • • • + r,,xF„ i dt II 2 M . .. -r dt l i t . 2 t d or — (m ] r,xv 1 + m 2 r.,xv 2 + • ■ • -f- /n„r„xv„) = ZrxF. (56) This equation shows that if the a real velocities of the different masses are multiplied by those masses, and all added together, the derivative of the sum obtained is equal to the moment of all the forces about the origin, the moments of the different forces being added as vector quantities. This result may be simplified and put in a different form. Consider again the resolution of F, into the sum F, n + F,i + • • • + F,„. and in particular consider the action F<; and the reaction F/, = — Fy between two particles. Let it be assumed that the action and reaction are not only equal and opposite, bat lie along the line connecting the two particles. Then the perpendicular distances from the origin to the action and reaction are equal and the moments of the action and reaction are Mjual and opposite, and may be dropped from the sum Zr,xF,-. which then reduces to Zr/xF,,,. < )n the other hand a term like iH,r,xV/ may be written as r,-x(w;V,-). This product is formed from the momentum in exactly the same way that: the moment is formed from the force, and it is called the moment of momentum, Hence the equation (56) becomes COMPLEX NUMBERS AND VECTORS 177 — (total moment of momentum) = moment of external forces. Hence the result that, as vector quantities : The rate of change of the moment of momentum of a system of particles is equal to the moment of the external forces (the forces between the masses being entirely neglected under the assumption that action and reaction lie along the line connecting the masses). EXERCISES 1. Apply the definition of differentiation to prove (a) d(u.v) = uWv + v.du, (p) d[u.(vxw)] = du.(vxw) + u.('Zvxw) + u.(vx + ja sin

with s = v«- + b- rf> ; show that the radius of curvature is (a- -f b-)/a. 5. Find the torsion of the helix. It is h/ (a- + b-). 6. Change the variable from s to some other variable /. in the formula for torsion. 7. In the following cases find the gradient either by applying the formula which contains the partial derivatives, or by using the relation rfr-YF = dF, or both : (a) r.r = x- + a- + z-. (0) log r, (7) r = Vr.r, (5) lo-(.r- + //-) = log[r.r - (k-r) 2 ], (e) (rxa).(rxb). 8. Prove these laws of operation with the symbol V : (a) V(F + G) = VF + VG, (j8) G-\(F/G) = GVF - FVG. 9. If r. rp are polar coordinates in a plane and r, is a unit vector along the radius vector, show that dr l /dt = nd

!') of the text. 178 DIFFERENTIAL CALCULUS 12. Show that the Cartesian expressions for the magnitude of the velocity and of the acceleration and for the rate of change of the speed dv/dt are / ,„ , — ^— — r, . / ,,,, , — 777— — r- y , x'x" + y'y" + z'z" v = Vx' 2 + V z + z'\ f = Vx" 2 + y" 2 + z"\ v' = — J J - - , vV 2 + y' 2 + z'- where accents denote differentiation with respect to the time. 13. Suppose that a body which is rigid is rotating about an axis with the angular velocity w = d

' 2 r toward the axis, (da./dt)xi perpendicular to the axis, under the assumption that the axis of rotation is invariable. 14. Let r denote the center of gravity of a system of particles and r/ denote the vector drawn from the center of gravity to the /th particle so that r,- = r + r- and Vj = v + v/. The kinetic energy of the £th particle is by definition i- m(vf = I mtVi-Vi — I nii (v + v/ ).(v + v/). Sum up for all particles and simplify by using the fact Z?m,t^ = 0, which is due to the assumption that the origin for the vectors r- is at the center of gravity. Hence prove the important theorem : The total kinetic energy of a system is equal to the kinetic energy iclticJi the total mass would have if moving with the center of gravity plus the energy computed from the motion relative to the center of gravity as origin, that is, T = i Sm,-i3? = i Mv 2 + \ ZniiVf. 15. Consider a rigid body moving in a plane, which may be taken as the xy- plane. Let any point r of the body be marked and other points be denoted rela- tive to it by r'. The motion of any point r' is compounded from the motion of r (l and from the angular velocity a = ku of the body about the point r n . In tact the velocity v of any point is v = V + axr'. Show that the velocity of the point denoted by r' = kxv (1 /w is zero. This point is known as the instantaneous center of rotation (ij 39). Show that the coordinates of the instantaneous center referred to axes at, the origin of the vectors r are x = r .i = ar ~° , y = r.j = y + . o) dt w dt 16. If several forces F x , F L> . • • ■. F„ act on a body, the sum R = SF; is called the resultant and the sum 2r,-xF,-. where r,- is drawn from an origin to a point in the line of the force F,-, is called the resultant moment about 0. Show that the resultant, moments Mo and Mr/ about two points are connected by the relation Mo- = M + Mr)-(Ro). where Mo-(Ro) means the moment about ()' of the resultant R considered as applied at 0. infer that moments about all points of any line parallel to the resultant are equal. Show that in any plane perpendicular to R there is a point ()' given by r = RxM^/R-R. where is any point of the plane, such that M v is parallel to R. PART IT. DIFFERENTIAL EQUATIONS CHAPTER VII GENERAL INTRODUCTION TO DIFFERENTIAL EQUATIONS 81. Some geometric problems. The application of the differential calculus to plane curves has given a means of determining some geometric properties of the curves. For instance, the length of the subnormal of a curve (§ 7) is ydy/dx, which in the case of the parabola if = ip.r is 2p, that is, the subnormal is constant. Suppose now it were desired conversely to find all curves for which the subnormal is a given constant in. The statement of this problem is evidently con- tained in the equation is found to be II = a 2 /3 r, that is, the radius of curvature varies inversely as the radius. If conversely it were desired to find all curves for which the radius of curvature varies inversely as the radius of the curve, the state- ment of the problem would be the equation +(£)' /■ f L + 2 (d^ r d- \d(f> / where /.• is a constant called a factor of proportionality.* Equations like these are unlike ordinary algebraic equations because, in addition to the variables x, y or /•, (f> and certain constants m or /.-, they contain also derivatives, as dy/dx or dr/d and d^r/dfy 1 , of one of the variables with respect to the other. An equation which contains * Many problems in geometry, mechanics, and physics are stated in terms of varia- tion. For purposes of analysis the statement x varies as ?/, or x x y, is written as ./• — ky, introducing a constant k called a factor of proportionality to convert the variation into an equation, in like manner the statement x varies inversely as >/. or x x \/y, becomes x — k/y, and x varies jointly with y and z becomes .<• = kyz. 17«> 180 DIFFERENTIAL EQUATIONS derivatives is called a differential equation. The order of the differential equation is the order of the highest derivative it contains. The equa- tions above are respectively of the first and second orders. A differen- tial equation of the first order may be symbolized as <£(.*', ij, if) = 0, and one of the second order as <&(.'•, y, ;/', y") = 0. A function y =f(x) given explicitly or defined implicitly by the relation F(x,y)=0 is said to be a solution of a given differential equation if the equation is true for all values of the independent variable x when the expressions for y and its derivatives are substituted in the equation. Thus to show that (no matter what the value of a is) the relation 4 ay — x- + 2 a- log x = gives a solution of the differential equation of the second order \dx) \dx-/ it is merely necessary to form the derivatives 2 a — = x , 2 a — = 1-1 dx x dx" x- and substitute them in the given equation together with y to see that 1 + - .''- — - = 1 + x- — 2 a 2 + — ) - — ( 1 + + — ) = \dxj \dx-l iiA x-J ±u-\ x- x*/ is clearly satisfied for all values of x. It appears therefore that the given relation for ij is a solution of the given equation. To integrate or salve a differential equation is to find all the functions which satisfy the equation. Geometrically speaking, it is to find all the curves which have the property expressed by the equation. In mechan- ics it is to find all possible motions arising from the given forces. The method of integrating or solving a differential equation depends largely upon the Ingenuity of the solver. In many cases, however, some method is immediately obvious. For instance if it be possible to separate the variables, so that the differential tly is multiplied by a function of // alone and dx by a function of ./■ alone, as in the equation {i/)di/ — \f/{.r)d.r, then J (j> ("//) dy = / \\i (./•) d.l' -f C (1) will clearly be the integral or solution of the differential equation. As an example, let. the curves of constant subnormal be determined. Here ijdy — mdx and ij- r- 2 mx -f C. The variables are already separated and the integration is immediate. The curves are parabolas with semi-latus rectum equal to the constant and with the axis GENERAL INTRODUCTION 181 coincident with the axis of x. If in particular it were desired to determine that curve whose subnormal was m and which passed through the origin, it would merely be necessary to substitute (0, 0) in the equation y- = 2mx + C to ascertain what particular value must be assigned to C in order that the curve pass through (0. 0). The value is C = 0. Another example might be to determine the curves for which the x-intercept varies as the abscissa of the point of tangency. As the expression (§ 7) for the x-intercept is x — ydx/dy, the statement is dx dx s — ii — = kx or (1 — k) x = y — dy dy Hence (1 — k) — = — '■ and (1 — k) log y = log x + G. •J x If desired, this expression may be changed to another form by using each side of the equality as an exponent with the base e. Then e (l-k)logy — e logx + C or yl-k = e C x — Q'x. As C'is an arbitrary constant, the constant C" = e c 'is also arbitrary and the solution may simply be written as y l ~ ,: = Cx, where the accent has been omitted from the constant. If it were desired to pick out that particular curve which passed through the point (1, 1). it would merely be necessary to determine C from the equation l 1 -* =6*1, and hence C = 1. As a third example let the curves whose tangent is constant and equal to a be determined. The length of the tangent is yVl + .'/'-///' and hence the equation is Vl + /- 1 + u'- Va 2 - y 2 , y = a or (j- — - — = a or 1 = y y' u"- U The variables are therefore separable and the results are Va 2 — y- ■.,., , ft + V« 2 — y 2 dx — — — '/// and x + ( = Va- — y- — (//') or f(y) (>/'), (i+Jpi =/ (,)* W or (l+j££= mH!l% (3) the variables x and y' or y and y' are immediately separable, and an integration may be performed. This will lead to an equation of the first order ; and if the variables are again separable, the solution may be completed by the methods of the above examples. In the first place consider curves whose radius of curvature is constant. Then 3 (1 + ij'-)± ^ _i,./^, .'• a 3 rt . x V3\ (*?) V — V = •<"-, y = L^' + e 2 •< I C l cos -— — + C, sin — — I — x-. 2. Determine the curves which have the following properties: (a) The subtangent is constant ; y m = CV. If through (2, 2). y>" - 2"'e x - 2 . (f3) The right triangle formed by the tangent, subtangent, and ordinate has the constant area fc/2 ; the hyperbolas xy + Cy + k = 0. Show that if the curve passes through (1, 2) and (2. 1). the arbitrary constant C is and the given k is — 2. (7) The normal is constant in length ; the circles (x — (')- + y 1 = k' 2 . (5) The normal varies as the square of the ordinate ; catenaries ky = cosh k(x— C). If in particular the curve is perpendicular to the y-axis, C = 0. (e ) The area of the right triangle formed by the tangent, normal, and ./'-axis is inversely proportional to the slope ; the circles (x — C) 2 + y' 2 = /. - . 184 DIFFERENTIAL EQUATIONS 3. Determine the curves which have the following properties: (cc) The angle between the radius vector and tangent is constant ; spirals r — CV*. (/3) The angle between the radius vector and tangent is half that between the radius and initial line ; cardioids ;• = C (1 — cos 0). (7) The perpendicular from the pole to a tangent is constant ; r cos (<£ — (!) = k. (5) The tangent is equally inclined to the radius vector and to the initial line ; the two sets of parabolas r = C/(l ± cos ). (e) The radius is equally inclined to the normal and to the initial line ; circles r = C cos or lines r cos

A.s = ; and if they now be divided by Ax and if Ax be allowed to approach zero, the result is the two equations of equilibrium where cos t and sin t are replaced by their values dx/ds and dy/ds. If the string is acted on only by forces parallel to a given direction, let the //-axis be taken as parallel to that direction. Then the component X will be zero and the first equation may be integrated. The result is ds * or C Vl + y'' 1 + pY = 0. (4') T+AT If this equation can be integrated, the form of the curve of equilibrium may be found. Another problem of a different nature in strings is to consider the variation of the tension in a rope wound around a cylinder without overlapping. The forces acting on the element An of the rope are the tensions T and T + AT, the normal pressure or reaction 7' of the cylinder, and the force of friction which is proportional to the pressure. It will be assumed that the normal reaction lies in the angle A(p and that the coefficient of friction is fj. so that the force of friction is fili. The components along the radius and along the tangent are 186 DIFFERENTIAL EQUATIONS (T + AT) sin A

) - fiR sin (0A) = 0, < < 1, (T + AT) cos A<£ + A' sin (0A or r=T eM>, where T is the tension when

( i+ £ r )i k \ ntg As a second example consider the motion of a particle vibrating up and down at the end of an elastic string held in the held of crravitv. Bv Hooke's Law for 188 DIFFERENTIAL EQUATIONS elastic strings the force exerted by the string is proportional to the extension of the string over its natural length, that is, F = kAl. Let I be the length of the string, A t) l the extension of the string just sufficient to hold the weight W= mg at rest so that kA l = mg, and let ./.• measured downward be the additional extension of the string at any instant of the motion. The force of gravity mg is positive and the force of elasticity — k (A,/ + x) is negative. The second form of the equation of motion is to be chosen. Hence do , , . , , dv , , , , my — = mg — k(AJ + x) or mo — = — kx, since mg — kAJ. clx dx Then mvdv = — kxdx or mv 2 = — kx- + C. Suppose that x = a is the amplitude of the motion, so that when x = a the velocity v = and the particle stops and starts back. Then C = ka 2 . Hence dx k /—, , dx k 7J v = — = \ V a 2 — x- or — = a - dt, dt \ m ~\/a 2 — x 2 * m and i- 1 - = -\—t+C or x = asin( \— 1+ Cj- a \m \ \ m / Now let the time be measured from the instant when the particle passes through the position x = 0. Then C satisfies the equation = a sin C and may be taken as zero. The motion is therefore given by the equation x — n sin \ k/mt and is periodic. While t changes by 2 ir Vm/k the particle completes an entire oscilla- tion. The time T=2iry, m/k is called the periodic time. The motion considered in this example is characterized by the fact that the total force — kx is propor- tional to the displacement from a certain origin and is directed toward the origin. Motion of this sort is called simple harmonic motion (briefly S. II. M.) and is of great importance in mechanics and physics. EXERCISES 1. The sum of S100 is put at interest at 4 per cent per annum under the condition that the interest shall be compounded at each instant. Show that the sum will amount to S200 in 17 yr. 4 ino., and to 81000 in 57' yr. 2. (liven that the rate of decomposition of an amount ./• of a given substance is proportional to the amount of the substance remaining undecomposed. Solve the problem of the decomposition and determine the constant of integration and the physical constant of proportionality if x = 5.11 when t = and ,/• = 1.48 when t = 40 min. Ans. k = .0309. 3. A substance is undergoing transformation into another at a rate which is assumed to be proportional to the amount of the substance still remaining untrans- formed. If that amount is 35.0 when t = 1 hr. and 13.8 when / = 4 lir.. determine the amount at the start when t = and the constant of proportionality and rind how many hours will elapse before only one-thousandth of the original amount will remain. 4. If the activity A of a radioactive deposit is proportional to its rate of diminution and is found to decrease to J its initial value in 4 days, show that A satisfies the equation -1/-I,, = c -0 - 173 '. GENERAL INTRODUCTION 189 5. Suppose that amounts a and b respectively of two substances are involved in a reaction in which the velocity of transformation dx/dt is proportional to the prod- uct (a — x) (b — x) of the amounts remaining un transformed. Integrate on the supposition that a ^ b. t I a — x I b — x log ] ^~J) _ („ _ y kt . and jf ~^ 0.4800 ; 0.2342 " ( 6 ~~ ^ 1205 1 0.3870 j 0.1354 determine the product A: (a — b). 6. Integrate the equation of Ex. 5 if a = b, and determine a and fc if x = 9.87 when t — 15 and x = 13.00 when t = 55. 7. If the velocity of a chemical reaction in which three substances are involved is proportional to the continued product of the amounts of the substances remaining, show that the equation between x and the time is / a \'>- c / b \ c ~"/ c \"-b lou (x = = — kt, where ■{ . ( a _6)(6_ c)(c- a) [« = 0. 8. Solve Ex. 7 if a = b ^ c ; also when a = b = c. Note the very different forms of the solution in the three cases. 9. The rate at which water runs out of a tank through a small pipe issuing horizontally near the bottom of the tank is proportional to the square root of the height of the surface of the water above the pipe. If the tank is cylindrical and half empties in 30 niin.. show that it will completely empty in about 100 min. 10. Discuss Ex. 9 in case the tank were a right cone or frustum of a cone. 11. Consider a vertical column of air and assume that the pressure at any level is due to the weight of the air above. Show that p =p e- kl ' gives the pressure at any height /;. if Boyle's Law that the density of a gas varies as the pressure be used. 12. Work Ex. 11 under the assumption that the adiabatic law pxp 1 - 4 repre- sents the conditions in the atmosphere. Show that in this case the pressure would become zero at a finite height. (If the proper numerical data are inserted, the height turns out to be about 20 miles. The adiabatic law seems to correspond better to the facts than Boyle's Law.) 13. Let I be the natural length of an elastic string, let Al be the extension, and assume Hooke's Law that the force is proportional to the extension in the form A/ = klF. Let the string be held in a vertical position so as to elongate under its own weight IT. Show that the elongation is ll'H'L 14. The density of water under a pressure of p atmospheres is p = 1 -f 0.00004 p. Show that the surface of an ocean six miles deep is about 000 ft. below the position it would have if water were incompressible. 15. Show that the equations of the curve of equilibrium of a string or chain are s( r D*' B =* £( r £) + '- in polar coordinates, where 11 and are the components of the force along the radius vector and perpendicular to it. 190 DIFFERENTIAL EQUATIONS 16. Show that dT + pSds = and T + pHX = are the equations of equilib- rium of a string if 11 is the radius of curvature and S and N are the tangential and norma] components of the forces. 17.* Show that when a uniform chain is supported at two points and hangs down between the points under its own weight, the curve of equilibrium is the catenary. 18. Suppose the mass dm of the element ds of a chain is proportional to the pro- jection dx of ds on the x-axis, and that the chain hangs in the held of gravity. Show that the curve is a parabola. (This is essentially the problem of the shape of the cables in a suspension bridge when the roadbed is of uniform linear density ; for the weight of the cables is negligible compared to that of the roadbed.) 19. It is desired to string upon a cord a great many uniform heavy rods of varying lengths so that when the cord is hung up with the rods dangling from it the rods will be equally spaced along the horizontal and have their lower ends on the same level. Required the shape the cord will take. (It should be noted that the shape must be known before the rods can be cut in the proper lengths to hang as desired.) The weight of the cord may be neglected. 20. A masonry arch carries a horizontal roadbed. On the assumption that the material between the arch and the roadbed is of uniform density and that each element of the arch supports the weight of the material above it, find the shape of the arch. 21. In equations (4') the integration may be carried through in terms of quadra- tures if pY is a function of //alone : and similarly in Ex. 15 the integration may be carried through if = and pR is a function of r alone so that the held is central. Show that the results of tints carrying through the integration are the formulas Cdu r < 'dr/r x + C /<

  • J ^(fpRdrf-C* 22. A particle falls from rest through the air. which is assumed to offer a resist- ance proportional to the velocity. Solve the problem with the initial conditions v = 0, x — 0. t = 0. Show that as the particle falls, the velocity does not increase indefinitely, but approaches a definite limit V = mg/k. 23. Solve Ex. 22 with the initial conditions v = v , x = 0. t = 0, where v is greater than the limiting velocity 1'. Show that the particle slows down as it falls. 24. A particle rises through the air. which is assumed to resist proportionally to the square of the velocity. Solve the motion. 25. Solve the problem analogous to Ex. 24 for a falling particle. Show that there is a limiting velocity V = \ mg/k. If the particle were projected down with an initial velocity greater than T. it would slow down as in Ex. 23. 26. A particle falls towards a point which attracts it inversely as the square of the distance and directly as its mass. Find the relation between x and t and determine the total time T taken to reach the center. Initial conditions v = 0, X = , kx = mV{l — e '" ), mV/k. 28. Solve Ex. 27 under the assumption that the resistance varies as vc. 29. A particle falls toward a point which attracts inversely as the cube of the distance and directly as the mass. The initial conditions are x = a, v = 0, t = 0. Show that x 2 = a- — kt-/a- and the total time of descent is T = tarts from rest at the top of the cycloid. 33. Two equal weights are hanging at the end of an elastic strinu'. ( die drops off. Determine completely the motion of the particle remaining. 34. One end of an elastic spring (such as is used in a spring balance) is attached rigidly to a point on a horizontal table. To the other end a particle is attached. If the particle be held at such a point that the spring is elongated by the amount a and then released, determine the motion on the assumption that the coefficient of friction between the particle and the table is /x ; and discuss the possibility of different cases according as the force of friction is small or large relative to the force exerted by the spring. 85. Lineal element and differential equation. The idea of a curve as made up of the points upon it is familiar. Points, however, have no extension and therefore must he regarded not as pieces of a curve hut merely as positions on it. Strictly speaking, the pieces of a curve are the elements An of are; hut for many purposes it is convenient to re- place the complicated element Ax by a piece of the tangent to the curve at some point of the arc As, and from this point of view a curve is made up of an infinite number of infinitesimal elements tangent to it. This is analogous to the point of view by which a curve is regarded as made 192 DIFFERENTIAL EQUATIONS up of an infinite number of infinitesimal chords and is intimately related to the conception of the curve as the envelope of its tangents (§ 65). A point on a curve taken with an infinitesimal portion of the tangent to the curve at that point is called a lineal element of the curve. These concepts and definitions are clearly equally available in two or three dimensions. For the present the curves under dis- cussion will be plane curves and the lineal elements will therefore all lie in a plane. ^ Axyn) To specify any particular lineal element three coordinates x, //, p will be used, of which the two (x, //) determine the point through which the element passes and of which the third p is the slope of the element. If a curve f(x, y) — is given, the slope at any point may be found by differentiation, dy of J cf P = ~T = ~ ~ / ~ ' ( b ) ax ox I cij K J and hence the third coordinate p of the lineal elements of this particular curve is expressed in terms of the other two. If in place of one curve f(x, y) — the whole family of curves f(x, y) = C, where C is an arbitrary constant, had been given, the slope p would still be found from (6), and it therefore appears that the third coordinate of the lineal elements of such a family of curves is expressible in terms of x and //. In the more general case where the family of curves is given in the unsolved form F(x, y, C) = 0, the slope p is found by the same formula but it now depends apparently on C in addition to on x and y. If, how- ever, the constant C be eliminated from the two equations F(*> Ih C) = ° and ■£ + j~ p = 0, (7) there will arise an equation $(./*, y, p) = which connects the slope /> of any curve of the family with the coordinates (./■, ■>/) of any point through which a curve of the family passes and at which the slope of that curve is p. Hence it appears that the three coordinates ('•'', //•/') of the lineal (dements of all the curves of a family are connected by an equa- tion <&(.'•, //. p) = 0, just as the coordinates (x, //. z\ of the points of a surface are connected by an equation &(x, //, ,*:) = 0. As the equation (.r, y, ,-.) = () is called the equation of the surface, so the equation (&(./•, //, ji) = is called the equation of the family of curves : it is, how- ever, not the finite equation F(x, >/, C) = but the differential equation of the family, because it involves the derivative p = ily/dx of y by x instead of the parameter C. GENERAL INTRODUCTION 193 As an example of the elimination (if a constant, consider the case of the parabolas ij- = Cx or //-/./• = C. The differentiation of the equation in the second form gives at once — y' 2 /x 2 + 2 yp/x = or y — 2 xp as the differential equation of the family. In the unsolved form the work is - UP = c, y- = '*■ >jpx, y = 2xp. The result is, of course, the same in either case. Tor the family here treated it makes little difference which method is followed. As a general rule it is perhaps best to solve for the constant if the solution is simple and leads to a simple form of the function /(j, y) ; whereas if the solution is not simple or the form of the function is complicated, it is best to differentiate first because the differentiated equation may be simpler to solve for the constant than the original equation, or because the elimination of the constant between the two equations can be con- ducted advantageously. If an equation <1> (>■, //. y) = connecting the three coordinates of the lineal element be given, the elements which satisfy the equation may be plotted much as a surface is plotted; that is, a pair of values (V, ij) may be assumed and substituted in the equation, the equation may then be solved for one or more values of p, and lineal elements with these values of y/ may be drawn through the point (./', //). In this manner the elements through as many points as desired may be found. The de- tached elements are of interest and significance chiefly from the fact that they can be assembled into curves, — in fact, into the curves of a family F(x, y, C) = of which the equation <&(./•,//, p) = is the differ- ential equation. This is the converse of the problem treated above and requires the integration of the differential equation <£(./•, //, //) = for its solution. In some simple cases the assembling may be accomplished intuitively from the geometric properties implied in the equation, in other cases it follows from the integration of the equation by analytic means, in other cases it can be done only approximately and by methods of computation. As an example of intuitively assembling the lineal elements into curves, take , . , , V f 2 — V" (x. >/. p) = y'-p- + y- — r- = or p = ± — " y The quantity Vr- — //- may be interpreted as one leg of a right triangle of which y is the other leu' and r the hypotenuse. The slope of the hypotenuse is then ± y/vr- — //- according to the position of the figure, and the differential equation * (x. ij. p) = states that the coordinate p of the lineal element which satisfies it is the negative reciprocal of this slope. Hence the lineal element is perpendicular io the hypotenuse. It therefore appears that the lineal elements are tangent to cir- cles of radius r described about points of the ./--axis. The equation of these circles is 194 DIFFERENTIAL EQUATIONS (,r — (')- + y 2 — r 2 . and this is therefore the integral of the differential equation. The correctness of this integral may be checked bv direct integration. For dy Vr 2 — y' 2 ydy p = — = ± or — — = dx or v /- — //- = x — C. dx y yV 2 - if 86. In geometric problems which relate the slope of the tangent of a curve to other lines in the figure, it is clear that not the tangent but the lineal element is the vital thing. Among such problems that of the orthogonal trajectories (or trajectories under any angle) of a given family of curves is of especial importance. If two families of curves are so related that the angle at which any curve of one of the families cuts any curve of the other family is a right angle, then the curves of either family are said to be the orthogonal trajectories of the curves of the other family. Hence at any point (./•, y) at which two curves belonging to the different families intersect, there are two lineal elements, one belonging to each curve, which are perpendicular. As the slopes of two perpendicular lines are the negative reciprocals of each other, it follows that if the coordinates of one lineal element are (./■, y,p) the coordinates of the other are (./•, >/, — l//i) ; and if the coordinates of the lineal ele- ment (./■, y, p) satisfy the equation <£(•'•, y, p) = 0, the coordinates of the orthogonal lineal element must satisfy <& (x, y, — 1/p) = 0. Therefore the rule for finding the orthogonal trajectories of the curves F(x, y, C)= is to find first the differential foliation <£(.'', y, />) = of the family, then to replace p hy — 1/p to find the differential equation of the orthogonal family, and finally to integrate this equation to find the family. It may be noted that if /•'(-■) = X (x, y) 4- ' F(>. //) is a function of z = x 4- iy (§ 73), the families X(,r, y) = C and Y(x, y) = K are orthogonal. As a problem in orthogonal trajectories find the trajectories of the semicubical parabolas (x — Cy' 1 = //-. The differential equation of this family is found as 3 {x - C) 2 = 2 i/p. x - C = (| yp)$, (J yp)% - y 2 or \p = y*. This is the differential equation of the given family. Replace p by — \/p and integrate : 2 i 3 i 3 i 4 — - - = i/s or 1 4- - -))i/3 = or fix + - irf dy = 0. and x + us = C. Sp ^2 2 8 Thus the differential equation and finite equation of the orthogonal family are found. The curves look something like parabolas with axis horizontal and vertex toward the right. Given a differential equation &(x, //. p) = Q or, in solved form, j, = <£ (./'. i/); the lineal element affords a means for ohtaining graphically and numerically an approximation to the solution udiidi passes through GENERAL INTRODUCTION 195 a "ssigned point PJr Q , l/J- For the value j/ Q of p at this point may be (•( puted from the equation and a lineal element PJ\ may be drawn, the length being taken small. As the lineal element is tangent to the curve, its end point will not lie upon the curve but will depart from it by an infinitesimal of higher order. Next the slope p of the lineal element which satisfies the equation and passes through I\ may be found and the element P,P, ^3^^^4. O 1 12 may be drawn. This element will not be tangent to the desired solution but to a solution lying near that one. Next the element P. 2 P 3 may be drawn, and so on. The broken line P P P P • • • is clearly an approximation to the solution and will be a better approximation the shorter the elements P(P i+ i are taken. If the radius of curvature of the solution at P Q is not great, the curve will be bending rapidly and the elements must be taken fairly short in order to get a fair approx- imation ; but if the radius of curvature is great, the elements need not be taken so small. (This method of approximate graphical solution indicates a method which is of value in proving by the method of limits that the equation p = <£(.'•, y) actually has a solution ; but that matter will not be treated here.) Let it be required to plot approximately that solution of yp + x = which passes through (0, 1) and thus to find the ordinate for x = 0.5. and the area under the curve and the length of the curve to this point. Instead of assuming the lengths of the successive lineal elements, let the lengths of successive increments 5/ of x be taken as 5x = 0.1. At the start -Co = °- Vo ~ !> and from p - — x /y [t follows that p Q = 0. The increment Sy of // acquired in moving along the tan- gent is 8y = pSx = 0. Hence the new- point of departure (./',. ?/ 1 ) is (0.1, 1) and the new slope is p l = — •i\/'j l — — 0.1. The results of the work, as it is contin- ued, may be grouped in the table. Hence it appears that the final ordinate is y — 0.00. By adding up the trapezoids the area is computed as 0.48. and by find- ing the elements 5s = ~v8x~ + Sy' 2 the length is found as 0.51. Now the particular equation here treated can be integrated. yp + x = 0, ydy + xdx = 0. x- + y- = < '. and hence x' 1 + y" = 1 is the solution which passes through (0. 1). The ordinate, area, and length found from the curve are therefore' 0.87. 0.48. 0.52 respectively. The errors in the approximate results to two places are therefore respectively 3, 0. 2 percent. If 5x had been chosen as 0.01 and four places had been kept in the computations, the errors would have been smaller. 1 ( ox 5// *i llr Pi 0. 1.00 0. 1 0.1 0. 0.1 1.00 -0.1 2 0.1 - 0.01 0.2 0.00 -0.2 3 0.1 - 0.02 0.3 0.97 - 0.31 4 0.1 - 0.03 0.4 0.04 - 0.43 5 0.1 - 0.04 0.5 0.00 '" 196 DIFFERENTIAL EQUATIONS EXERCISES 1. In the following cases eliminate the constant C to find the differential equa- tion of the family given : (a) x 2 = 2Cy+ C' 2 , (fi) y = Cx + Vl - C 2 , (7) x- — y 2 = Cx, (8) y = x tan (x + C), (e) _*_ + -*- = 1, An,. W + ^^!H^)^. 1= 0. a 2 — 6' 6 2 — 6' W/ xij dx 2. Plot the lineal elements and intuitively assemble them into the solution : (a) yp + x = 0, (/3) xp - y/ = 0, (7) r -^ = 1. Check the results by direct integration of the differential equations. 3. Lines drawn from the points (± c, 0) to the lineal element are equally in- clined to it. Show that the differential equation is that of Ex. 1 (e). What are the curves ? 4. The trapezoidal area under the lineal element equals the sectorial area formed by joining the origin to the extremities of the element (disregarding infinitesimals of higher order), (a) Find the differential equation and integrate, (p) Solve the same problem where the areas are equal in magnitude but opposite in sign. What are the curves ? 5. Find the orthogonal trajectories of the following families. Sketch the curves. (a) parabolas y 2 = 2 Cx, Ann. ellipses 2x 2 + if- = C. (P) exponentials y = Ce lx , Ans. parabolas l ky' 2 + x = (.'. (7) circles (x — C)' 2 + y 2 — a' 2 , Ans. tractrices. (5) x 2 - y 2 = C 2 , (e) Ci/ 2 = X s , (f) A + IP = Ct. 6. Show from the answer to Kx. 1 (e) that the family is self-orthogonal and illustrate with a sketch. From the fact that the lineal element of a parabola makes equal angles with the axis and with the line drawn to the focus, derive the differ- ential equation of all coaxial confocal parabolas and show that the family is self- orthogonal. 7. If <£ (x, y, p) = is the differential equation of a family, show / p — m \ I p + m \ ( x, y, - — — ) = and $ ( x, y, 1 = \ 1 + mp) \ 1 - mpj are the differential equations of the family whose curves cut those of the given family at tan -1 m. What is the difference between these two cases '.' 8. Show that the differential equations •(£•'•♦)=< ' •(-'£"*)= o define orthogonal families in polar coordinates, and write the equation of the family which cuts the first of these at the constant angle tan- 1 //*. 9. Find the orthogonal trajectories of the following families. Sketch. (a) r = <■<'•'■'. (p) r = C'(l - cos0), (7) r = (70, (5) /•'-' = 6'- cos 2 0. GENERAL INTRODUCTION 197 10. Kecompute the approximate solution of yp + x = under the conditions of the text hut with 5/ = 0.05, and carry the work to three decimals. 11. Plot the approximate solution of p = xy between (1, 1) and the ?/-axis. Take 5x = — 0.2. Pind the ordinate, area, and length. Check by integration and comparison. 12. Plot the approximate solution of p = — x through (1, 1). taking 8x = 0.1 and following the curve to its intersection with the x-axis. Pind also the area and the length. 13. Plot the solution of p = Vx' 2 + y- from the point (0, 1) to its intersection with the x-axis. Take 8x = — 0.2 and find the area and length. 14. Plot the solution of p = s which starts from the origin into the first quad- rant (s is the length of the arc). Take 8x = 0.1 and carry the work for five steps to find the final ordinate, the area, and the length. Compare with the true integral. 87. The higher derivatives ; analytic approximations. Although a differential equation <£(.'', //, //') = does not determine the relation between x and y without the a] (plication of some process equivalent to integration, it does afford a means of computing the higher derivatives simply by differentiation. Thus Hi .'/') = 0. A further differentiation gives the equation (/-<£> f-* r~ r~ c~<$> c~<& , = 5 + l> . i/ + 2 -/---- f + A >r- + 2 ^ v y ax- c.i- excy ' cxCij cy- ' Cl J c !l c~<$> c c<£> dy» J cy J cy' J which may be solved for y'" in terms of x, y, //, y"; and hence, by the preceding results, y'" is expressible as a function of x and y ; and so on to all the higher derivatives. In this way any property of the inte- grals of <&(./', y. y') = which, like the radius of curvature, is expressi- ble in terms of the derivatives, may be found as a function of x and y. As the differential equation <£(.'', y, //') = defines y' and all the higher derivatives as functions of x, //. it is clear that the values of the derivatives may be found as y' , //,", //,", ••■ at any given point (x ., y). Hence it is possible to write the series y = .'/„ + y'o (x - ,•„) + 1 „:; (x - x o y + y ,/;; (x - x o y + ■■■. (S) If this power series in x — x converges, it defines y as a function of x for values of x near x : it is indeed the Taylor development of the 198 DIFFERENTIAL EQUATIONS function y (§ 167). The convergence is assumed. Then y' = y'* + !/'■>' - - r o) + h !h (■'• - ' r )' 2 h — • It may be shown that the function y denned by the series actually satisfies the differential equation <£(.'', y, y') = 0, that is, that Q(x) =' ^ = 0.110. The series for y to four terms is therefore y - 1 + 0.402 (./• - 2) - 0.0070 (x - 2)- + 0.018 (x - 2) 3 . It may be noted that it is generally simpler not to express the higher derivatives in terms of x and //. but to compute each one successively from the preceding ones. 88. Picard has given a method for the integration of the equation //' = (.'', y) by sifcrrsstrt' fijtpro.runtttinnx which, although of the highest theoretic value and importance, is not particularly suitable to analytic GENERAL INTRODUCTION 199 uses in finding an approximate solution. The method is this. Let the equation y' = cj>(x, y) be given in solved form, and suppose (./'., y () ) is the point through which the solution is to pass. To find the first approximation let y be held constant and equal to y , and integrate the equation y' = (x, //„). Thus dy = (x, y ) dx ; y = y Q + J cf> (x, y Q ) dx = /](,,■), (9) where it will lie noticed that the constant of integration has been chosen so that the curve passes through (,t , //j. For the second approximation let y have the value just found, substitute this in (f>(x, y), and integrate again. Then V = .% + / 4> ■'-, .'/ + j <$> (*, %) dx -m- (9 f ) With this new value for y continue as before. The successive deter- minations of y as a function of ./■ actually converge toward a limiting function which is a solution of the equation and which passes through (./• Q , ?/ ). It may be noted that at each step of the work an integration is required. The difficulty of actually performing this integration in formal practice limits the usefulness of the method in such cases. It is clear, however, that with an integrating machine such as the integraph the method could be applied as rapidly as the curves c£ (x, /](■>')) could be plotted. To see how the method works, consider the integration of if — x -f y to find the integral through (1, 1). For the first approximation y = 1. Then dy - (x + I) dx, y = I x- + .;• + ' ', y - \ .'■- + X— I -f x {x). From this value of y the next approximation may be found, and then still another : dy = [x+ (I x 2 + x - ])] dx, y = Jx 3 + x 2 - \x + l ; = f,(x), dy = [x -f / 2 (x)] dx, 2/ = 2? jA + -J *'" + i x ' + 1' + *V In this ease there are no difficulties which would prevent any number of appli- cations of the method. In fact it is evident that if //' is a polynomial in x and y, the result of any number of applications of the method will be a polynomial in ./:. The method of undetermined coefficient* may often be employed to advantage to develop the solution of a differential equation into a series. The result is of course identical with that obtained by the application of successive differentiation and Taylor's series as above ; the work is sometimes shorter. Let the equation be in the form y' = (/>(.'■. //) and assume an integral in the form y = i/o + «i ( •'• - * ) + a , (•'• - ^o) 2 + «■ (•'' - O 8 + ■■■ ( 10 ) 200 DIFFERENTIAL EQUATIONS Then <£ (x, y) may also be expanded into a series, say, (•'•, y) = A o + M x - a 'o) + A -2 ( x - *oY + A z ( x - * ) 3 + — • But by differentiating the assumed form for y we have ,/ = a x + 2 % (x - x ) + 3 a s (x - x f + 4 % (x - ,r ) 3 + . . . . Thus there arise two different expressions as series in x — x for the function y\ and therefore the corresponding coefficients must be equal. The resulting set of equations a 1 = A Q , 2a z = A v 3a. 6 = A. 2 , 4a 4 = ^ 3 , ••• (11) may be solved successively for the undetermined coefficients a , #.„ a , a , ■ ■ • which enter into the assumed expansion. This method is partic- ularly useful when the form of the differential equation is such that some of the terms may be omitted from the assumed expansion (see Ex. 14). As an example in the use of undetermined coefficients consider that solution of the equation y' = Vx' 2 + Sy' 2 which passes through (1, 1). The expansion will pro- ceed according to powers of x — 1, and for convenience the variable may be changed to t = x — 1 so that dy V{t + 1)'- + 3 y\ y = 1 + a x t + a 2 i 2 + a s t s + a 4 t* + ■■■ are the equation and the assumed expansion. One expression for y' is y' = a l + 2 a J, + 3 a 3 fi + 4 rq£ 3 + • • • . To find the other it is necessary to expand into a series in t the expression y' = V(l + if + 3(1 + "0 :] GENERAL INTRODUCTION 201 Hence the successive equations for determining the coefficients are cq = 2 and 2 a.-, = I (1 + 3 «j) or a 2 = J, 3 a 3 = £ (1 + 6 a 2 + 3 af) - T ^ (1 + 3 oq) 2 or a 3 = }£, 4 «4 = 1 («i a 2 + « 3 ) ~ is (1 + 3 (x, y, //) and its derivatives J ox cy J dy' J y* = pl + 2pty' + 2^,y" + ^y>* + 2j%- l yy ex- gxcij excy' cy C 'J C U cy'- J dy J cy' * Evidently the higher derivatives of y may be obtained in terms of x, y, y' ; and y itself may be written in the expanded form y = v + ti(x - x ) + * >/;;(..- - x o y + > y ;> - * ) 3 (m where any desired values may be attributed to the ordinate y at which the curve cuts the line x = a* , and to the slope y of the curve at that point. Moreover the coefficients //,', y'^', ■ ■ ■ are determined in such a way that they depend on the assumed values of y n and y . It therefore is seen that the solution (12) of the differential equation of the second order really involves two arbitrary constants, and the justification of writing it as F(x, y, <' v ('.,) = is clear. In following out the method of undetermined coefficients a solution of the equation would be assumed in the form y = y + y'«(x - *o) + a -2 (■'• - -'\f + " 3 (■'• - 'o) 3 + «4 (■'' - ^r + • ■ ■ i (13) from which y' and //" would be obtained by differentiation. Then if the series for y and //' be substituted in y" = (x, y, y 1 ) and the result arranged as a series, a second expression for y" is obtained and the comparison of the coefficients in the two series will afford a set of equa- tions from which the successive coefficients may be found in terms of y and y by solution. These results may clearly be generalized to the case of differential equations of the nth. order, whereof the solutions will depend on n arbitrary constants, namely, the values assumed for ii and its first n — 1 derivatives when x — x . 202 DIFFERENTIAL EQUATIONS EXERCISES 1. Find the radii and circles of curvature of the solutions of the following equa- tions at the points indicated : (a) y' = Vx- + y* at (0, 1), (8) yy' + x = at (x , y ). 2. Find y'H 1} = (5 V2 - 2)/4 if y' = Vx 2 + y 2 . 3. Given the equation y 2 y' 3 + xyy" 2 — yy' + x- = of the third degree in y' so that there will be three solutions with different slopes through any ordinary point (x, y). Find the radii of curvature of the three solutions through (0. 1). 4. Find three terms in the expansion of the solution of y' — <■"' about (2, l). 5. Find four terms in the expansion of the solution of y = logsinxy about (^ 7r, 1).. 6. Expand the solution of y' = xy about (1, y ) to five terms. 7. Expand the solution of y' = tan (y/x) about (1. 0) to four terms. Note that here x should be expanded in terms of y, not y in terms of x. 8. Expand two of the solutions of y 2 y' 3 + xyy" 2 — yy' + x 2 = about (— 2, 1) to four terms. 9. < )btain four successive approximations to the integral of y' = xy through (1, 1). 10. Find four successive approximations to the integral of y' = x + y through (0, y ). 11. Show by successive approximations that the integral of y' — y through (0, y ) is the well-known y = y e x . 12. Carry the approximations to the solution of y' = — x/y through (0, 1) as far as you can integrate, and plot each approximation on the same figure with the exact integral. 13. Find by the method of undetermined coefficients the number of terms indi- cated in the expansions of the solutions of these differential equations about the points given : (a) y' — vx + y, five terms, (0, 1), (3) y' = Vx + y, four terms, (1, 3), (7) V' = X + y. n terms, (0, //,,). (5) y' = xx- + y-. four terms, (f. i). 14. If the solution of an equation is to be expanded about (0. // ) and if the change of x into — x and y' into — y' does not alter the equation, the solution is necessarily symmetric with respect to the y-axis and the expansion may be assumed to contain only even powers of x. If the solution is to be expanded about (0. 0) and a change of x into — x and y into — y docs not alter the equation, the solution is symmetric with respect to the origin and the expansion may be assumed in odd powers. Obtain the expansions to four terms in the following cases and compare the labor involved in the method of undetermined coefficients with that which would be involved in performing the requisite six or seven differentiations for the application of Madauriifs series: (a) y' = — — about (0. 2), (p) y' = sin xy about (0. 1), x x- + //-' (7) y' = e-'." about (0, 0), (5) y' = x*y + xy-' about (0. 0). 15. Expand to and including the term x 4 : ((X) y" — //"- -f xy about x,, = 0. y u = n t) . y' tl — a x (by both methods). (3) xy" + y' + y = about x = 0. y = u , y' = - u (by mid. coeffs.). CHAPTER VIII THE COMMONER ORDINARY DIFFERENTIAL EQUATIONS 89. Integration by separating the variables. If a differential equa- tion of the first order may be solved for y' so that y' = (x, y) or M(x, >,) dx + N(x, y) dy = (1) (where the functions , M, X are single valued or where only one spe- cific branch of each function is selected in case the solution leads to multiple valued functions), the differential equation involves only the first power of the derivative and is said to be of the first degree. If, furthermore, it so happens that the functions <£. M. X are products of functions of x and functions of y so that the equation (1) takes the form y' = ^(x) Jy) or M^x) M 2 (y) dx + X^x) Xjy) dy = 0, (2) it is clear that the variables may be separated in the manner and the integration is then immediately performed by integrating each side of the equation. It was in this way that the numerous problems considered in Chap. VII were solved. As an example consider the equation yy' + xy- = x. Here ydy + x {y 2 - \)dx = or -^~- + xdx = 0, V- - 1 and \ log {y- — 1) + \ x- = C or (y- — 1) e* 2 = C. The second form of the solution is found by taking the exponential of both sides of the first form after multiplying by 2. In some differential equations (1) in which the variables are not immediately separable as above, the introduction of some change of variable, whether of the dependent or independent variable or both, may lead to a differential equation in which the new variables are sepa- rated and the integration may be accomplished. The selection of the proper change of variable is in general a matter for the exercise of ingenuity : succeeding paragraphs, however, will point out some special 203 204 DIFFERENTIAL EQUATIONS types of equations for which a definite type of substitution is known to accomplish the separation. As an example consider the equation xdy — ydx = x Vx 2 + y- dx, where th e varia- bles are clearly not separable without substitution. The presence of Vx 2 + y 1 suggests a change to polar coordinates. The work of finding the solution is : x = r cos 6, y = r sin 9, dx = cos 6dr — r sin Bdff, dy = sin 6dr + r cos 6d6 ; then xdy — ydx = r-dO, x Vx- + y- dx — r- cos 6d (r cos 6). Hence the differentia) equation may be written in the form r-d6 = r 2 cos 6d (r cos 6) or sec 9d0 = d (r cos #), and loc tan (i 6 + }tt) = r cos + C or log — — — = x + C. cos 6 Hence — = C'e' ; (on substitution for 6). x Another change of variable which works, is to let y = vx. Then the work is x (fcZx + xdv) — vxdx = x 2 ~v 1 + v 2 dx or dv = v 1 + v 2 dx. dv Then , = dx, sinh- 1 ?; = x + C, y = x sinh (x + C). V 1 + w 2 This solution turns out to be shorter and the answer appears in neater form than before obtained. The great difference of form that may arise in the answer when different methods of integration are employed, is a noteworthy fact, and renders a set of answers practically worthless ; two solvers may frequently waste more time in trying to get their answers reduced to a common form than each would spend in solving the problem in two ways. 90. If in the equation //' = 4>(- T , ,'/) the function <£ turns out to be tfi(j//x) } a function of y/x alone, that is, if the functions M and .V are homogeneous functions of x, y and of the same order (§ 53), the differ- ential equation is said to be homogeneous and the change of variable y = vx or x = ry will always result in separating the variables. The statement may be tabulated as : if *¥ = +(-)> substitute! V = VX (3) Ox r \xj [ or x = ry. A sort of corollary case is given in Ex. 6 below. As an example take 7/^1 + c")dx + c" (y — x)dy — 0, of which the homogeneity is perhaps somewhat disguised. Here it is better to choose x = ry. Then (1 + e v ) dx + e r (1 — v) dy = and dx = vdy + ydv. Hence (v + e v )dy + y(\ + C')dv = or — + du = 0. y v + e v Hence log y + log (u + e') = C or x + yc" — C. COMMONER ORDINARY EQUATIONS 205 If the differential equation may be arranged so that % + XJz) y = X 2 (x) r or ~ + Y^y) x = Y+jj) x», <4) where the second form differs from the first only through the inter- change of x and y and where A' and A'., are functions of ./■ alone and Y and Y functions of y, the equation is called a Bernoulli equation; and in particular if n = 0, so that the dependent variable does not occur on the right-hand side, the equation is called linear. The substitution winch separates the variables in the respective cases is y = V6 -S x iV> d * or x = ve-f T W*. (5) To show that the separation is really accomplished and to find a general formula for the solution of any Bernoulli or linear equation, the sub- stitution may be carried out formally. For dx r- I /- dv i y"- yx = — e" + vje- - i/^e- = — e- dy dy dy and dv lu- „ „ , dv , l'f- — e. =y 3 v-c'J' or — = ife- dy, dy t- 1 ] 5 >/= 1 - 1 -r- and =('/" — 2)e + C or - = 2 — ;/- + Ce . v x This result could have been obtained by direct substitution in the formula x i- » = (l _ n) e ( " - 1 '/ 3V/ " [ f Yj 1 - ">/ r '' / " dy] , but actually to carry the method through is far more instructive. * If 7i=l, the variables are separated in the original equation. 206 . DIFFEKENTIAL EQUATIONS EXERCISES 1. Solve the equations (variables immediately separable) : (a) (1 + x) y + (1 — y)xy' = 0, Ans. xy = Ce'J- (/3) a (xdy + 2 //dx) = xydy, (7) Vl — a; 2 d// + Vl — y 2 dx = 0, (5) (1 + 2/2) dx - 0/ + Vl + i/)(l + x)i # = 0. 2. By various ingenious changes of variable, solve : (a) (x + y)' 2 y' = a 2 , ^l?i,s. x + y = a tan (?//a + (7). (/3) (x - //-) dx + 2 22/dy = 0, (7) xdy - ?/dx = (x 2 + y 2 ) dx, (8) y' =x- y, (e) yy' + if- + x + 1 = 0. 3. Solve these homogeneous equations : (a) (2 Vxy — x) y' + y = 0, yln.s. Vx/2/ + log y = (7. (/3) xe* + ?/ — x// = 0, u4?is. y + x log log C/x = 0. (7) (^' 2 + 2/ 2 ) dy = xydx, (8) xy' — y = Vx 2 + y 2 . 4. Solve these Bernoulli or linear equations : (a) y' + y/x = y 2 , Aits, xy log Cx +1 = 0. (;3) //' — // esc x = cosx — 1, yln.s. // = sin x + C tan £-x. (7) xy' + 2/ = ?/ 2 logx, Ans. y~ l = logx + 1 + Cx. (8) (1 + y 2 ) dx = (tan- 1 y — x) dy, (e) ydx + (ax 2 y n — 2 x) d;/ = 0, (f ) x//' - ay = x + 1, (t?) J/?/' + I // 2 = cos x. 5. Show that the substitution y = vx always separates the variables in the homogeneous equation y' = (y/x) and derive the general formula for the integral. 6. Let a differential equation be reducible to the form dy 6 ( a i x + \v + c i\ o,'a, — a„ft] 7^ 0, dx \a 2 x + b 2 y + cj or a.fi 2 — a % b x — 0. Id case a x b 2 — a 2 &j ^ 0, the two lines a x x + b x y + c x = and a„x + 6 2 2/ 4- c 2 = will meet in a point. Show that a transformation to this point as origin makes the new equation homogeneous and hence soluble. In case a x b 2 — a b l = 0, the two lines are parallel and the substitution z = a.,x + b 2 y or z — a v t + b x y will separate the variables. 7. Hy the method of Ex. 6 solve the equations : (a) (3y — 7x + 7)dx + (7y — 3x + 3)dy = 0, Ans. (y — x + l) 2 (y + x — l) 5 = C. (P) (2x + 3y-b)y' + (3x + 2y-5)=0, (7) (4x + 3 y+ l)dz+ {x + y + l)dy = 0, dy J x — y — ] \2 x - 2 y + h (8) (2x + y) = ;/(4 x + 2y-l), (e) ^ = ( dx \i 8. Show that if the equation may be written as yf(xy)dx + xg(xy)dy = 0, where /and g are functions of the product xy, the substitution v = xy will sepa- rate the variables. 9. By virtue of Ex. 8 integrate the equations: (a) (y + 2xy 2 — x 2 y z )dx + 2x 2 ydy = 0, Ans. s + x-y = C(1 — .r//). (/3) (y + xy 2 ) dx + (x - x 2 i/) (/// = 0, (7) (1 + xy) xy 2 dx + (xy - 1 ) xdy = 0. COMMONER ORDINARY EQUATIONS 207 10. By any method that is applicable solve the following. If more than one method is applicable, state what methods, and any apparent reasons for choos- ing one : (a) y' + V cos x = y" sin 2x, (/3) (2 x-y + 3 y s ) dx = (x 3 + 2 xy 2 ) dy, (?) (4x + 2y - \)y' + 2x + y + 1 = 0, (5) y y' + xy 2 = x, (e) y'sin'y + since cosy = sin x, (f) Va' 2 + x' 2 (1 — y') = x + y, (v) («V + &y' 2 + W + I) y + (« 3 2/ 3 — xhj- — xy + 1) xy', (0) y' = sin (x — ?/), _ ?/ ( t ) xydy — ?/ 2 dx = (x + yf e •<• dx, (k) (1 — 2/ 2 ) dx = axy (x + 1) dy. 91. Integrating factors. If the equation Mdx + iVW?/ = by a suita- ble rearrangement of the terms can be put in the form of a sum of total differentials of certain functions u, v, ■ • ■ , say dzc + dv-\ = 0, then u + v -\ = C (7) is surely the solution of the equation. In this ease the equation is called an exact different Lai equation. It frequently happens that although the equation cannot itself be so arranged, yet the equation obtained from it by multiplying through with a certain factor fi(x, y) may be so arranged. The factor fi(x, y) is then called an integrating factor of the given equation. Thus in the case of variables separable, an integrating factor is 1/1/ ' N ; for — — \M3I„ dx + N,N dyl = ^^ dx -f- ^^ dy = ; (8) and the integration is immediate. Again, the linear equation may be treated by an integrating factor. Let dy + Xjjdx = X 2 dx and fi = ef x >' !r ; (9) then ef^ dx dy + X x el^ dx ydx = ef^ dx X 2 dx (10) d\jjel^ dx \ = e!^ dx X 2 dx, and yef x > dx = Cef^ dx X 2 da (11) In the case of variables separable the use of an integrating factor is therefore implied in the process of separating the variables. In the case of the linear equation the use of the integrating factor is somewhat shorter than the use of the substitution for separating the variables. In general it is not possible to hit upon an integrating factor by inspec- tion and not practicable to obtain an integrating factor by analysis, but the integration of an equation is so simple when the factor is known, and the equations which arise in practice so frequently do have simple integrating factors, that it is worth while to examine the equation to see if the factor cannot be determined by inspection and trial. To aid in the work, the differentials of the simpler functions such as 208 DIFFERENTIAL EQUATIONS dxy — xdy + ydx, \ d (./•- + //-) = xdx + y d tan : - = - — z ~ , (12) x x- y x z + f should be borne in mind. Consider the equation (x 4 e* — 2 mxy 2 ) dx + 2 mx 2 ydy = 0. Here the first term x*e x dx will be a differential of a function of x no matter what function of x may be assumed as a trial n. With fx = 1/x 4 the equation takes the form /tydv y 2 dx\ , , >/' 2 e*dx + 2 m - - - - — = de* + md — = 0. \ X 2 X° / X 2 The integral is therefore seen to be e T + my-fx- — C without more ado. It may be noticed that this equation is of the Bernoulli type and that an integration by that method would be considerably longer and more tedious than this use of an integrating factor. Again, consider (x + y) dx — (x — y)dy = and let it be written as xdx + ydy + ydx — xdy = ; try n = l/(x 2 + y~) ; xdx + ydy ydx — xdy n 1 , , . „ ,, , , x then + = or - d log (j- 2 + ?/) + d tan- 1 - = 0, x 2 + y 2 x 2 + 2/ 2 2 j/ and the integral is log V x 2 + y 2 + tan- 1 (x/ty) = C Here the terms xdx + ?/d# strongly suggested x 2 + y 2 and the known form of the differential of tan -1 (x/y) corroborated the idea. This equation comes under the homogeneous type, but the use of the integrating factor considerably shortens the work of integration. 92. The attempt has been to write Mdx + Ndy or /u. (Mdx -f- Xdy) as the sum of total differentials du + dv -+- ■ • • , that is, as the differential dF of the function u + " + •••, so that the solution of the equation Mdx -f- Ndy = could be obtained as F = C. When the expressions are complicated, the attempt may fail in practice even where it theoreti- cally should succeed. It is therefore of importance to establish condi- tions under which a differential expression like Pdx + Qdy shall be the total differential dF of some function, and to find a means of obtaining F when the conditions are satisfied. This will now be done. (13) Suppose Pdx + Qdy = dF = ex Cy then cF p = — , y ^ dp -JP (x, y) dx +2 y JQ (*„ //) d 'J = YijJ MX + Q ( "'o' y) - These differentiations, applied to the first form of F, require only the fact that the derivative of an integral is the integrand. The first turns out satisfactorily. The second must be simplified by interchanging the order of differentiation by y and integration by x (Leibniz's Rule, § 119) and by use of the fundamental hypothesis that P' y = Q' x . oy f Pdx + Q(x , y) = f C ~£ dx + Q(x , y) r x do \ x = y: d x + ( i (*„> y) = Q (*> y) + Q (*<» y) = <2 fo ?)• The identity of P and Q with the derivatives of F is therefore estab- lished. The second form of F would be treated similarly. Show that (x- + log y) dx + x/ydy = is an exact differential equation and obtain the solution. Here it is first necessary to apply the test P' = (J x . Now — (x- + log ?/) = - and — ■ - = - . cy y cxy y Hence the test is satisfied and the integral is obtained by applying the formula : r x r 1 | (x 2 + log y)dx + { - cfy = -x 3 + x log ?/ = G Jo J y 6 J - * '' x c 1 - '?'/ + I (x 2 + log 1) dx = x log y + - x° = C. l y J 3 It should be noticed that the choice of x = simplifies the integration in the first case because the substitution of the lower limit is easy and because the second integral vanishes. The choice of y Q = 1 introduces corresponding simplifications in the second case. 210 DIFFERENTIAL EQUATIONS Derive the partial differential equation which any integrating factor of the differ- ential equation Mdx + Ndy = must satisfy. If ft is an integrating factor, then IxMdx + fiNdy = dF and -!— = -^— . cy dx ,,5/* T. T dn (dN cM\ Hence M — — A 7 — = /* ) (15) d// gx \dx cy I is the desired equation. To determine the integrating factor by solving this equa- tion would in general be as difficult as solving the original equation ; in some special cases, however, this equation is useful in determining ft. 93. It is now convenient to tabulate a list of different types of dif- ferential equations for which an integrating factor of a standard form can be given. With the knowledge of the factor, the equations may then be integrated by (14) or by inspection. Equation Mdx -f Ndy = : Factor /a : I. Homogeneous Mdx + Ndy = 0, II. Bernoulli dy + Xjjdx = Xjfdx, lII.M=yf(*y),N = M*y), Mx _ Ny Mx + Ny y -n (><■'- - n) fjridx^ 1 dM dN V If dy dx \. it y dN ?.\1 v. if Tx ~ dy VI. Type x a y B (mydx + nxdy) = 0, {''' "' , . "" " ' [_k arbitrary. f ..km — 1 — a .Jen — 1 . — fi VII. x a y B (7nydx + nxdii) -f- xyifiipiidx -f- \ , Ni/ where — dy Mx + Ny dx \Mx + Ny/ By \x 1 + ' — yd/left dy \x 1 +

    )- x' 2 dx \y 1 + As this is an evident identity, the theorem is proved. To find the condition that the integrating factor may be a function of x only and to find the factor when the condition is satisfied, the equation (15) which /x satisfies may be put in the more compact form by dividing by it. M-^-N-^=—- — or M g l0 " M N d ]o " ^ = <' X d M . n&) /xdy it dx dx cy cy ex dx dy Now if ix (and hence log fx) is a function of x alone, the first term vanishes and diog M K-K dx N -.f(x) or log ix = jf(x)dx. This establishes the rule of type IV above and further shows that in no other case can /x be a function of x alone. The treatment of type V is clearly analogous. Integrate the equation x 4 y (3 ydx + 2 xdy) + x' 2 (4 ydx + 3 xdy) = 0. This is of type VII ; an integrating factor of the form fx = xPy* will be assumed and the ex- ponents p, a will be determined so as to satisfy the condition that the equation be an exact differential. Here P = fxM = 3 xP + i y (T + 2 + 4 xp + 2 y' T + 1 , Q = /xN =2xP + *>y a + 1 + 3 xP + 3 y°\ Then P' y - 3 (er + 2) xP + *y° + 1 + 4 (a + 1) xP + 2 y cr = 2 (p + 5) xP + 4 // (F) is a solution, consider the equation M* y - X*; = (MF' y -NF X ) $'. As F is a solution, the expression MF' y —NF' x vanishes by (16), and hence M$ y — N$' x also vanishes, and is a solution of the equation as is desired. The first half of 2 is proved. Next, if fi and v are two integrating factors, equation (15') gives glog/* N d\0<*H _ ^ glogP ^. glogp Qr ^ ( log fi/v Y (' 1' )g n/p __ cy ex vy ex By ex On comparing with (16) it then appears that log^/v) must be a solution of the equation and hence p/v itself must Vie a solution. The inference, however, would not hold if ix/v reduced to a constant. Finally if /j. is an integrating factor leading to the solution F= ('. then dF = fj. (Mdx + Ndy), and hence pi> (F)(Mdx + Ndy) = d f$(F) dF. It therefore appears that the factor /u(F) makes the equation an exact differen- tial and must be an integrating factor. Statement 2 is therefore wholly proved. COMMONER ORDINARY EQUATIONS 213 The third proposition is proved simply by differentiation and substitution. For dF cF dx cF dy ,,dx ^ T dy = 1 = /J.M 1- /J.JSI — • dn dx dn cy dn dn dn And if t denotes the inclination of the curve F = G, it follows that dy M . dy N dx M tan t = — = , sin t = — ^ = , — cos t = — = . dx N dn -y/M' 2 + N 2 dn Vi¥ 2 + N' 2 Hence dF/dn = y. Vm 2 + JV' 2 and the proposition is proved. EXERCISES 1. Find the integrating factor by inspection and integrate : (a) xdy — ydx = (x 2 + y 2 ) dx, (/3) (y 2 — xy) dx + x 2 dy — 0, (7) ydx — xdy + logxdx = 0, (5) y (2 xy + e x ) dx — e?dy — 0, ( e ) (1 + xy) ydx + (1 - xy) xdy = 0, (f ) (x — y 2 ) dx + 2 xydy = 0, (17) (xy 2 + y) dx — xdy = 0, (8) a {xdy + 2 ydx) = xydy, ( 1 ) (x 2 + y 2 ) (xdx + ydy) + Vl + (x 2 + y 2 ) (ydx — xdy) = 0, (k) xhjdx — (x s + y 3 ) dy = 0, (X) xdy — ydx — xVx 2 — y 2 dy. 2. Integrate these linear equations with an integrating factor : (a) y' + ay — sin hx, (/3) y' + y cotx = sec x, (7) (x + 1) y' -2y = (x+ l) 4 , (5) (1 +x 2 ) 1/ + y = etan-i^ and 05), (5), (f) of Ex. 4, p. 200. 3. Show that the expression given under II, p. 210, is an integrating factor for the Bernoulli equation, and integrate the following equations by that method : (a) y' — y tan x = y* sec x, ((3) 3 y 2 y' + y 3 = x — 1, (7) y' + y cos x = if sin 2 x, (S) dx + 2 xydy = 2 ax 3 y 3 dy, and (a), (7), (e), (ij) of Ex. 4, p. 200. 4. Show the following are exact differential equations and integrate : (a) (3x 2 + Gxy 2 )dx+(6x 2 y + 4iy 2 )dy = 0, (/3) sin x cos ydx + cos x sin ydy — 0, (y) (6x-2y + l)dx + (2y-2x-3)dy = 0, (5) (x 3 + 3 xy 2 ) dx + (y 3 + 3 x 2 y) dy = 0, ( 6 ) ^+_ 1 dx 4- V —^dy = 0, (0 (l 4- S) dx + ev (l --)dy = 0, y y- \ y) (■q) e x (x 2 4- y 2 + 2 x) dx + 2 yt^dy = 0, (6) (y sin x — 1) dx + (?/ — cos x) dy = 0. 5. Show that (Mx—Ny)- 1 is an integrating factor for type III. Determine the integrating factors of the following equations, thus render them exact, and integrate : (a) (y 4- x) dx 4- xdy = 0, (/3) (y 2 — xy) dx 4- x 2 dy = 0, (7) (x 2 4- y 2 ) dx — 2 xydy = 0, (5) (x 2 y 2 4- xy) ydx 4- (x 2 ?/ 2 — 1) xdy = 0, ( e ) (Vxy - l) xdy - (Vx^4- 1) ydx = 0, ( f) x 3 dx 4- (3 x 2 y 4- 2 y 3 ) dy = 0, and Exs. 3 and 9, p. 200. 6. Show that the factor given for type VI is right, and that the form given for type VII is right if k satisfies k (qm — pn) = q (a — 7) — p (/3 — 5). 214 DIFFERENTIAL EQUATIONS 7. Integrate the following equations of types IV-VII : (a) (y* + 2y)dx + (xf + 2 if - 4 x) dy = 0, (/3) {x 1 + if- + 1) dx - 2 xydy = 0, ( 7 ) (3j 2 + o^ + 3y-)cic-f (2x 2 + 3xy)dy = 0, (5) (2xV 2 + 2/)- (^ - 3x) y'= 0, ( e) (2 ^^ _ 3 y 4) dx + ( 3 x s + 2 ^) dy _ 0> ( f ) (2 - ?/) sin (3 x - 2 ?/) + y' sin (a; - 2 y) = 0. 8. By virtue of proposition 2 above, it follows that if an equation is exact and homogeneous, or exact and has the variables separable, or homogeneous and under types IV-VII, so that two different integrating factors may be obtained, the solu- tion of the equation may be obtained without integration. Apply this to finding the solutions of Ex. 4 (£), (5), (7) ; Ex. 5 (a), (7). 9. Discuss the apparent exceptions to the rules for types I, III, VII, that is, when Ms -f Xy = or Mx — Ny = or qm — pn = 0. 10. Consider this rule for integrating Mdx + Ndy=0 when the equation is known to be exact : Integrate Mdx regarding y as constant, differentiate the result regard- ing y as variable, and subtract from N; then integrate the difference with respect to y. In symbols, C = f {Mdx + Ndy) = C Mdx + f (x - f- f Mdx\dy. Apply this instead of (14) to Ex. 4. Observe that in no case should either this formula or (14) be applied when the integral is obtainable by inspection. 95. Linear equations with constant coefficients. The type + «,)y X. (20) Let a and a be the roots of the equation If- + a^D -f- a 2 — so that the differential equation may be written in the form [lf—(a l -\-a 2 )D + a 1 a 2 ]y — X or (D — cc~) (1> — aA y = X. The solution may now be evaluated by a succession of steps as {D - ag y = jj±-^ X = e^je-^Xdx, (20') 1) — a, !/ D - a. Xdx ./>»•■ / j ai -a t U j e -«i*Xdx (20") The solution of the equation is thus reduced to quadratures. The extension of the method to an equation of any order is immediate. The first step in the solution is to solve the equation ]>» + „ ]7 ,»-i + . . . + anxD + an _ o so that the differential equation may be written in the form (j) _ a j ( D _ ( q . . . (Z) - a n _X (I) - a„) y = X • (17") whereupon the solution is comprised in the formula y e «n* f e («n-l-«n)* C ... C.l^-ajr C e ~ «l* X (dx) n , (17'") where the successive integrations are to be performed by beginning upon the extreme right and working toward the left. Moreover, it appears that if the operators D — a n , T) — ) = 0, the exponentials aris- ing from the pure imaginary parts of the roots may be concerted into trigonometric functions. As an example take (1)* — 2 If' + D' 2 ) y = 0. The roots are 1, 1, 0, 0. Hence the solution is .,, , „ . , ,„ , „ . V = & C (C 1 + C 2 x) + (C, + C\x). Again if (Z) 4 + 4) y = 0, the roots of 7>* + 4 = are ±1 ±i and the solution is y = C 1 e< 1 +0^+ c. 2 eV-V*+ O 3 e(- a +0*+ C' 4 e ( - 1 - ,)j 218 DIFFERENTIAL EQUATIONS or y = e x (G^ + C 2 e~ ix ) + e~ • r (C.^e ix + C 4 e~ ix ) = e x (C x cos x + C. 2 sin x) + e~ x (C 3 cos x + 6' 4 sin x), where the new C's are not identical with the old C's. Another form is y = e x A cos (x + 7) + e~ x B cos (x + 5), where 7 and 5, A and B, are arbitrary constants. For C 1 <7 9 C l cos x + C 2 sin x = Vcf+cj COSX + and if 7 = tan - 1 I -f j, then C^ cos x -f C., sin x = v^ 4. £'| cos (x + 7). Next if A' is not zero but i/" any one solution I can be found so that P(Z))/=A', then a solution containing n arbitrary constants 'may be found by adding to I the solution of P(D\y = 0. For if p(D) I = X and P(D) y = 0, then P(D) (T + y) = A. It therefore remains to devise means for finding one solution I. This solution / may be found by the long method of (17'"), where the inte- gration may be shortened by omitting the constants of integration since only one, and not the general, value of the solution is needed. In the most important eases which arise in practice there are, however, some very short cats to the solution 7. The solution I of P(D)y = X is called the particular integral of the equation and the general solu- tion of P(T))y = is called the complementary function for the equa- tion P(1>) y = A. Suppose that A' is a. polynomial in x. Solve symbolically, arrange /'(//) in ascending powers of 7), and divide out; to powers of J) e< pial to the order of the polynomial A'. Then P(D)I X, I = A = P(D) ( m+ n(D) ^ > P(D) X, (22) where the remainder 7.' (7>) is of higher order in /> than X in x. Then P(D)I = P(D)Q(D) X + R ( 7>) A, R (1>) X = 0. Hence Q(D)x may be taken as /, since ]>(!)) Q(D) X = ]>(]>) I = X. By this method the solution / may be found, when A is a polynomial, as rapidly as /'(/>) can be dl elded into 1 ; the solution of P(D) // = may be written down by (21); and the sum of /and this will be the required solution of P( If) y = A containing n constants. As an example consider (//' + 4 7/- + 31))y = x-. The work is as follows: I = - x 2 3 d + i D± + j/i 1 1 1) 3 + 4 1) + lfi P- 4 |_3 9 13 T , I!(I>)1 a D + ]/2 + y J Ua. 21 l\I))\ COMMONER ORDINARY EQUATIONS 219 Hence I = Q(D)x 2 = -(- - -D + —iA x 2 = -x 3 - - x 2 + — x. * v ; D\3 9 27 / 9 9 27 For Z> 3 4-47) 2 + 3Z> = the roots are 0,-1,-3 and the complementary function or solution of P(D)y = would be C\ + C 2 e~ x + C 3 e~ 3x . Hence the solution of the equation P (D) y = x* 2 is y = C\ + C 2 e~* + <7 3 e-3* + |x 3 - £x 2 + if X. It should be noted that in this example D is a factor of P ( D) and has been taken out before dividing ; this shortens the work. Furthermore note that, in interpreting 1/1) as integration, the constant may be omitted because any one value of I will do. and P{D)e ax = P(a)e ax \ hence P(D) e" Ce a But P(D) I = Ce ax , and hence I = — j-r e ax (23) 97. Next suppose that X = Ce ax . Now De ax = ae ax , D l \ C_ IP (a) C_ ■(«) is clearly a solution of the equation, provided a is not a root of P(D) = 0. If P(a) = 0, the division by P(a) is impossible and the quest for I has to be directed more carefully. Let a be a root of multiplicity m so that P(D) = (D — ayP^D). Then I\(D) (D — a)'"I = Ce ax , (D — a) m I = — — e ax , n r T rt p ax T m and / = —7- e™ \ \ ( 2 + l)y = e x . The roots are 1, 1, — 1, — 1, and a = 1. Hence the solution for I is written as (D + 1)2 (D - I) 2 1 = e* (D - l) 2 1 = \ e x , I = \ e*x 2 . Then y = e x (G x + C 2 x) + e- x (C s + C 4 x) + \ e*x 2 . Again consider (I) 2 — 5JD + 6)y = x + e mx . To find the I x corresponding to x, divide. , r. + w7.£>+--- x = -x + 1 6 - 5 D + Z> 2 \6 36 / 6 36 To find the I 2 corresponding to e inx , substitute. There are three cases, I 2 = — e""-', J„ = xe 3 *, Z„ = — xe 2x , vi- -5m + 6 220 DIFFERENTIAL EQUATIONS according as in is neither 2 nor 3, or is 3, or is 2. Hence for the complete solution, 2/ = (V 8 * + C 2 e 2 * + -x + — + 6 36 m 2 — 5 hi + 6 when to is neither 2 nor 3 ; but in these special cases the results are y - C^ x + C. 2 c 2:r + I x + 3% — xe 2x , y = C x e Sx + GjP- x + -\ x + /,, + xe s - r . The next ease to consider is where X is of the form cos fix or sin fix. If these trigonometric functions he expressed in terms of exponentials, the solution may he conducted by the method above ; and this is per- haps the best method when +_ fit are roots of the equation 7 ) (/)) = 0. It may be noted that this method would apply also to the case where X might he of the form e^cos fix or e ax sin fix. Instead of splitting the trigonometric functions into two exponentials, it is possible to combine two trigonometric functions into an exponential. Thus, consider the equations P (D) ,j = e ax cos fix, P(D) y = e ax sin fix, and P (D) y = e ax (cos fir +• i sin fix) = e (a + ^ x . (24) The solution / of this last equation may be found and split into its real and imaginary parts, of which the real part is the solution of the equation involving the cosine, and the imaginary part the sine. "When X has the form cos fix or sin fix and +_ fit are not roots of the equation P(D) = 0, there is a very short method of finding /. For Z) 2 cos fix — — fi- cos fix and D 2 sin fix = — fi 2 sin fix. Hence if P(J>) be written as I\(D'-) -f- J>PjD' 2 ) by collecting the even terms and the odd terms so that P. and P n are both even in D, the solution may be carried out symbolically as 11 1 p{D) ' ;y //-> + Di\{if-) ' ' 1\(- fi-) + dp 2 (- fi-) T P l (-fi 1 )-DP a (-fiT-) cos .r, , -^cos.r. (25 s ) By this device of substitution and of rationalization as if D were a surd, the differentiation is transferred to the numerator and can be performed. This method of procedure may be justified directly, or it may be made to depend upon that of the paragraph above. Consider the example (7/ 2 + \)y — cosx. Here /3/ = i is a root of I)' 2 + 1=0. As an operator l)' 1 is equivalent to — 1, and the rationalization method will not work. If the first solution be followed, the method of solution is 1 e ix 1 c- ix 1 c> 1 e- ''-' 1 r . . ., 1 . H = ■ ; ; — — - — = — ; [.EC''' — XC~ ,X J = - X Sill X. I)- + 1 2 //- + 12 D- Hi I) + j 4 i 4 1 2 If the second suggestion be followed, the solution may be found as f COMMONER ORDINARY EQUATIONS 221 1 xe^ (I)' 1 + 1) I = cos x + i sin x = e' x , I = e te = v ; 7) 2 + 1 2 i J\ow I = — (cosx + i sin x) = - x sin x ix cosx. 2i v ' 2 2 Kence 7 = |xsinx for (7) 2 + 1)1 = cosx, and 7 = — \ x cosx for (D- + 1) I = sinx. The complete solution is y — C\ cos x + C 2 sin x + J x sin x, and for (7) 2 + l)y = sin x, y = C x cosx + C 2 sinx — \x cosx. As another example take (7J 2 — 3D + 2)y = cosx. The roots are 1, 2, neither is equal to ± /3t = ± i, and the method of rationalization is practicable. Then 1 1 1 + 3 2) 1 2) 2 -32) + 2 1-37) 10 10 (cosx — 3 sinx). The complete solution is y = C,e _x + C'„e-' 2x + ^(cosx— 3sinx). The extreme simplicity of this substitution-rationalization method is noteworthy. EXERCISES 1. By the general method solve the equations : , % d 2 v . du „ „ ,„. d 3 ?/ „ d' 2 y n dy a -~+4~ + 3y = 2e 2 - /3 _|-3-| + 3/-2, = e* dx 2 dx dx 3 dx 2 dx D 2 — 4D + 2)y=: x, (5) (7> 3 + 2) 2 - 4 2) - 4) j/ = x, 2)8 + 5 2)2 + 6% = z, (f) (X* 2 + D + 1) y = ze* 7/2 + 2) + l)y = sin 2 x, (0) (D 2 - 4)y = z + e** L» + SD + 2)y = x+ cos x, (k) (7>* - 4 2J 2 ) y = 1 - sin x, If- + i)y = cosx, {/j) (D 2 + \)y = secx, (v) (2) 2 + l)y = tanx. By the rule write the solutions of these equations : Jfi + 3 D + 2) y = 0, (/3) (2> 3 + 3 //- + 2) - 5) // = 0, 2) - l) 3 y = 0, (5) (D* + 2 I/ 2 + 1)2/ = 0, 2)3 _ 3 Z/2 + 4) j, = o, (f) (D* - 7> 3 - 9 2) 2 - 11 2)- 4)y = 0, 2)3 _ (5 7)2 + 9 7;) y _ o, (0) (2> - 4 Z> 3 + 8 2) 2 - 82) + 4) y = 0, 7^ _ 2 2> + 2> 3 ) y = 0, (k) (7> 3 - 2) 2 + I)) y = 0, 7)4 - 1)2 y = o, ( M ) (2)5 - 132) 3 + 20 7)2 + 82 7) + 104) y = 0. By the short method solve (7), (5), (e) of Ex. 1, and also : 2H _ 1) y - X \ (|8) (Z> 3 - 6 J/ 2 +11/)- (J) y = X . 7>3 + 3 7)2 + 2 7>) y = x 2 , (5) (7> 3 - 3 7/ 2 _ 6 2) + 8) y = x, If- + 8) y = x 4 + 2 x + 1, (f) (7> 3 - 3 Ifi - I) + 3) // = x 2 , 7>* - 2 7> 3 + 7)2) ?/ = x, (0) (7)4 + 27> 3 + 37/ 2 + 27;+l)y = l + x + .J 2 , 7> 3 - l)y = x 2 , ( K ) (7)4 _ 2 7> 3 + 2) 2 ) 2/ = x 3 . By the short method solve (a), (£). (0) of Ex. 1, and also : 7)2 _ 3 7) + 2) y = e* (/3) (7> - 7> 3 - 3 D 2 + 5 2) - 2)y = e 3 *, 7) 2 - 2 7) + 1) ?y = e* (5) (7> 3 - 3 2> 2 + 4) 2/ = e 3 *, 7/ 2 + 1) y = 2 e* + x 3 - x, (f) (7> 3 + 1) 2/ = 3 + e~ x + 5 e 2a; , 2)4 + 2 7/ 2 + 1) ?/ = e* + 4. (0) (7> 3 +SL* + 8D + l)y = 2 e-* 7/ 2 _ 2 7^) 2/ = e 2x + 1, (k) (7) 3 + 2 2) 2 + D) ?/ = c 2 ^ + x 2 + x, 7> 2 - a 2 ) y = e"* + e'-\ ( M ) (7) 2 - 2 aD + a 2 ) y = e* + 1. 222 DIFFERENTIAL EQUATIONS 5. Solve by the short method (77), (t), (k) of Ex. 1, and also : (a) (D 2 - 2) - 2) y = sin x, (/3) (2) 2 + 2 2) + 1) V = 3 e 2 * - cos x, (7) (2> 2 + 4) y = a: 2 + cos X) (5) (2> 3 + I/ 8 - 2) - 1) y = cos 2 x, (e) (D 8 + l) 2 ?/ = cosx, (f) (D 3 - D 2 + 2) - 1)?/ = cos x, (7?) (2)a - 5 D + 6) y = cos x - e 2 *, (0) (D 8 -22)2-3 D) y = 3 x 2 + sin x, ( 1 ) (2)2 - 1)2 y - .sin x, (/c) (2) 2 + 3 2) -f 2) y = e 2 * sin x, (X) (2>* - 1) y = e* cosx, (/*) (2>" - 3 2) 2 + 4 2) - 2) ;/ = e- + cosx, ( v) (2) 2 — 2 2) + 4) y = e x sin x, (0) (i> 2 + 4) 1/ = sin 3 x + e* + x 2 , - x V-3 (tt) (2) 6 + 1) y = sin § x sin J x, (p) (If 1 + l)y = e- x sin x + e 2 sin — — , (a) (D 2 + 4)?y = sin 2 x, (t) (2)4 + 322)4- 48)2/ = xe~ 2 * + e 2 *cos2f x. 6. If X has the form ^X , show that 2 = e<™ A' = e"* A", . 1 P(IJ) ! P(I)+a) 1 This enables the solution of equations where X 1 is a polynomial to be obtained by a short method ; it also gives a way of treating equations where X is e ax cos /3x or e ax sin /3x, but is not an improvement on (24) ; finally, combined with the second suggestion of (24), it covers the case where X is the product of a sine or cosine by a polynomial. Solve by this method, or partly by this method, (f) of Ex. 1 ; (k). (X), (v), (p), (t) of Ex. 5 ; and also (a) (2) 2 - 2 2) + 1) y = x 2 e 3 *, (/3) (IP + 3 2) 2 + 3 2) + l)y = (2 - x 2 ) e~* (7) (2) 2 + n 2 ) 2/ = x 4 e*, (5) (2X 1 - 2 IJ-" - 3 2) 2 + 4 1) + 4) 7/ = x 2 e*, (e) (2> 3 -72)- 6) y = e 2 * (1 + x), ( f ) (2) - lyhj = e* + cos x + x 2 e*, (7;) (2) — 1) 3 2/ = x — x 3 e x , (0) (I) 2 + 2)?/ = x 2 e 3a; + e T cos2x, ( 1 ) (2> 3 — 1) y = xe* + cos 2 x, (k) (2> 2 — 1) 7/ = x sin x + (1 + x 2 ) e x , (X) (2) 2 + 4)y = x sin x, (m) (2/ 1 + 2 2J 2 + 1) y = x 2 cos ax._ ( p) (2> 2 + 4) 2/ = (x sin x) 2 , (0 ) (2) 2 - 2 1) + 4) 2 j/ = xe* cos V3 x. 7. Show that the substitution x = c f . Ex. 0. p. 152, changes equations of the type x«2»»2/ + a x x" -i.D» -hj + ... + a „ _ ix2>2/ + a„y = A* (x) (26) into equations with constant coefficients ; also that ax + b = e' would make a simi- lar simplification for equations whose coefficients were powers of ax + b. Hence integrate : (a) (x 2 2) 2 - xU + 2) 2/ = x log x, (/3) (x 3 2> 3 — x 2 Ifi + 2 xl) - 2) y = x 3 + 3 x, (7) [(2x-l) 3 2> 3 +(2x-l)2)-2]2/=:0, (5) (x 2 2) 2 + 3x2) + \)y = (1 - x)- 2 , ( (x 3 2> 3 + x2) - 1) 2/ = x log x, (f) [(x + 1) 2 2> 2 - 4 (x + 1)2) + <>] 2/ = x, (77) (x 2 2) 2 + 4 x2) + 2) 2/ = e^, ((9) (x 3 2) 2 -3 x 2 2J + x) // = log x sin log x + 1 , ( 1 ) (x 4 2>' + 0x 3 2) 3 + 4 x-I)" - 2 x2) - 4) 2/ = x 2 + 2 cos log x. 8. If L be self-induction, /■' resistance, C capacity, i current, q charge upon the plates of a condenser, and /(f) the electromotive force, then the differential equa- tions for the circuit are <«•) % + f I + -h. = r •«')• <» S + 7 t, + 7^ = 7 no. at- L at LL L at- L dt L C L Solve (a) when/(<) = e~ at sin bt and (/?) when/(Q = sin6i. Reduce the trigonometric X>art of the particular solution to the form K sin (6i + 7). Show that if II is small and h is '/.early equal to 1/x LC. the amplitude 2l is large. COMMONER ORDINARY EQUATIONS 223 98. Simultaneous linear equations with constant coefficients. If there be given two (or in general ri) linear equations with constant coefficients in two (or in general ri) dependent variables and one inde- pendent variable t, the symbolic method of solution may still be used to advantage. Let the equations be (u/r + tt] />" - 1 + ■ • • + « n ) x + (b D»> + hjr - 1 + • • • + b m ) y = R(f), (c D*> + c x D» - 1 + • • • + c p ) x + Q/ D" + JJP' 1 + ■■■ + d q ) y = S (, ), ^ ' > when there are two variables and where D denotes differentiation by t. The equations may also be written more briefly as PjD) x + QjD) y = R and PjD) x + QjD) y = S. The ordinary algebraic process of solution for x and y may be employed because it depends only on such laws as are satisfied equally by the symbols 1), PjD), QJD), and so on. Hence the solution for x and y is found by multiplying by the ap- propriate coefficients and adding the equations. QjD) PJD) l\(D),r + q i (D) ! ,=R, /'.,( D) x + QJ D) ij = S. Then [PjD) QjD) - PjD) QjD)] x = QjD) R - QjD) S, [PjD) Q(D) - PjD) QjD)-],, = PjD) S - PjD) R. It will be noticed that the coefficients by which the equations are multi- plied (written on the left) are so chosen as to make the coefficients of x and y in the solved form the same in sign as in other respects. It may also be noted that the order of P and Q in the symbolic products is im- material. By expanding the operator PjD) Q. 2 (D) — PJD) QJD) a certain polynomial in D is obtained and by applying the operators to R and S as indicated certain functions of t are obtained. Each equation, whether in x or in y, is quite of the form that has been treated in §§ 95-97. As an example consider the solution for x and y in the case of 2 ^_^_4x = 2i, 2^ + 4^-3v = 0; at- at at at or (2 B- - 4) x - By = 2t. 2 Dx + (4 /J - 3) y = 0. Solve 4 B - 3 -21) \ (2D-- 4) x - By = 2 1 ]) -2 B- - 4 ' 2 Bx + (4 B - 3) y = 0. Then [(ID - 3) (2 IP - 4) + 2 IX 2 ] x = (4 B - 3)2 1, [2 B"- + (2 IP - 4) (4 B - 3)] y = - (2 B) 2 t, or 4 (2 If' - B"- - i B + Pj) x = 8 - Gt, 4 (2 IP - i» 2 - 4 B + 3) 7/ = - 4. The roots of the polynomial in I) are 1, 1. — H ; and the particular solution 7 X for x is — I f. and I, for ;/ is — J,. Hence the solutions have the form 224 DIFFERENTIAL EQUATIONS The arbitrary constants which are introduced into the solutions for x and // are not independent nor are they identical. The solutions must be substituted into one of the equations to establish the necessary relations between the constants. It will be noticed that in general the order of the equation in D for x and for y is the sum of the orders of the highest derivatives which occur in the two equations, — in this case, 3 = 2+1. The order may be diminished by cancellations which occur in the formal algebraic solutions f or x and y. In fact it is conceivable that the coeffi- cient P,Q 2 —P - | C. s e~ I* - j) + 4 (A x e l + K„e f + KJc' - § K s e~ 1 ') - 3 (h\e> + K. 2 te> + Jv' 3 e _ 2 ' - i) = 0, or e'(2 C\ + 2 C, + A, + A 2 ) + te\2 C 2 + A 2 ) - 3 e~ i '(C s + 3 A 3 ) = 0. As the terms e', te*, e~ - ' are independent, the linear relation between them can hold only if each of the coefficients vanishes. Hence C 3 + 3 A 3 = 0, 2 C 2 + A 2 = 0, 2 C\ + 2 C, + K 1 + K 2 = 0, and 6*3 = - 3 A 3 , 2 C, = - A 2 , 2C, = -A' r Hence x = (C\ + CJ) e> - 3 K s e~ i i ' - {t, y = - 2 (C, + CJ) d + K 3 e~ l ' - \ are the finished solutions, where C\ . C 2 , A 3 are three arbitrary constants of inte- gration and might equally well be denoted by <\. ('.,. ('.,. or A',. K n . A,. 99. One of the most important applications of the theory of simultaneous equa- tions with constant coefficients is to the theory of small vibrations about a state of equilibrium in a conservative* dynamical system. If q x . q„ . ■ ■ ■ . q n are n coordinates (see Exs. 19-20, p. 112) which specify the position of the system measured relatively * The potential energy V is defined as — dV — dW = Q\'lq\ + Qodq* + ■ • • + Qndqn, t'li CQi I'li Cqi Cqi Cq; This is tjie immediate extension of Q x as given in Ex. 19, p. 112. Here dW denotes the differential of work and dW — ZF.-'/r, ■= 2 (A>/.--,- + )>/.'/,+ /,'/?,). To find Q,- it is generally quickest to compute 'MI" from this relation with dxi, dyi. dz t expressed in terms of the differentials <>q l , • ■ • , dq n . The generalized forces Q, are then the coefficients of dq;. If there is to be a potential V, the differential '/If must be exact. It is frequently easy to find V directly in terms of q x , ■••, q n rather than through the mediation of Q\ . • ■ • , Qn'- when this is not so, it is usually better to leave the equations in the form d cT cT , , rv „— = Q t rather than to introduce band L. dt cqi cqi COMMONER ORDINARY EQUATIONS 225 to a position of stable equilibrium in which all the q's vanish, the development of the potential energy by Maclaurin's Formula gives V(

  • 2 x - 3 x - 4 y = 0, Jf-y + x + y = 0, dx — dy ,. . . ,. dx dy (5) = - = dt, (e) — dt = = , y — 7 x 2 x + 6 y 3 x + 4 y 2 x + 5 y • (f) tl)x + 2 (x - ?/) = 1, tDy + x + by = t, (t?) Dx = ny — mz, By = Iz — ??x, JJz = mx — ly, (9) Ifix - 3 x - 4 y + 3 = 0, Jfiy + x-8y + 5 = 0, ( i ) D*x - 4 ]) 3 y + 4 TPx - x = 0, l^y - 4 If'x + 4 Ifiy - y = 0. 2. A particle vibrates without friction upon the inner surface of an ellipsoid. Discuss the motion. Take the ellipsoid as X 2 v i ( z _ c \2 / Vcr/ \ / V CO ±- + ^ + ±1 ±-=1; then x = Cmi( — »t+C,), y = Ksm(—^-t + K a? b* c* \ o V \ b 3. Same as Ex. 2 when friction varies with the velocity. 4. Two heavy particles of equal mass are attached to a light string, one at the middle, one at one end, and are suspended by attaching the other end of the string to a fixed point. If the particles are slightly displaced and the oscillations take place without friction in a vertical plane containing the fixed point, discuss the motion. 5. If there be given two electric circuits without capacity, the equations are T di, -, r di„ _ . _ _ di„ ,,'?', ,, . L. — ! + M - 2 + ILL = E. , X., — 2 + M — + HJ., = E„, 1 dt dt ll l - dt dt 22 where i x , i 2 are the currents in the circuits, L i , L., are the coefficients of self- induction, B t , ll 2 are the resistances, and M is the coefficient of mutual induction. (a) Integrate the equations when the impressed electromotive forces E r , E 2 are zero in both circuits. (/3) Als27 6. If the two circuits of Ex. 5 have capacities C 1? C 2 and if g n g., are the charges on the condensers so that i 1 = dq,/dt, 1. 2 = dq 2 /dt are the currents, the equations are 1 (It* dP l dt C l x 2 dt 2 dP 2 dt C 2 " Integrate when the resistances are negligible and E t = E 2 = 0. If T 1 = 2ir VC l L l and T,, = 2 ir V C 2 L 2 are the periods of the i ndividua l separate circuits and G = 2ttMVC 1 C 2 , and if T 1= T,, show that Vr 2 + e 2 and Vr 2 - e 2 are the independent periods in the coupled circuits. 7. A uniform beam of weight G lb. and length 2 ft. is placed orthogonally across a rough horizontal cylinder 1 ft. in diameter. To each end of the beam is suspended a weight of 1 lb. upon a string 1 ft. long. Solve the motion produced by giving one of the weights a slight horizontal velocity. Note that in finding the kinetic energy of the beam, the beam may be considered as rotating about its middle point (§ 39). CHAPTER IX ADDITIONAL TYPES OF ORDINARY EQUATIONS 100. Equations of the first order and higher degree. The degree of a differential equation is defined as the degree of the derivative of highest order which enters in the equation. In the ease of the equation *(■''> !h !/') = °f ^ ie nrs ^ order, the degree will he the degree of the equation in ij . From the idea of the lineal element (§ 85) it appears that if the degree of \f in y' is n, there will be n lineal elements through each point (x, y). Hence it is seen that there are n curves, which are compounded of these elements, passing through each point. It may be pointed out that equations such as // = x\l + //", which are apparently of the first degree in //', are really of higher degree if the multiple value of the functions, such as Vl -f- y 2 , which enter in the equation, is taken into consideration; the equation above is replaceable by y' 2 = cc 2 -f- x 2 if, which is of the second degree and without any multiple valued function.* First suppose that the differential equation * (■'-, y, !/') = [//' - .'/)] x [.'/ - ^(*» //>] • • = o (l) may be solved for //'. It then becomes equivalent to the set //' - ^(.r, y) = 0, y' - , y) = 0, • • • (1') of equations each of the first order, and each of these may be treated by the methods of ( 'hap. VIII. Thus a set of integrals t F^y, C) = 0, Fj,; !h C) = 0, • • • (2) may be obtained, and the product of these separate integrals !■(,; y, C) = /•;<>, y, ■ Fj.r, y, C) • • ■ = (2') is the complete solution of the original equation. Geometrically speak- ing, each integral /•',•(.''• y, C) = represents a family of curves and the product represents all the families simultaneously. * It is therefore apparent that the idea of decree as applied in practice is somewhat indefinite. t The same constant C or any desired function of <_' may he used in the different solutions because C is an arbitrary constant and no specialization is introduced by its repeated use in this way. 228 ADDITIONAL ORDINARY TYPES 229 As an example consider y" 1 + 2 y'y cot x = y-. Solve. y' 2 + 2 y'y cot x + y" 1 cot 2 x = ?/ 2 (l + cot 2 x) = y 2 esc 2 x, and (?/' + ?/ cot x — y. esc a - ) (?/' + y cot x + y esc x) = 0. These equations both come under the type of variables separable. Integrate dy 1 — cosx , cZcosx .„ , ,. — = dx = , y (1 + cos x) = C , 2/ sinx 1 + cosx dy 1 + cosx , dcosx ,„ , _, and — = dx — , y (1 — cosx) = 6. y sinx 1 — cosx Hence 0(1 + cosx) + C][y(l — cosx) + C] = is the solution. It may be put in a different form by multiplying out. Then i/ 2 sin 2 x + 2C'y + C 2 = 0. If the equation cannot be solved for y' or if the equations resulting from the solution cannot be integrated, this first method fails. In that case it may be possible to solve for y or for x and treat the equation by differentiation. ~Lety'=p. Then if y=f^v)> d^ =p= -dx + YpTx- (3 ) The equation thus found by differentiation is a differential equation of the first order in dp/dx and it may be solved by the methods of Chap. VIII to find F(p, x, C) = 0. The two equations y=f(x,p) and F(p,x,C) = Q (3') may be regarded as defining x and y parametrically in terms of p, or p may be eliminated between them to determine the solution in the form Q, (x, y, C) = if this is more convenient. If the given differential equa- tion had been solved for x, then -, a 1 dx 1 df df dp x=f(y,p) and — = — = — + — -f- • (I) dy P c !l °1> dy The resulting equation on the right is an equation of the first order in dp/dy and may be treated in the same way. As an example take xp 2 — 2 yp -f ax — and solve for y. Then ax n dy „ dp ax dp a 2y = xp+—, 2^ = 2p=p + x^----^ + -, p dx dx pj dx p \ p — - — + ( p ) = 0, or xdp — pdx = 0. |_ p\ dx \p J x or — P The solution of this equation is x = Cp. The solution of the given equation is 2y = xp -\ . X — Gp P when expressed parametrically in terms of p. If p be eliminated, then x 2 2y — — \- aG parabolas. 230 DIFFERENTIAL EQUATIONS As another example take p 2 y -f 2 px = y and solve for x. Then (I \ c (lx 2 1 / 1 A dp 2x = y(--p), 2 — = - = --p + y l- i)JL, \P I dt/ p p \ p* ) dy 1 . /I . ,\dp + P + V ( — + 1 )-r = °> or ^ + ^ = °- p \p- / dy The solution of this is py = C and the solution of the given equation is 2 x = y (- - p j , py = C, or y 2 = 2 C'x + C\ Two special types of equation may be mentioned in addition, although their method of solution is a mere corollary of the methods already given in general. They arc the equation homogeneous in (x, y) and Clairavt's equation. The general form of the homogeneous equation is ^(.I'y y/ x ) = 0- This equation may be solved as P = *(£) oras l=f^ V = */(?)•> ( 5 ) and in the first case is treated by the methods of Chap. VIII, and in the second by the methods of this article. Which method is chosen rests with the solver. The Clairaut type of equation is y=px+f{p) (6) and comes directly under the methods of this article. It is especially noteworthy, however, that on differentiating with respect to x the result- ing equation is d P _ A „„ (1 1 a > P+f(p)l±-o or jl-o. ( &) Hence the solution for p is p — C, and thus y = Cx +/(C) is the solu- tion for the Clairaut equation and represents a family of straight lines. The rule is merely to substitute C in place of p. This type occurs very frequently in geometric applications cither directly or in a disguised form requiring a preliminary change of variable. 101. To this point the only solution of the differential equation &(x, y, /A = which has been considered is the general solution F(x,y, C)=0 containing an arbitrary constant. If a special value, say 2, is given to C, the solution F(x, y, 2) = is called a ptn'tirulai- solution. It may happen that the arbitrary constant C enters into the expression F(x, //, C)= in such away that when C becomes positively infinite (or negatively infinite) the curve F(x, //, C) = approaches a definite limiting position which is a solution of the differential equation ; such solutions are called infinite solutions. In addition to these types of solution which naturally group themselves in connection with the general solution, there is often a solution of a different kind which is ADDITIONAL ORDINARY TYPES 231 known as the singular solution. There are several different definitions for the singular solution. That which will he adopted here is : A singu- lar solution is the envelope of the family of curves defined by the general solution. The consideration of the lineal elements (§ 85) will show how it is that the envelope (§ 65) of the family of particular solutions which constitute the general solution is itself a solution of the equation. For consider the figure, which represents the particular solutions broken up into their lineal elements. Note that the envelope is made up of those lineal elements, one taken from each particular so- lution, which are at the points of contact of the envelop e^ envelope with the curves of the family. It is seen st^ y^]> that the envelope is a curve all of whose lineal elements satisfy the equation ^ (x, y, p) = for the reason that they lie upon solutions of the equation. Now any curve whose lineal elements satisfy the equation is by definition a solution of the equation; and so the envelope must he a solution. It might conceivably happen that the family F(x, y, C) = was so constituted as to envelope one of its own curves. In that case that curve would be both a particular and a singular solution. If the general solution F(x, y, C) = of a given differential equation is known, the singular solution may be found according to the rule for finding envelopes (§ 65) by eliminating C from F(x, y, C) = and —, F(x, y, C) = 0. (7) It should be borne in mind that in the eliminant of these two equations there may occur some factors which do not represent envelopes and which must be discarded from the singular solution. If only the singu- lar solution is desired and the general solution is not known, this method is inconvenient. In the case of Clairaut's equation, however, where the solution is known, it gives the result immediately as that obtained by eliminating C from the two equations y=Cx+f(C) and = x + /'(C). (8) It may be noted that as p = C, the second of the equations is merely the factor x +f'(p) = discarded from (6'). The singular solution may therefore be found by eliminating p between the given Clairaut equa- tion and the discarded factor x -\-f\p s )= 0. A reexamination of the figure will suggest a means of finding the singular solution without integrating the given equation. For it is seen that when two neighboring curves of the family intersect in a point P 232 DIFFERENTIAL EQUATIONS near the envelope, then through this point there are two lineal elements which satisfy the differential equation. These two lineal elements have nearly the same direction, and indeed the nearer the two neighboring curves are to each other the nearer will their intersection lie to the envelope and the nearer will the two lineal elements approach coinci- dence with each other and with the element upon the envelope at the point of contact. Hence for all points (x, y) on the envelope the equa- tion ty(x, y, p) = of the lineal elements must have double roots for p. Now if an equation has double roots, the derivative of the equation must have a root. Hence the requirement that the two equations s -(2x + y 2 )p 2 + (x 2 -y 2 + 2xy 2 )p-(x 2 -y 2 )i/ 2 = 0, (y) xp 2 -2yp-x = 0, {8) p 3 (x + 2y) + 3p 2 {x + y) + p (y + 2 x) = 0, (e) y 2 + p 2 = 1, (f) p 2 - ax 3 = 0, ( v ) p = (a- x) Vl + p 2 . 2. Integrate the following equations by solving for y or x : (a) 4 xp 2 + 2 xp — y = 0, (/3) y = — xp + x*p 2 , (y) p + 2 xy — x 2 — y 2 = 0, (5) 2px — y + log p = 0, (e) x — yp — ap 2 , (f) y = x + a tan _1 p, (77) x = y + a logp, {6) x + py (2p- + ?>) = 0, ( 1) a-yp 2 - 2 xp + y = 0, (k) p % — \xyp + 8 y 2 = 0, (X) x =p + logp, (/*.) p 2 (x 2 + 2 ax) = a 2 . 3. Integrate these equations [substitutions suggested in (t) and (k)] : (a) xy 2 {p 2 + 2) =2 pi/ + x\ (0) (?u + py) 2 = (1 + P 2 ) (y 2 + nx 2 ), (7) U" + -ryp - X'P 1 = 0, (5) y = yp 2 + 2px, (e) y=px+ sin-ip, (f) y = p (x - b) + a/p, (t?) y=px + p(l- p 2 ), (0) y 2 - 2pxy - 1 = p 2 (1 -x 2 ), (1) 4 e 2 v p 2 + 2 xp — 1 = 0, z = e 2 !>, (k) y = 2 px + y 2 p". y' 2 — z. (X) AerVp 2 + 2e 2x p — e x = 0, (n) x 2 (y — px) = yp 2 . 4. Treat these equations by the p method (9) to find the singular solutions. Also solve and treat by the C method (7). Sketch the family of solutions and examine the significance of the extraneous factors as well as that of the factor which gives the singular solution : (a) p 2 y 4- p (x — y) — x = 0. (/3) pry 2 cos' 2 a — 2 pxy sin 2 a + y 2 — x 2 sin 2 (p) +f(p) is given parametrically by the given equation and the solution of the linear equation: dx (P)-P P_-)■ Show that the orthog- onal trajectories of any family of lines leads to an equation of the type of Ex. ,T1 !I (1 P /im Tx=*> d^ = Tx ° V d? =P dy (10) rendered the differential equations integrable by reducing; them to in- tegrable equations of the first order. These substitutions or others like them are useful in treating certain cases of the differential equation ADDITIONAL ORDINARY TYPES 235 *(- r j Vi y\ //"> "'? y (n) )=0 of the nt\\ order, namely, when one of the variables and perhaps some of the derivatives of lowest order do not occur in the equation. / d'u d i+1 y d n y\ Incase *(*, ^ ^, . . , -£) = 0, (11) y and the first i — 1 derivatives being absent, substitute ^■. = q so that *( M) i-,f^Lo. (11') dx l 2 \ ' "" dx ' dx n - 1 ) K ' The original equation is therefore replaced by one of lower order. If the integral of this be F(x, q\ = 0, which will of course contain n — i arbitrary constants, the solution for q gives ?==/(*) and y=f--ff(x)(dxy. (12) The solution has therefore been accomplished. If it were more con- venient to solve F(x, q) = for x = (q), the integration would be V =f- ■ fl (dx)* =[■ ■ Jq L4>'Qt)dqJ ; (12') and this equation with x — ('/) would give a parametric expression for the integral of the differential equation. Incase *L g, % . . ., g) = 0, (13) x being absent, substitute p and regard p as a function of y. Then dq d 2 // dp c/ 3 y d I dp\ and *^ 7>| _,..., __j = . In this way the order of the equation is lowered by unity. If this equa- tion can be integrated as F(y, p) = 0, the last step in the solution may be obtained either directly or parametrically as It is no particular simplification in this case to have some of the lower derivatives of y absent from \1> = 0, because in general the lower deriva- tives of p will none the less be introduced by the substitution that is made. 23tf DIFFERENTIAL EQUATIONS / (F'y d 2 y\ 2 /d 3 w\ 2 As an example consider ( x ) = ( — 1+1, \ (/x ;i dx 1 ) \dx s / . , . ( dq V / \dx/ for the equation is a Clairaut type. Hence, finally, y=ff[c i x± Vcf+l~\(dx) 2 = I C^ ± 3 x 2 VC' 2 + 1 + C\x + C 3 . As another example consider y" — y' 2 = y 2 log?/. This becomes p -/- - p 2 = y 2 log ?y or -^—L - 2 p- = 2 // 2 log ?/. dy d?y The equation is linear in p 2 and has the integrating factor a-' 1 '-'. - p2 e - 2 y = j2/2e- 2 » log ydy, — = p = \c 2 vjyH- 2 v log ydy , and C — r = V2 x. I e 2 » f?y 2 e- 2 ■" log ?/d?y The integration is therefore reduced to quadratures and becomes a problem in ordinary integration. If an equation is homogeneous with respect to y and its derivatives, that is, if the equation is multiplied by a power of k when y is replaced by ky, the order of the equation may be lowered by the substitution y = e z and by taking z' as the new variable. If the equation is homo- geneous with respect to x and, dx, that is, if the expiation is multiplied by a power of /.• when x is replaced by lex, the order of the equation may be reduced by the substitution x = <■'. The work may be simplified (Ex. 9, p. 152) by the use of D*y = e~ nt D t (D t - 1) • • ■ (D t -n + 1) y. (15) If the equation is homogeneous with respect to x and y and, the dif- ferentials dx, dy, d 2 y, ■ ■ ■ , the order may be lowered by the substitution x = e\ y = e'z, where it may be recalled that />;.'/ = e~ nt D t {D t - 1) • • • (!) t - n + 1) y = r -(»-^>(l), ; + 1) I), ■ ■ ■ (/), - n + 2),-:. Finally, if the equation is homogeneous with respect to x considered of dimensions 1, and y considered of dimensions m, that is, if the equation is multiplied by a power of k when kx replaces x and /.'"// replaces y, the substitution x = e', y = e m, z will lower the degree of the equation. It may be recalled that ir r y = ,.-<'»-»>'(/>, + in) (I), + m — 1) • • • (D t + m - n + 1) z. (15") ADDITIONAL OKDLNARY TYPES 237 Consider xyy" — xy" 2 = yy' + bxy' 2 /Va 2 — x 2 . If in this equation y be replaced by ky so that y' and y" are also replaced by ky' and ky" ', it appears that the equation is merely multiplied by k 2 and is therefore homogeneous of the first sort mentioned. Substitute y = e z , y' = e 2 z', ?/' = e*(z" + z' 2 ). Then e 2 * will cancel from the whole equation, leaving merely xdz' 1 , bxdx s" = z + vxz u /\ a- — x- or — ax = k // * * V a- — x- The equation in the first form is Bernoulli ; in the second form, exact. Then x , r~z : „ . , xdx — = h V «- — x- + and dz 6 Va 2 — x' 2 + C The variables are separated for the last integration which will determine z = logy as a function of x. Again consider x 4 — — = (x 3 + 2 xy) — — 4 y 2 . If x be replaced by kx and y by dx 2 dx k 2 y so that ?/' is replaced by ky' and y" remains unchanged, the equation is multi- plied by k* and hence comes under the fourth type mentioned above. Substitute x = e', y = e 2 'z, I) x y = c'(D t + 2)z, Dpj = (D t + 2) (D t + l)z. Then c if will cancel and leave z" + 2(1 — z)z' = 0, if accents denote differentiation with respect to t. This equation lacks the independent variable t and is reduced by the substitution z" = z'dz'/dz. Then dz' dz -, dz — + 2 (1 - z) = 0, z' = — = (1 - z) 2 + C, = eft. dz K ' dt v ; (i_ z -') +G . There remains only to perform the quadrature and replace z and t by x and y. 103. If the equation may be obtained by differentiation, as / (hi y n) + 4» 2 > where the coefficient of //'° is collected into <£ r Now integrate 1} par- tially regarding only y (H_1) as variable so that / d eft eft Then * - —J dx r,/» — i In-Tcy dx n — k + That is, the expression ^ — ft/ does not contain ?/'° and may contain no derivative of order higher than n — 1\ and may be collected as 238 DIFFERENTIAL EQUATIONS indicated. Now if * was an exact derivative, so must * — Q[ be. Hence if m =£ 1 } the conclusion is that ^ was not exact. If m = 1, the process of integration may be continued to obtain fi., by integrating partially with respect to yC"-*- 1 ). And so on until it is shown that * is not exact or until ^ is seen to be the derivative of an expression O x -f 2 + • • • = C. As an example consider Sk = xry'" + xy" + (2 xy — 1) y' + y 2 = 0. Then fy = jxMy" = x 2 y", * - fi x ' = - xy" + (2 xy — 1) y' + 2/ 2 , &, = f - xdy' = - xy', * - ttj - fi 2 ' = 2 a^' + r = (xy 2 )'. As the expression of the first order is an exact derivative, the result is * - P.[ _ o 2 ' _ (xyty = ; and * l = x"-y" - xy' + xy- - C\ = is the new equation. The method may be tried again. Q 1 = fx-dy' = x 2 y', * t — n' t = -Sxy' + xy 2 - C x . This is not an exact derivative and the equation <& x = is not exact. Moreover the equation ^ = contains both x and y and is not homogeneous of any type except when C\ = 0. It therefore appears as though the further integration of the equation ^ = were impossible. The method is applied with especial ease to the case of d n y . 'I"~ l n . fhf A '° dx n + 'V ■•■ + A'»_ 1 ^ + A„ 2/ - J R(.r)=0, (17) where the coefficients are functions of x alone. This is known as the linear equation, the integration of which has been treated onlv when the order is 1 or when the coefficients are constants. The application of successive integration by parts gives fij = A'^o-n n a = (a\- A' ')y»-% o 3 = (x. 2 - a; + A' ")y— S >, • • • : and after n such integrations there is left merely (A„ - X' n _, + • ■ ■ +(_l)»-iA' ] +(-l)«A o ) ;/ _ R, which is a derivative only when it is a function of .'■. Hence x„ - A-;., + ...+(- 1 y-K\\ + ( - 1 )"A = o (18) is the condition that the linear equation shall be exact, and V n - !) +(A a - A' ')//«- 2 > + (A 2 - A'i + AiV'"^ + • ■ • = f A''/.'' (19) is the first solution in case it is exact. As an example take y'" + y" ens./- — 2 //'sin.r — ycosx = sin 2x. The test A'j — X.,' + A'j" — A" '" = — cosx + 2 cos a; — cosx = ADDITIONAL ORDINARY TYPES 239 is satisfied. The integral is therefore y" + y'cosx — ysinx = — \ cos2x + C\. This equation still satisfies the test for exactness. Hence it may be integrated again with the result y' + y cos x = — ^sin2 x + C\x + C 2 . This belongs to the linear type. The final result is therefore V dnx je Biax (C 1 x + C 2 )dx + C 3 e- sinx + \ (1 — sinx). EXERCISES 1. Integrate these equations or at least reduce them to quadratures : (a) 2xy'"y" = y" 2 - a 2 , (/3) (1 + x 2 ) y" + 1 + y' 2 = 0, (7) l/ iv + «V =0, (5) y v - m-y"' = e«*, (e) x 2 y iv + a-y" = 0, (j") a 2 l/'Y = *■ (v) ■''!/" + >/'_= °, (#) v'"y" = 4, (0 (1 - x"-)y" - xy' = 2, (/t) y* = A y"\ (A) y" =/(y), (At) 2 (2 a — y) y" = 1 + y' 2 , (v) yy" - y' 2 - y 2 y' = 0, (o) yy" + y' 2 +1 = 0, (ir) 2 y" = c", (p) y"y" = a. 2. Carry the integration as far as possible in these cases: (a) x"y" = (mx 2 y' 2 + ny 2 )-, (/3) mxnj" = (y — xy') 2 , (y) x*y" = (y - W) s , (*) x*v" - *V - - r V 2 + 4 y 2 = o, (e) x~ 2 y" + x~ 4 y = iy' 2 , (f) a////" + by' 2 = yy'(c 2 + x 2 )~ a. 3. Carry the integration as far as possible in these cases: (or) (y 2 + x) y'" + 6 yy'y" + y" + 2 y' 2 = 0. (/3) y'j/" - y/'V = xy 2 , (7) &W" + z&y'y" + 9*W + '>-' - V 2 + I8xyy' + 3y 2 = o, (5) y + 3xy' + 2 yy' 3 + (x 2 + 2 y 2 y') y" = 0, (e ) (2 x 3 y' + .r 2 t x + y esc 2 ,r = c< >s x, (5) (x 2 - x) y" + (?> x — 2) y' + y = 0, (e ) (x - x s ) y'" + (1 — 5 .'■-) y" - 2 xy' + 2 y = 6 x, (O (z 3 + J' 2 - 3x + 1) y'" + (it./- 2 + 6x- !»)y" + (18 x + 6) y' + fl y = x\ (i?) (x + 2) 2 y'" + (x + 2) y" + y' = 1, (0) x 2 y" + 3x?/ + y = x, ( ( ) (x 3 - x) y'" + (8 x 2 - 3) y" + 14 xy' + 4 y = 0. 5. Note that Ex. 4 (0) comes under the third homogeneous type, and that Ex. 4 (77) may be brought under that type by multiplying by (x + 2). Test sundry of Exs. 1, 2. 3 for exactness. Show that any linear equation in which the coefficients are polynomials of degree less than the order of the derivatives of which the} 7 are the coefficients, is surely exact. 6. Sometimes, when the condition that an equation be exact is not satisfied, it is possible to find an integrating factor for the equation so that after multiplication by the factor the equation becomes exact. For linear equations try x'". Integrate (a) shj" + (2 x* - x) y' - (2 x 3 - X)y = 0, (/3) (x 2 - x 4 ) y" - xhf - 2 y = 0. 7. Show that the equation y" + Ty' + Qy' 2 = may be reduced to quadratures 1° when P and Q are both functions of y. or 2° when both are functions of x, or 3° when P is a function of / and Q is a function of y (integrating factor 1/y'). In each case find the general expression for y in terms of quadratures. Integrate y" + 2 y' cot x + 2 y' 2 tan y — 0. 240 DIFFERENTIAL EQUATIONS 8. Find and discuss the curves for which the radius of curvature is proportional to the radius r of the curve. 9. If the radius of curvature 7.' is expressed as a function R = R(s) of the arc s measured from some point, the equation R = R(s) or s — $(R) is called the intrinsic equation of the curve. To find the relation between x and y the second equation may be differentiated as ds = s'(R)dR, and this equation of the third order may be solved. Show that if the origin be taken on the curve at the point s — and if the x-axis be tangent to the curve, the equations express the curve parametrically. Find the curves whose intrinsic equations are (a) R = a, (/3) nR = .s 2 + " 2 , (7) R 2 + s 2 = 16 « 2 . 10. Given F = ?/(»> + X^(» - 1 ' + X.,y(" - 2) 4. ... 4. X n _i?/ + X n y = 0. S) ow that if it, a function of x alone, is an integrating factor of the equation, then * = M ("> - (2»<»-« + (J»0— » + (- l)—i(X n -w)' + (- l)»X BJ u = is the equation satisfied by /x. Collect the coefficient of it to show that the condition that the given equation be exact is the condition that this coefficient vanish. The equation = is called the adjoint of the given equation F = 0. Any integral n of the adjoint equation is an integrating factor of the original equation. Moreover note that fpFdx = fiyin-l) 4. ( fX ,A\ - n') yin-2) 4 . . . 4 (_ 1)« JyQdX, or d[/xF - (- l)»y] = d [>. In other words, any solution y of the original equation is an integrating factor for the adjoint equation. 104. Linear differential equations. The equations X D»,, + Ay;"- 1 // + • • • + X„_ x l>,, + X n y = R(x), X D">/ + XJ)" -hj 4- • • • + X n _ x l>y + A'„y = are linear differential equations of the nth order; the first is called the complete equation and the second the rehired equation. If y., //.,, y„, ■ ■ ■ are any solutions of the reduced equation, and C , ('.,, C g , ••■ are any constants, then y = C 1 y l -f- C'.,//., + C // -| is also a solution of the reduced equation. This follows at once from the linearity of the reduced equation and is proved by direct substitution. Furthermore if / is any solution of the complete equation, then y -f- / is also a solution of the complete equation (of. §90). As tln> equations (20) are of the nth order, they will determine //"' and, by differentiation, all higher derivatives in terms of the values of ■ r > !/) .'/\ ' ' ' j .'/" _1) - i fence if the values of the n quantities y , y' , • • ■ , y§" _1) which correspond to the value ./■ = x be given, all the higher derivatives are determined (§§ 87~88). Hence there are n and no more than n arbi- trary conditions that may be imposed as initial conditions. A solution ADDITIONAL ORDINARY TYPES 241 of the equations (20) which contains n distinct arbitrary constants is called the general solution. By distinct is meant that the constants can actually be determined to suit the n initial conditions. If y , y , ■ ■ ■ , y n are n solutions of the reduced equation, and V = C ll/l + r V/-2 H 1" ( 'n!ln, I/' = Cii/x + C02 H h Cn!/n, (21) y0-« = C\y{"-v + C^—" + • ■ • + C n y^\ then ?/ is a solution and y', • • • , // ( " _1) are its first n — 1 derivatives. If x be substituted on the right and the assumed corresponding initial values y , y' , ■ ■ ■ , _>/,," _1) be substituted on the left, the above n equations become linear equations in the n unknowns C , C n , ■ ■ ■ , C n ; and if they are to be soluble for the C's, the condition w Q/v y*>'--> ?/n) = !/l I/o ■■■ Vn \y\ ifi ••• Vn ._„ ^-i) ... ^-d =£ (22) must hold for every value of x = x . Conversely if the condition does hold, the equations will be soluble for the C's. The determinant W(y , //.,, •••, //„) is called the Wronskian of the n functions y , ;/.,, •••, y n . The result may be stated as: If n functions y v v/.,, • • •, y n which are solutions of the reduced equation, and of which the Wronskian does not vanish, can be found, the general solution of the reduced equation can be written down. In general no solution of the equation can be found, whether by a definite process or by inspection; but in the rare instances in which the n solutions can be seen by inspec- tion the problem of the solution of the reduced equation is completed. Frequently one solution may be found by inspection, and it is therefore important to see how much this contributes toward effecting the solution. If y is a solution of the reduced equation, make the substitution y — yz. The derivatives of y may lie obtained by Leibniz's Theorem (§ 8). As the formula is linear in the derivatives of z, it follows that the result of the substitution will leave the equation linear in the new variable z. Moreover, to collect the coefficient of z itself, it is necessary to take only the first term yf>z in the expansions for the derivative y (l) . HenCe («° + X&1" " ,} + • • • + X n _ x y\ + X nVl ) z = is the coefficient of z and vanishes by the assumption that y is a solu- tion of the reduced equation. Then the equation for z is p o ^0 + 7 y„-D + . . . + p n _ iZ n + p^^ = Q . (23) 212 DIFFERENTIAL EQUATIONS and if z' be taken as the variable, the equation is of the order n — 1. It therefore appears that the knowledge of a solution y reduces the order of the equation by one. Now if ?/.,, y , ■ • ■ , y p were other solutions, the derived ratios (23') yj w ' * Wx/ would be solutions of the equation in z' ; for by substitution, v = y x «, = !/,> y = Vr-z = Uv • • • j y = y x * P _i = y P are all solutions of the equation in y. Moreover, if there were a linear relation C x z\ -f- C 2 .~ 2 + • • • + C p _ 1 z' p _ 1 = connecting the solutions ;:•, an integration would give a linear relation c V/ 2 + c itt + • • • + Cp-iy- + ^ = ° connecting the ^» solutions y { . Hence if there is no linear relation (of which the coefficients are not all zero) connecting the p solutions y { of the original equation, there can be none connecting the p — 1 solutions z[ of the transformed equation. Hence a knowledge of p solutions of the original reduced equation gives a new reduced equation of which p — \ solutions are known. And the process of substitution may be continued to reduce the order further until the order n — p is reached. As an example consider the equation of the third order (1 - x) y'" + (x 2 -Y)y"— x-y'+ *V = 0. Here a simple trial shows that x and e x are two solutions. Substitute y = c'z, y' = e x (z + z / ), y"-e-{z^-lz' + z"), y"' = e x (z + 3z' + 3z" + z'"). Then (1 - x) z'" + (x 2 - 3 x + 2) z" + (x 2 - 3 x + 1) z' = is of the second order in z'. A known solution is the derived ratio (x/e* - )'. z' = (xe~ ■')' = e- x (1 — x) . Let z' = e~ x (1 — x) w. From this, z" and z'" may be found and the equation takes the form du/ 2 (1 — ./■) w" + (1 + x) (x — 2) v/ = or = xdx dx ■ w' X — 1 This is a linear equation of the first order and may be solved. log w" = I x 2 — 2 log (x — 1) + C or w' — C\t l - x \x — 1)--. Hence w = C, fd x \x — l)~-dx + C 2 , ^=C 1 /(^)7ei-(x-l)- 2 ( C Zx) 2 + C 2 ^+C 3 , y = C *z = C 1 e x f(^ ) Cti '\x - l)- 2 (dx) 2 + C 2 x + C 3 e*. ADDITIONAL ORDINARY TYPES 243 The value for y is thus obtained in terms of quadratures. It may be shown that in cast' the equation is of the nth degree with p known solutions, the final result will call for p(n — p) quadratures. 105. If the general solution y = C l >/ 1 + C. ,//., -f- • • • + C n y n of the reduced equation has been found (called the complementary function for the complete equation), the general solution of the complete equation may always be obtained in terms of quadratures by the important and far- reaching method of the variation of constants due to Lagrange. The question is : Cannot functions of x be found so that the expression y = C/.r) y x + ( './.-• ) //., + • ■ ■ + C H (x) y n (24) shall be the solution of the complete equation '.' As there are n of these functions to be determined, it should be possible to impose n — 1 condi- tions upon them and still find the functions. Differentiate y on the supposition that the C's are variable. !/' = <\>A + C 2 y'. 2 +■■■ + C n y' n + ili r \ + V£'. 2 + • • • + y n C n . As one of the conditions on the C's suppose that yiCi + y*c: 2 + ••■ + y„c; = o. Differentiate again and impose the new condition yiC' 1 + y,C' ii + ... + y' ii C' n =0, so that y" = C\y'( + C. 2 y. 2 -\ \- C H ;£. The differentiation may be continued to the (n — 1 )st condition /: - ' 1)( "n = o, and y (n - ]) = <\:i["' u + < V/.i" _1) -\ h < 'j/i" _1) - Then y (l,) = < \y{"> + CV/a° H h r „//I" ) Now if the expressions thus found for y, //, //". •••, y° l ~ l) , y°° be substituted in the complete equation, and it be remembered that y , .'/.,• • ' ■ ? .'/„ are solutions of the reduced equation and hence give when substituted in the left-hand side of the equation, the result is y?- l) c-'\ + » { r l)( ", + ■■■ + !i\r" )( "n = r- Hence, in all, there arc n linear equations vS'\ +y*c* +■■■ + !//': =0, y[c\ +{,:/•: +■■■ + u',<" n = o, (25) 244 DIFFERENTIAL EQUATIONS connecting the derivatives of the Cs ; and these may actually be solved for those derivatives which will then be expressed in terms of x. The Cs may then be found by quadrature. As an example consider the equation with constant coefficients (D 3 4- I)) y — sec x with y — G x + C a cos x + C 3 sin x as the solution of the reduced equation. Here the solutions y x , y 2 , y 3 may be taken as 1, cosx, sinx respectively. The conditions on the derivatives of the C 1? s become by direct substitution in (25) C'j + cosxC +sinxC'g =0, — sinxC^ +C0SXC3 =0, — cos xC' 2 — sinxC' 3 = secx. Hence G\ = sec x, ('', = — 1, C' s = — tan x and C x = log tan (\ x + J tt) + c x , C, 2 = — x + c 2 , C 3 = log cos x + r, . Hence ?/ = c t + log tan (| x + |- 7r) + (c 2 — x) cos x + (c 3 + log cos x) sin x is the general solution of the complete equation. This result could not be obtained by any of the real short methods of $S 96-97. It could be obtained by the general method of § 95, but with little if any advantage over the method of variation of constants here given. The present method is equally available for equations with variable coefficients. 106. Linear equations of the second order are especially frequent in practical problems. In a number of cases the solution may be found. Thus 1° when the coefficients are constant or may be made constant by a change of variable as in Ex. 7, p. 222, the general solution of the reduced equation may be written down at once. The solution of the complete equation may then be found by obtaining a particular integral / by the methods of §§95-97 or by the application of the method of variation of constants. And 2° when the equation is exact, the solution may be had by integrating the linear equation (19) of § 103 of the first order by the ordinary methods. And 3° when one solution of the re- duced equation is known (§ 104), the reduced equation may be com- pletely solved and the complete equation may then be solved by the method of variation of constants, or the complete equation may be solved directly by Ex. 6 below. Otherwise, write the differential equation in the form tPu (III /r% ^ -,,+ 1' f + Qi/ = R. (26) ax ax The substitution y = uz gives the new equation d-z /2du \dz 1 „ R (l.i- \ 11 dx I il.r 11 11 If u be determined so that the coefficient of .-:' vanishes, then «=e-lS**< and l ^.J Q -\ t If.-\A z=iR(i \S^. (27) ax" \ J, ax 4 / ADDITIONAL ORDINARY TYPES 245 Now 4° if Q — \ P' — \ P- is constant, the new reduced equation in (27) may be integrated : and 5° if it is k/x 2 , the equation may also be integrated by the method of Ex. 7, p. 222. The integral of the com- plete equation may then be found. (In other cases this method may be useful in that the equation is reduced to a simpler form where solu- tions of the reduced equation are more evident.) Again, suppose that the independent variable is changed to z. Then #y z» + Pz' dy Q R dz" 2 + z" dz + z'* y -z* ( ^ 8) Now 6° if z n = ± Q will make z" + Pz' = kz' 2 , so that the coefficient of dy/dz becomes a constant /.-, the equation is integrable. (Trying if z'' 2 = ± Qz 2 will make z" + Pz' = kz' 2 /:: is needless because nothing in addition to 6° is thereby obtained. It may happen that if z be deter- mined so as to make z" -4- 1''-' = 0, the equation will be so far simpli- fied that a solution of the reduced equation becomes evident.) ^ , d 2 y 2 du a 2 , . . Consider the example 1 1 y = 0. Here no solution is apparent. dx 2 x dx x 4 Hence compute Q — \ P' — I P 2 . This is « 2 /x 4 and is neither constant nor propor- tional to 1/x 2 . Hence the methods 4° and 5° will not work. From z" 2 = Q = a 2 /x* or z' = a/x 2 , it appears that z" + Pz' = 0, and 0° works ; the new equation is d 2 y . , a — + y = with z = dz 2 x The solution is therefore seen immediately to be y = C\ cosz — C 2 sin z or y = C\ cos (a/x) + C 2 sin (a/x). If there had been a right-hand member in the original equation, the solution could have been found by the method of variation of constants, or by some of the short methods for finding a particular solution if R had been of the proper form. EXERCISES 1. If a relation C\y x + C„y. 2 + • • • + C„y„ — 0, with constant coefficients not all 0. exists between n functions y x , y n , • • • , y n of x for all values of x. the functions are by definition said to be linearly dependent ; if no such relation exists, they are said to be linearly independent. Show that the nonvanishing of the Wronskian is a criterion for linear independence. 2. If the general solution y = C\y l + C.,y., + • • • + C„y„ is the same for X o2 /00 + X l2 /<»-D + ■ ■ ■ + X n y = and P o2 /C) + P^C— « + ■ ■ • + P n y = 0, two linear equations of the nth order, show that y satisfies the equation (X,P - X Q P t ) y(» -i) + • • • + (X n P - X P n ) y = of the (n — l)st order: and hence infer, from the fact that y contains n arbitrary constants corresponding to n arbitrary initial conditions, the important theorem: If two linear equations of the ?, then the solution is obtainable in terms of quadratures. Show that Qilh + P1P2 + lh ( k = X i an < 1 ( h'k + lh'l-i = x -i ■ In this manner integrate the following equations, choosing p x and p., as factors of X and determining q x and r/„ by inspection or by assuming them in some form and applying the method of undetermined coefficients : (a) xy" + (1 - x) y -y = e*, (p) 3 xhj" + (2 - (i x 2 ) if - 4 = 0, (7) 3x 2 y"+(2 + Gx-Gx 2 )y / -4y = 0, (5) (x 2 - \)y"- (3x + 1) y'- x (x-l)y = 0. ( e ) axy" + (3 a + bx) y' + 3 by = 0, ( f ) xy" - 2 x (1 + x) y' + 2 (1 + x) y = x 3 . 11. Integrate these ecpiations in any manner: (a) y" - - _ y '+■■ —— y = 0, (/3) y" - - 2/ + a- + - )y = 0, Vx 4x x \ x-/ ADDITIONAL ORDINARY TYPES 247 (7) V" + V' tan x + y cos" 2 x = 0, (5) y"— 2\n \y'+ In 2 — 2— \y = e tt *, ( e) (1 - x 2 ) y" - xy' - c 2 y = 0, (f ) (a 2 - x 2 ) y" - 8 xy' - 12 y = 0, , v „ ! /2 , \ ,^ „ 9- 4a; , 6-3x (17)2/ + -71 rl/ = e a - + logx , (#) 2/ - — — -y + 2/ = 0, x 2 log x \x / 3 — x 3 — x ( £ ) ?/' + 2 x- V - h, 2 */ = 0, (k) y" -4xy' + (4 x 2 - 3) y = e 3 ' 2 , (X) «/" + 2 ny' cot ?ix + (m- — n 2 ) y = 0, (/x) y" + 2 (x- 1 + L'x-" 2 ) y' + Ax~±y = 0. 12. If y 1 and ?/ 2 are solutions of y" + ~Py' + R = 0, show by eliminating Q and integrating that ,, y{y-i - 2/o2/i = Ce J "What if C = ? If C ^ 0, note that y x and ?/j cannot vanish together ; and if y x {a) = y x (b) = 0, use the relation (y„y[) a : (2/22/1)5 = &>0 to show that as ?/j a and ?/j 6 have opposite signs, 2/ 2a and 2/95 have opposite signs and hence 2/ 2 (£) = where a < £ < 6. Hence the theorem : Between any two roots of a solution of an equation of the second order there is one root of every solution independent of the given solution. What conditions of continuity for y and if are tacitly assumed here ? 107. The cylinder functions. Suppose that C n (x) is a function of x which is different for different values of n and which satisfies the two equations d In C n -,(x) - C n+1 (x) = 2 - C n {x), C.^Cr) + C n+1 (x) = — C n (x). (29) Such a function is called a cylinder function and the index n is called the order of the function and may have any real value. The two equa- tions are supposed to hold for all values of n and for all values of x. They do not completely determine the functions but from them follow the chief rules of operation with the functions. For instance, by addi- tion and subtraction, CJx) = C n _,(,•) - I C H (x) = I C n (x) - C n +1 (x). (30) Other relations which are easily deduced are D x [^C n {ax)-] = ax-C^^ax), 1 >.,{.<-" C Jax)] = - ax~ n C n+l (x), (31) D x [Jc n ( V^)] = I V^./^C n _ 1 ( V^), (32) c 'o(*) = - f \(:'')> C -n(*) = (- l) n C n (x), n integral, (33) C n (x)K(x) - C' n (x)K n (x) = C n+1 (x)K n (x) - C n (x)K n+1 (x) = ±, (34) where C and K denote any two cylinder functions. The proof of these relations is simple, but will be given to show the use of (29). In the first case differentiate directly and substitute from (29). D x [x n C n {ax)~\ aD ax C n (ax) + - C n (ax) aC n -i(ax) — a — C„(ax) + - C„(ax) . ax X 248 DIFFERENTIAL EQUATIONS The second of (31) is proved similarly. For (32), differentiate. " -I » _i w i ~ D x [x 2 C„(Vax)] = - »x- C„(Vax)+ x- - -i '- 7J .— C„(Vax) LVux yV „(2 V- 6c|)]. * If n is not integral, both ?i! and (n + i) ! must be replaced (§ 147) by T(n + 1) and r(»+i + l). ADDITIONAL ORDINARY TYPES 251 The value of y may be found by substitution and use of (29). n n bc/n) — A3 „(2x 2 V— bc/n) bc/n) + A3 a (2x 2 V— bc/n) where A denotes the one arbitrary constant of integration. It is noteworthy that the cylinder functions are sometimes expressible in terms of trigonometric functions. For when n = \ the equation (35) has the integrals y = A sinx + B cos a; and y = x*[AC\{x) + BC_ i(x)]. Hence it is permissible to write the relations x 2 Cj (x) = sin x, x ^ C'_ i (x) = cos x, (45) where C is a suitably chosen cylinder function of order \. From these equations by application of (29) the cylinder functions of order p + J, where p is any integer, may be found. Now if Riccati's equation is such that b and c have opposite signs and a/n is of the form p + |, the integral (44) can be expressed in terms of trigonometric functions by using the values of the functions G . i just found in place of the J"'s. Moreover if b and c have the same sign, the trigonometric solution will still hold formally and may be converted into exponential or hyperbolic form. Thus Riccati's equation is integrable in terms of the elementary functions when a/n = p + \ no matter what the sign of be is. EXERCISES 1. Prove the following relations: (a) 4 C'; = C n _ 2 - 2 C n + C n + 2 , (/3) xC n =2{n+ 1) C, +i - xG n + 2 , ( 7 ) 2 3 (?;" = C„ _ 3 - 3 C n _i + 3 C +i - C„ + 3 , generalize, (5) xC n = 2(n + 1) C„ +i-2 (n + 3) C n + 3 + 2 (n + 5) C„ + 6 - xC n + 6 . 2. Study the functions defined by the pair of relations F n _ x (x) + F n + l (x) = 2^-F n (x), F„_i(x) - F„ +1 (x) = 2 ^„(x) ax x especially to find results analogous to (30)-(35). 3. Use Ex. 12, p. 247, to obtain (34) and the corresponding relation in Ex. 2. /(lx 1- BI n . Xn^ll 5. Write out five terms in the expansions of 7 , I 1 , I_ a , J Q , J x . R 1 6. Show from the expansion (42) that 4 l\ -Ji (x) = - sin x. \X 2 X 7. From (45), (29) obtain the following : i sinx i /3 A . 3 x^ Cs (x) = cos x, x 2 (7.5 (x) = 1 1 sin x cos x, 2 X 2" \X 2 / X i^, / . cosx l _ . x 3 . /3 A x?C_ s (x) = — sin x , xa C_ 5 (x) = - sin x + ( 1 ) cos x. - X 2 x \X 2 } 252 DIFFERENTIAL EQUATIONS - 2 — dx = -^ + 6-4 + 6-8 | —^— • x 3 x- ! X 4 «/ x- J 9. Suppose C„(x) and K n (x) so chosen that ^4 = 1 in (34). Show that y = A C n (x) + BK„{x) + L \x n {x) f -^ dx - C„(x) f ^^ dxl is the integral of the differential equation x-y" + xy' + (x 2 — n 2 )?y = Lx~ 2 . 10. Note that the solution of Kiccati's equation has the form y= i i ! C ] + A i g ^% ' and show that -; V + P (x) y. + Q (x) y 2 = E (x) F(x) + AG (x) dx will be the form of the equation which has such an expression for its integral. 11. Integrate these equations in terms of cylinder functions and reduce the results whenever possible by means of Ex. 7 : (a) xy' - 5 y + y 2 + x 2 = 0, (/3) xy' - 3 ?/ + ?/ 2 = x 2 . (7) 2/" + ye 2x = 0, (5) xV + nxy' + (6 + ex 2 »') y = 0. 12. Identify the functions of Ex. 2 with the cylinder functions of ix. 13. Let (x 2 - 1) P; = (n + 1) (P„ + 1 - xP„), P' n +1 = xP' n + (n + 1) P n (46) be taken as defining the Legendre functions P„(x) of order n. Trove (a) (x 2 - 1) P; = n (xP n - P n _i), (/3) (2 n + 1) xP n = (n + 1) P„ +i + « P„ _i , (7) (2 n + 1) P H = P' n +1 - P' n _ x , (5) (1 - x 2 ) P;; - 2 xP; + n (n + 1) P n = 0. ^4 ^4 14. Show that P n q n — P n Q n = and P n Q n +i — P H + i +i M ri+i _ 7>+i( m< ») = uJ)» +hi» + 2(n + l)r/J»«» + ?! (n + 1) 7;»-i»", J> +l u n+l = I)»I_) u n+l - 2 ( }l + 1) Pn(xU») = 2 (?l + 1) xZ>"!<» + 2 H (?l + 1) P" "hi". Hence show that the derivative of the second equation and the eliminant of D n - 1 ic n between the two equations give two equations which reduce to (46) if _, , 1 d" , „ (When n is integral these are P„(x) = — (x 2 — l) n . i T , . ,' . , 2" • n '. dx" ^Legendre s polynomials. 16. Determine the solutions of Ex. 13 (5) in series for the initial conditions (a) P,(0) = 1, P'JO) = 0, (/3) P„(0) = 0. P'JO) = 1. 17. Take P (l — 1 and /', = x. Show that these are solutions of (46) and compute P.,. /':.. P 4 from Ex. 13 (13). If x = cos 0. show P 2 = f cos 2 (9+ i, 7 1 ,. = % cos 0(9 + fcostf, P 4 = |Jcos4 + f£cos2 + &. d 18. Write Ex. 13 (5) as [(] - x-) 7'„] + n(n + 1) P„ = and show dx r +i r +i r d (1 - j- 2 ) P' d (1 - J 2 ) P' I [»i(m+l)-n(n + l)]J P n P m dx = J ^ \P m — "- - P H — \dx. ADDITIONAL ORDINARY TYPES 253 Integrate by parts, assume the functions and their derivatives are finite, and show I fjjPmd-C = 0, if 'i ^ ?»• 19. By successive integration by parts and by reduction formulas show J-i 2 2 »(?i!) 2 ./_i dx" dx» 2»-nlJ-i /» +1 2 and I P^dx = — — . n integral. J_x 2)1 + 1 r \\ r +l d"(x" — \\n 20. Show f x'*P H dx= ( x>» — =0, if m + • • • . 2!2!3!3!4! By assuming w = a,x -f <(.,.<.'' J + • •'• , determine the «"s and hence obtain and (A -f 7? logx) 7 () 4- w> is then the complete solution containing two constants. As AI Q is one solution. 7Jlogx- /,, + w is another. From this second solution for »i = 0. the second solution for any integral value of n may be obtained by differ- entiation ; the work, however, is long and the result is somewhat complicated. CHAPTER X DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES 109. Total differential equations. An equation of the form P (x, y, z) dx + Q (x, y, z) dy + R (x, y, z) dz = 0, (1) involving the differentials of three variables is called a total differen- tial equation. A similar equation in any number of variables would also be called total; but the discussion here will be restricted to the case of three. If definite values be assigned to x, y, z, say a, b, c, the equation becomes Adx + Bdy + Cdz = A (x - a) + B (y - b) + C (z - c) = 0, (2) where x, y, z are supposed to be restricted to values near a, b, c, and represents a small portion of a plane passing through (a, b, c). From the analogy to the lineal element (§ 85), such a portion of a plane may be called & planar element. The differential equation therefore repre- sents an infinite number of planar elements, one passing through each point of space. Xow any family of surfaces F(x, y, z) = C also represents an infinity of planar elements, namely, the portions of the tangent planes at every point of all the surfaces in the neighborhood of their respective points of tangency. In fact dF = F' x dx + F' y dy + F'Az = (3) is an equation similar to (1). If the planar elements represented by (1) and (3) are to be the same, the equations cannot differ by more than a factor fi(x, y, z). Hence F' x = fiP, F' y = fiQ, F' z = fiR. If a function F(x, y, z) = C can be found which satisfies these condi- tions, it is said to be the integral of (1), and the factor /x (s. y, z) by which the equations (1) and (3) differ is called an integrating factor of (1). Compare § 91. It may happen that /x = 1 and that (1) is thus an exact differential. In this case the conditions P' y =Qx, Q: = n' y , K = p:, W 251 MORE THAN TWO VARIABLES 255 which arise from F'J y = F' y ' x , Fy Z = F'Jy, F zx = F xz , must be satisfied. Moreover if these conditions are satisfied, the equation (1) will be an exact equation and the integral is given by P (as, y, z) dx -f / Q (a? , y, z) dy + I R (x , y , z) da = C, where x , y , z may be chosen so as to render the integration as simple as possible. The proof of this is so similar to that given in the case of two variables (§ 92) as to be omitted. In many cases which arise in practice the equation, though not exact, may be made so by an obvious integrating factor. As an example take zxdy — yzdx + x 2 dz = 0. Here the conditions (4) are not fulfilled but the integrating factor l/x' 2 z is suggested. Then xdy — ydx dz , (y + — = d( J - + \ogz x- z \x is at once perceived to be an exact differential and the integral is y/x + log z — C. It appears therefore that in this simple case neither the renewed application of the conditions (4) nor the general formula for the integral was necessary. It often happens that both the integrating factor and the integral can be recognized at once as above. If the equation does not suggest an integrating factor, the question arises, Is there any integrating factor ? In the case of two variables (§ 94) there always was an integrating factor. In the case of three variables there may be none. For cii cP ,, cix cQ J J du cQ „ du dR du cR „ cu cP F' r = R^ + u — = F: = P^ + u — : ex ex dz cz R, P, Q. If these equations be multiplied by A', P, Q and added and if the result be simplified, the condition "\ + e (|K_|PW§£_|i\ = (5) Cz cy ) \cx cz J \cy exj v J is found to be imposed on P, Q, R if there is to be an integrating fac- tor. This is called the condition of integr ability. For it may be shown conversely that if the condition (5) is satisfied, the equation may be integrated. Suppose an attempt to integrate (1) be made as follows : First assume that one of the variables is constant (naturally, that one which will 256 DIFFERENTIAL EQUATIONS make the resulting equation simplest to integrate), say z. Then Pdx + Qdy = 0. Now integrate this simplified equation with an inte- grating factor or otherwise, and let F(x, //, '-) = (■-) be the integral, where the constant C is taken as a function <£ of z. Next try to deter- mine <£ so that the integral F(x, y, '-) = <£('-') will satisfy (1). To do this, differentiate ; F' x dx + Fydij + Fjlz = d. Compare this equation with (1). Then the equations* F' x = XP, F' v = XQ, (/•': - XII) dz = t/cji must hold. The third equation (F' z — XR) dz = d may be integrated provided the coefficient S = F' z — Xll of dz is a function of z and <£, that is, of :: and F alone. Tins is so in case the condition (5) holds. It therefore appears that the integration of the equation (1) for which (5) holds reduces to the succession of two integrations of the type discussed in Chap. VIII. As an example take (2x 2 + 2xi/ + 2xz' 2 + 1)dx + dy + 2zdz = 0. The condition (2 xr + 2 xy + 2 xz 2 + 1) + 1 (- 4 xz) + 2 z (2 x) = of integrability is satisfied. The greatest simplification will be had by making x constant. Then dy + 2 zdz — and y + z 2 =

    ; or — (2 x 3 + 1 + 2 x0) dx = d'p or + 2 x^tZx = - (2 x 2 + l)dx. This is the linear type with the integrating factor c x ~. Then c<'\d

    /> = — f c- r "(2 x' 2 + 1 ) dx + C. Hence y + z 2 + e-* 2 J e a "(2x 2 + l)dx = C'c-' 2 or c' J () is satisiied, it is necessary to show that when the condition is satisfied the coefficient S = F'„ — \/i' is a function of z and F alone. Let it be regarded as a function of x, /•', z instead of x, y, z. It is necessary to prove that the derivative of 8 by x when F and z are constant is zero. By the formulas for change of variable (•>''n,z \(J''r,z \cl'/ ex \(i//,;z \cF/ XtZ cy * Here the factor X is not an integrating factor of (1), but only of the reduced equation /'•+ Qdy 0. MOKE THAN TWO VARIABLES 257 But F' = \P and F' = \Q, and hence Q ( ( - ) - P ( C - S ) = Q ( — ) . \cx/y, s \cy/ x ,z \cx/ FiZ \cx/, /tZ cx\cz J czcx cx cz ex Hence (^ = X (^-^Wp--£-, \CX/ ;/ ,z \£z cx/ CZ CX and H .X^-^ + Q^-Z^. \cy/x,z \8z By J cz dy \8x/ Ut ~ \cylx,z L \C3 ex/ \c// cz/J L cx cijA and g(£) = xf«(?-a + p(?-£S\+J.(^_?)l \Bx/ Ft ~ L \cs cx/ W cz/ \cx cy/J _ E rexQ_exp-| ( L ex dy J where a term has been added in the first bracket and subtracted in the second. Now as X is an integrating factor for Pdx + Qdy, it follows that (XQ)^. = (XP)' ; and only the first bracket remains. By the condition of integrability this, too, vanishes and hence S as a function of x, F, z does not contain x but is a function of F and z alone, as was to be proved. 110. It lias been seen that if the equation (1) is integrable, there is an integrating factor and the condition (5) is satisfied; also that con- versely if the condition is satisfied the equation may be integrated. Geometrically this means that the infinity of planar elements defined oy the equation can be grouped upon a family of surfaces F(x, y, z) = C to which they are tangent. If the condition of integrability is not satis- fied, the planar elements cannot be thus grouped into surfaces. Never- theless if a surface G(x, y, z) = be given, the planar element of (1) which passes through any point (,r , y , z ) of the surface will cut the surface G = in a certain lineal element of the surface. Thus upon the surface G (x, y, z) = there will be an infinity of lineal elements, one through each point, which satisfy the given equation (1). And these elements may be grouped into curves lying upon the surface. If the equation (1) is integrable, these curves will of course be the intersections of the given surface G = with the surfaces F = C defined by the integral of (1). The method of obtaining the curves upon G(x, //, z) = which are the integrals of (1), in case (5) does not possess an integral of the form F(x, y, z) = C, is as follows. Consider the two equations Pdx + Qdy + Rdz = 0, G' x dx + G'jhj + G'Jz = 0, of which the first is the given differential equation and the second is the differential equation of the given surface. From these equations 258 DIFFERENTIAL EQUATIONS one of the differentials, say dz, may be eliminated, and the correspond- ing variable z may also be eliminated by substituting its value obtained by solving G (x, y, z) — 0. Thus there is obtained a differential equa- tion Mdx + Ndy = connecting the other two variables x and y. The integral of this, F(x, y) = C, consists of a family of cylinders which cut the given surface G = in the curves which satisfy (1). Consider the equation ydx + xdy — (x + y + z) dz = 0. This does not satisfy the condition (5) and hence is not completely integrable ; but a set of integral curves may be found on any assigned surface. If the surface be the plane z = x + y, then ydx + xdy — (x + y + z) dz = and dz = dx + dy give (x + z) dx + (y + z) dy = or (2 x + y) dx + (2 y + x) dy = by eliminating dz and z. The resulting equation is exact. Hence x 2 + xy + y 2 — G and z = x + y give the curves which satisfy the equation and lie in the plane. If the equation (1) were integrable, the integral curves may be used to obtain the integral surfaces and thus to accomplish the complete integration of the equa- tion by Mayer's method. For suppose that -F^x, y,z) = G were the integral surfaces and that F(x, y, z) — F(0, 0, z ) were that particular surface cutting the z-axis at z . The family of planes y — Xx through the z-axis would cut the surface in a series of curves which would be integral curves, and the surface could be regarded as generated by these curves as the plane turned about the axis. To reverse these considerations let y = Xx and dy = \dx ; by these relations eliminate dy and y from (1) and thus obtain the differential equation Mdx + Ndz = of the intersections of the planes with the solutions of (1). Integrate the equation as/(x, z, X) = C and determine the constant so that/(x, z, X) =/(0, z , X). For any value of X this gives the intersection of F(x, y, z) = F(0, 0, z () ) with y — Xx. Now if X be eliminated by the relation X — y/x, the result will be the surface fix, z, -J =/IO, z , -J, equivalent to F(x, y, z) = F(0, 0, z ), which is the integral of (1) and passes through (0, 0, z ). As z is arbitrary, the solution contains an arbitrary constant and is the general solution. It is clear that instead of using planes through the z-axis, planes through either of the other axes might have been used, or indeed planes or cylinders through any line parallel to any of the axes. Such modifications are frequently necessary owing to the fact that the substitution /(0, z , X) introduces a division by or a log or some other impossibility. For instance consider y 2 dx + zdy — ydz = 0, y = Xx, dy = \dx, \ 2 x 2 dx + Xzdx — \xdz = 0. Then \dx + — — - = 0, and Xx =/(x, z, X). x 2 x But here /(0, z , X) is impossible and the solution is illusory. If the planes (y — 1) = Xx passing through a line parallel to the z-axis and containing the point (0, 1, 0) had been used, the result would be dy = Xdx, (1 + \x) 2 dx + Xzdx — (1 + Xx) dz = 0, MORE THAN TWO VARIABLES 259 , , Xzdx — (1 + \x)dz . . z ,. % . or dx -\ i - — = 0, and x — =/(x, z, X). (1 + Xx) 2 ' 1 + \x V ' ' ' Hence x = — z n or x = — z n = C, 1 + Xx ° y ° is the solution. The same result could have been obtained with x = Xz or y = X (x — a). In the latter case, however, care should be taken to use/(x, z, X) =/(«, s , X). EXERCISES 1. Test these equations for exactness ; if exact, integrate ; if not exact, find an integrating factor by inspection and integrate : (a) (y + z) dx + (z + x)dy + (x + y) dz = 0, (/3) y 2 dx + zdy - ydz = 0, (7) xdx + ydy — Va 2 — x 2 — y 2 dz = 0, (5) 2 z (dx — cZy) + (x — y) dz = 0, (e ) (2 x + y' 1 + 2 xz) (Zx + 2 xyefy + x 2 cZz = 0, (f) z.yrtx = zxdy + ?/' 2 tZz, (7,) X (y-l)(z- l)dx + y(z-l)(x- l)dy + z (x - 1) (y - 1) dz = 0. 2. Apply the test of integrability and integrate these: (a) (x 2 - y 2 — z 2 ) dx + 2 xydz/ + 2 xzdz = 0, (j3) (x + y 2 + z 2 + 1) (Zx + 2 ydy + 2 ztZz = 0, (7) (y + a) 2 dx + zdy = [y + a)dz, (S) (1 — x 2 — 2 y 2 z) dz = 2 xztZx + 2 yz 2 dy, ( e ) x 2 cZx 2 + y 2 dy 2 — z 2 (Zz 2 + 2 xydxdy = 0, (f) z(xdx + ytZi/ + zcZz) 2 = (z 2 - x 2 - y 2 ) (xdx + ydy + zdz)dz. 3. If the equation is homogeneous, the substitution x = uz, y = vz, frequently shortens the work. Show that if the given equation satisfies the condition of inte- grability, the new equation will satisfy the corresponding condition in the new variables and may be rendered exact by an obvious integrating factor. Integrate : (a) [y 2 + yz) dx + (xz + z 2 ) dy + (y 2 - xy) dz = 0, (p) (x 2 y — y 3 — y 2 z) dx + (xy 2 — x 2 z — x 3 ) dy + (xy 2 + x 2 y) dz = 0, (7) (Z/' 2 + yz + z 2 ) dx + (x 2 + xz + z 2 ) dy + (x 2 + xy + y 2 ) dz = 0. 4. Show that (5) does not hold ; integrate subject to the relation imposed : (a) ydx + xdy — (x + y + z) dz = 0. x + y + z = k or y = kx, (/3) c (xdy + ydy) + Vl — a 2 x 2 — b 2 y 2 dz = 0, « 2 x 2 + b 2 y 2 + c 2 z 2 = 1, (7) dz =■ aydx + bdy, y — kx or x 2 + y 2 + z 2 — 1 or y =f(x). 5. Show that if an equation is integrable, it remains integrable after any change of variables from x, y, z to u, v, w. 6. Apply Mayer's method to sundry of Exs. 2 and 3. 7. Find the conditions of exactness for an equation in four variables and write the formula for the integration. Integrate with or without a factor : (a) (2 x + y 2 + 2 xz) dx + 2 xydy + x 2 dz + du = 0, (/3) yzudx + xzudy + xyudz + xyzdu =0, (7) (y + z + u) (Zx + (x + z + u) dy + (x + y + u) dz + (x + y + z) iZu = 0, (5) u (y + z) cZx + u (y + z + 1) dy + udz - (y + z)du = 0. 8. If an equation in four variables is integrable, it must be so when any one of the variables is held constant. Hence the four conditions of integrability obtained by writing (5) for each set of three coefficients must hold. Show that the conditions 260 DIFFERENTIAL EQUATIONS are satisfied in the following cases. Find the integrals by a generalization of the method in the text by letting one variable be constant and integrating the three remaining terms and determining the constant of integration as a function of the fourth in such a way as to satisfy the equations. (a) z(y + z) dx + z(u — x) dy + y (x - u) dz + y (y + z) du = 0, (j3) uyzdx + uzx log xdy + uxy log xdz — xdu = 0. 9. Try to extend the method of Mayer to such as the above in Ex. 8. 10. If G(x, y, z) = a and II (x, y. z) — b are two families of surfaces defining a family of curves as their intersections, show that the equation {G'yK - G' z n' y )dx + (cur, - (;'ji: : )dy + (a'n; - a'ji'^dz = o is the equation of the planar elements perpendicular to the curves at every point of the curves. Find the conditions on G and II that there shall be a family of sur- faces which cut all these curves orthogonally. Determine whether the curves below have orthogonal trajectories (surfaces) ; and if they have, find the surfaces : (a) y = x + . (f) x- + 2 //'-' + Sz- = y> -)' d~ = •' ( •'"' y> z ) ( 6 ) in the two dependent variables // and z and the independent variable ./■ constitute a set of simultaneous equations of the first order. It is more customary to write these equations in the form j}x ' / .'/ »> dt dt) dt- V dt dt d?r /dd>\ 2 I dr dd>\ Id/ ,dd>\ ( dr dd> 1? - '■(*) " " ('•' * Jt ' T,)' rdt^^r 9 ['■' +'Tt'dt especially those of the second order like these, are of constant occur- rence in mechanics ; for the acceleration requires second derivatives with respect to the time for its expression, and the forces are expressed in terms of the coordinates and velocities. The complete integration of such equations requires the expression of the dependent variables as functions of the independent variable, generally the time, with a num- ber of constants of integration equal to the sum of the orders of the equations. Frequently even when the complete integrals cannot be found, it is possible to carry out some integrations and replace the given system of equations by fewer equations or equations of lower order containing some constants of integration. Xo special or general rules will be laid down for the integration of systems of higher order. In each case some particular combinations of the equations may suggest themselves which will enable an integration to be performed.* In problems in mechanics the principles of energy, momentum, and moment of momentum frequently suggest combinations leading to integrations. Thus if x" = X, y"=Y, x" = Z, where accents denote differentiation with respect to the time, be multi- plied by dx, (///, dx and added, the result x"dx + y"dy + z"dz = Xdx + Ydy + Zdz (11) contains an exact differential on the left ; then if the expression on the right is an exact differential, the integration i_0" + if + x' 2 ) = J Xdx + Ydy + Zdx + C (11') * It is possible to differentiate the given equations repeatedly and eliminate all the dependent variables except one. The resulting differential equation, say in .<• and f. may then lie treated by the methods of previous chapters ; but this is rarely sueeessf ul except when the equation is linear. 204 DIFFERENTIAL EQUATIONS can be performed. This is the principle of energy in its simplest form If two of the equations are multiplied by the chief variable of the other and subtracted, the result is yx"-xy»=yX-xY (12) and the expression on the left is again an exact differential ; if the right-hand side reduces to a constant or a function of t, then yx'-xy'=ff(t) + C (12') is an integral of the equations. This is the principle, of moment of momentum. If the equations can be multiplied by constants as Ix" + my" + nz" = IX + m Y + nZ, (13) so that the expression on the right reduces to a function of t, an inte- gration may be performed. This is the principle of momentum. These three are the most commonly usable devices. As an example : Let a particle move in a plane subject to forces attracting it toward the axes by an amount proportional to the mass and to the distance from the axes; discuss the motion. Here the equations of motion are merely d 2 x , d-i/ , d 2 x , iJ-i/ m — = — kmx, in — - = — kmy or — = — kx, — = — kv. dt 2 dt* dt 2 dt 2 Then dx \ /. = — — , ( /- = = 0. //- \dt! in 2 r r dt \ dt (<■- ) MOKE THAX TWO VARIABLES 2G5 The second integrates directly as r 2 d/dt = h where the constant of integration h is twice the areal velocity. Now substitute in the first to eliminate 0. d 2 r It- _ r h 2 _ d 2 r _ r _ A'''\"'_ r ~ r , dl 2 )'•> in 2 r> dt 2 m 2 \dt/ rrfi Now as the particle is projected perpendicularly to the radius, dr/dt = at the start when ;• =v?u/i. Hence the constant C is li/m. Then dr „ . r 2 dd> ,, . ~vmhdr dt and = dt give - — = j^. ft_4 * , ,, Hence Vwi/i-*/— = + C or = -^— — ^-. V r 2 h r 2 Inn mh Now if it be assumed that = at the start when r = \mh, we find G = 0. Hence r 4 = is the orbit 1 + 0- To find the relation between and the time, r 2 dd) = hdt or — — — dt or £ = mtan- 1 rf>, 1 + 0~ if the time be taken as t = when — 0. Tims the orbit is found, the expression of as a function of the time is found, and the expression of r as a function of the time is obtainable. The problem is completely solved. It will be noted that the constants of integration have been determined after each integration by the initial conditions. This simplifies the subsequent integrations which might in fact be impossible in terms of elementary functions without this simplification. EXERCISES 1. Integrate these equations: . dx dy dz . dx d>/ dz (a) = - =— , yz xz xy , . dx di/ dz (y) — = — = —, xz yz xy dx dy dz (0 - ^ = -,-,-.7' y ■'• 1 + z- 2. Integrate the equations: bz — cy ex — az ay — bx dx dy dz dx dy dz 03) -.-- — 7, = - n — = — ; — - (y) - y 2 x 2 x 2 y 2 z 2 dx dy dz (5) — = — = — ■ — , yz xz x + y /« (? ''' a v - 1 3 y + 4 z dz 2y + bz' , v fa dy dz x 2 + y 2 2 xy xz + yz y + z x + z x + y dx dy ._ dz dx dy dz ifx - 2 x* 2 y* - x 3 y z {x 3 - y 3 )' x (y - z) y (z - x) z (x - y) . dx dy dz , , dx — dt/ dz (O — -- — o: = —y-r—.r, = —r^> — : ^ > (v) x(y 2 -z 2 ) y(z 2 -x 2 ) z{x 2 -y 2 ) x(y 2 -z 2 ) y{z 2 + x 2 ) z{x 2 + y 2 ) dx dy dz , . dx dy dz ,, — = — — = = dt, (t ) - — = = — - = dt. y - z x + y x + z y — z x + y + t x + z + t 266 DIFFERENTIAL EQUATIONS 3. Show that the differential equations of the orthogonal trajectories (curves of the family of surfaces F (x, y, z) = C are dx-.dy.dz = F' x : F' : F' z . Find the curves which cut the following families of surfaces orthogonally : (a) a 2 x 2 + b 2 y 2 + c 2 z 2 = C, (j8) xyz = C, (7) y 2 = Cxz, (8) y = x tan (z + C), (e) y — x tan Cz, (f) z = Cxy. 4. Show that the solution of dx : dy : dz — X : Y ': Z, where X. Y, Z are linear expressions in x, y, z, can always be found provided a certain cubic equation can be solved. 5. Show that the solutions of the two equations — + T(ax + by) = T 1 , ^ + T(a'x + b'y) = T 2 , where T, T v T 2 are functions of t, may be obtained by adding the equation as j (x + ly) + \T(x+ ly) = T 1 + IT„ after multiplying one by I, and by determining X as a root of X 2 - {a + b') \ + ab'— a'b = 0. 6. Solve : (a) t — + 2 (x - y) = t, t -^ + x + 5 y = P, dt dt (/3) tdx = (t—2 x) dt, tdy = (tx + ty + 2 x - t) dt, . , Idx mdy ndz dt (7) — = — — . m.11 (y — z) nl (z — x) Im (x — y) t 7. A particle moves in vacuo in a vertical plane under the force of gravity alone. Integrate. Determine the constants if the particle starts from the origin with a velocity V and at an angle of a degrees with the horizontal and at the time t = 0. 8. Same problem as in Ex. 7 except that the particle moves in a medium which resists proportionately to the velocity of the particle. 9. A particle moves in a plane about a center of force which attracts proportion- ally to the distance from the center and to the mass of the particle. 10. Same as Ex. 9 but with a repulsive force instead of an attracting force. 11. A particle is projected parallel to a line toward which it is attracted with a force proportional to the distance from the line. 12. Same as Ex. 11 except that the force is inversely proportional to the square of the distance and only the path of the particle is wanted. 13. A particle is attracted toward a center by a force proportional to the square of the distance. Find the orbit. 14. A particle is placed at a point which repels with a constant force under which the particle moves away to a distance n where it strikes a peg and is deflected off at a right angle with undiminished velocity. Find the orbit of the subsequent motion. 15. Show that equations (7) may be written in the form drxF = 0. Find the condition on F or on X. Y. Z that the integral curves have orthogonal surfaces. MORE THA^s TWO VARIABLES 267 113. Introduction to partial differential equations. An equation which contains a dependent variable, two or more independent varia- bles, and one or more partial derivatives of the dependent variable with respect to the independent variables is called a partial differential equation. The equation is clearly a linear partial differential equation of the first order in one dependent and two independent variables. The discussion of this equa- tion preliminary to its integration may be carried on by means of the concept of planar elements, and the discussion will immediately suggest the method of integration. When any point (x Q , y , s fj ) of space is given, the coefficients P, Q, R in the equation take on definite values and the derivatives p and q are connected by a linear relation. Xow any planar element through (x , y n , z Q ) may be considered as specified by the two slopes p and q ; for it is an infinitesimal portion of the plane z — z Q = p (x — x ) + q (y — y ) in the neighborhood of the point. This plane contains the line or lineal element whose direction is dx : dy : dz = P:Q:R, (15) because the substitution of P, Q, R for dx = x — x , dy = y — y , dz = z — z in the plane gives the original equation Pp + Qq = R. Hence it appears that the planar elements defined by (14), of which there are an infinity through each point of space, are so related that all which pass through a given point of space pass through a certain line through that point, namely the line (15). Xow the problem of integrating the equation (14) is that of grouping the planar elements which satisfy it into surfaces. As at each point they are already grouped in a certain way by the lineal elements through which they pass, it is first advisable to group these lineal elements into curves by integrating the simultaneous equations (15). The integrals of these equations are the curves defined by two families of surfaces F(x, y, z) = C x and G(x, y, z) — C n . These curves are called the charac- teristic curves or merely the characteristics of the equation (14). Through each lineal element of these curves there pass an infinity of the planar ele- ments which satisfy (14). It is therefore clear that if these curves be in any wise grouped into surfaces, the planar elements of the surfaces must satisfy (14) : for through each point of the surfaces will pass one of the curves, and the planar element of the surface at that point must there- fore pass through the lineal element of the curve and hence satisfy (14). 268 DIFFERENTIAL EQUATIONS To group the curves F(x, y, z) — C v G(x, y, z) = C 2 which depend on two parameters C v C„ into a surface, it is merely necessary to intro- duce some functional relation C. 1 =/(C , 1 ) between the parameters so that when one of them, as C , is given, the other is determined, and thus a particular curve of the family is fixed by one parameter alone and will sweep out a surface as the parameter varies. Hence to integrate (14), first integrate (15) and then vrite G(x, y, z) = *[F(x, y, *)] or $>(F, G) = 0, (16) where <& denotes any arbitrary function. This will be the integral of (14) and will contain an arbitrary function $. As an example, integrate (y — z)p + (z — x)q = x — y. Here the equations dx dy dz . n „ ., „ _, — = — — = give x 2 + y 2 + z 2 = C v x+y + z = C 2 y — z z — x x — y as the two integrals. Hence the solution of the given equation is x + y + z - 4> (x 2 + y' 1 + z 2 ) or (x 2 + y 2 + z 2 , x + y + z) = 0, where $ denotes an arbitrary function. The arbitrary function allows a solution to be determined which shall pass through any desired curve ; for if the curve be f(x, ?/, z) — 0, g(x, y, z) — 0, the elimination of x, y, z from the four simultaneous equations F(x, y, z) - C' v G (x, y, z) = C 2 , f(x, y, z) = 0, g (x, y, z) = will express the condition that the four surfaces meet in a point, that is, that the curve given by the first two will cut that given by the second two ; and this elimi- nation will determine a relation between the two parameters C 1 and C„ which will be precisely the relation to express the fact that the integral curves cut the given curve and that consequently the surface of integral curves passes through the given curve. Thus in the particular case here considered, suppose the solution were to pass through the curve y = x 2 , z = x ; then x 2 + y 2 + z 2 = C„ x + y + z = C„, y - x 2 , z = x give 2 x 2 + x 4 = C v x 2 + 2 x = C 2 , whence (Cl + 2 C 2 - C^) 2 + 8 C| - 24 C, - 16 <\(\, = 0. The .substitution of C, = x 2 + y 2 4- z 2 and C, = x + // + 2 in this equation will give the solution of (y — z)p+ (z — x)q = x — y which passes through the parabola ,// = ,."' 2 . z — x. 114. It will be recalled that the integral of an ordinary differ- ential equation fQ>',y,y',--- , y ( ' ,y ) = of the ?ith order contains h con- stants, and that conversely if a system of curves in the plane, say /•'(.'•, y, < ', ■ • •, (■„) — 0, contains n constants, the constants may be eliminated from the equation and its first n derivatives with respect to ./•. It has now been seen that the integral of a certain partial differential equation contains an arbitrary function, and it might be MORE THAN TWO VARIABLES 269 inferred that the elimination of an arbitrary function would give rise to a partial differential equation of the first order. To show this, suppose F(x, y, z) = &\_G(x, y, z)~\. Then K + KP = <*>' • (/Ir + r 'J>), K + K'l = *' • (G'y + G' t q) follow from partial differentiation with respect to x and //; and (y.-r/; _ F'/Qp + (JF' X G' Z - F' z G x )q = F' y G' x - F' x G' y is a partial differential equation arising from the elimination of <£>'. More generally, the elimination of n arbitrary functions will give rise to an equation of the nth order; conversely it may be believed that the integration of such an equation would introduce n arbitrary func- tions in the general solution. As an example, eliminate from z = <&(xy) + ♦ (x + y) the two arbitrary func- tions <£ and 4'. The first differentiation gives p — <£' • y + &', q = $' • x + ' + (y — x) and ^ . The general integral of this equation would be z — <&(xy) + & (x + y). A partial differential equation may represent a certain definite type of surface. For instance by definition a conoidal surface is a surface generated by a line which moves parallel to a given plane, the director plane, and cuts a given line, the directrix. If the director plane be taken as z = and the directrix be the ."-axis, the equations of any line of the surface are z = C v y = 'Y> with c 'i = $ ( c 2 ) as the relation which picks out a definite family of the lines to form a particular conoidal surface Hence z = <£>(///./■) may be regarded as the general equation of a conoidal surface of which z = is the director plane and the 2-axis the directrix. The elimination of <£ gives jjx -f- qy = as the differential equation of any such conoidal surface. Partial differentiation maybe used not only to eliminate arbitrary func- tions, but to eliminate constants. For if an equation f(x, //, z, ( \. < '.,) =0 contained two constants, the equation and its first derivatives with respect to x and y wotdd yield three equations from which the constants could 270 DIFFERENTIAL EQUATIONS be eliminated, leaving a partial differential equation F(x, y, z, p, q) = of the first order. If there had been five constants, the equation with its two first derivatives and its three second derivatives with respect to x and y would give a set of six equations from which the constants could be eliminated, leaving a differential equation of the second order. And so on. As the differential equation is obtained by eliminating the constants, the original equation will be a solution of the resulting dif- ferential equation. For example, eliminate from z = A x 2 + 2 Bxy + Cy 2 + Dx + Ey the five con- stants. The two first and three second derivatives are p = 2 Ax + 2 By + X>, q = 2 Bx + 2 Cy + E, r = 2 A, s = 2 B, t = 2 C. Hence z = — \ rx- — \ ty 2 — sxy + px + qy is the differential equation of the family of surfaces. The family of surfaces do not constitute the general solution of the equation, for that would contain two arbitrary functions, but they give what is called a complete solution. If there had been only three or four constants, the elimination would have led to a differential equation of the second order which need have contained only fine or two of the second derivatives instead of all three ; it would also have been possible to find three or two simultaneous partial differential equations by differentiating in different ways. 115. If f(x, y,z,C v Cj=0 and F(x, y, z, p, q) = (17) are two equations of which the second is obtained by the elimination of the two constants from the first, the first is said to be the complete solu- tion of the second. That is, any equation which contains two distinct arbitrary constants and which satisfies a partial differential equation of the first order is said to lie a complete solution of the differential equa- tion. A complete solution has an interesting geometric interpretation. The differential equation F = defines a series of planar elements through each point of space. So does f(x, y, z, C v C'.,) = 0. For the tangent plane is given by C.r with f(x , y , * , c v r 2 ) = o as the condition that C and C., shall be so related that the surface passes through (./■.., y , .-■/). As there is only this one relation between the two arbitrary constants, there is a whole series of planar elements through the point. As f(x, //. z, C . C.,) = satisfies the differential equa- tion, the planar elements defined by it are those defined by the differen- tial equation. Thus a complete solution establishes an arrangement of the planar elements defined by the differential equation upon a family of surfaces dependent upon two arbitrary constants of integration. ) c !/.o c -\c l MORE THAN TWO VARIABLES 271 From the idea of a solution of a partial differential equation of the first order as a surface pieced together from planar elements which satisfy the equation, it appears that the envelope (p. 140) of any family of solutions "will itself be a solution ; for each point of the envelope is a point of tangency with some one of the solutions of the family, and the planar element of the envelope at that point is identical with the planar element of the solution and hence satisfies the differential equa- tion. This observation allows the general solution to be determined from any complete solution. For if in f(x, y, z, C' v ('.,) = any relation C 2 = 3>(C' 1 ) is introduced between the two arbitrary constants, there arises a family depending on one parameter, and the envelope of the family is found by eliminating C 1 from the three equations cf (l<& df c.-*VJ, W + ^tF^' /=a (18) As the relation C„ = ^(C.) contains an arbitrary function 4>, the result of the elimination may be considered as containing an arbitrary func- tion even though it is generally impossible to carry out the elimination except in the case where <£ has been assigned and is therefore no longer arbitrary. A family of surfaces f(x, >/, z, C , C.,) = depending on two param- eters may also have an envelope (p. 139). This is found by eliminat- ing Cj and C 2 from the three equations f(x,ij,z,C l ,CJ = 0, ^ = 0, ^ = 0. This surface is tangent to all the surfaces in the complete solution. This envelope is called the singular solution of the partial differential equation. As in the case of ordinary differential equations (§ 101), the singular solution may be obtained directly from the equation;* it is merely necessary to eliminate p and q from the three equations F(», y, .=, p, .,) = 0, ¥ = 0, ^ = 0. The last two equations express the fact that F( p. (./'- + y- + z-), (/?) * (./■- + //-. z - sy) = 0, (7) z — 4> (x + y) 4- ^ (x — y), (0) z - c""4> (x — y), (e) z = //- + 2 (x- 1 + log y), (f) * ^ , V ~, i\ = 0. MORE THAN TWO VARIABLES 278 8. Find the differential equations of these types of surfaces: (or) cylinders with generators parallel to the line x = az, y = bz, (/3) conical surfaces with vertex at (a. b, c), (7) surfaces of revolution about the line x :y :z = a:b :c. 9. Eliminate the constants from these equations: (a) z = U + a) {y + b), (/3) a {x 2 + y°-) + hz 2 = 1, (7) (x - a) 2 + (y- hf + (z- cf = 1, (8) (x - a) 2 + (y - 6j 2 + (2 - c) 2 = d 2 , (e) .4x 2 + Bxy + Cy 2 + JJxz + Eyz = z 2 . 10. Show geometrically and analytically that F(x, y, z) + aG (x, y, z) — b is a complete solution of the linear equation. 11. How many constants occur in the complete solution of the equation of the third, fourth, or nth order'.' 12. Discuss the complete, general, and singular solutions of an equation of the first order F(x, y, z, u, u' x , u, n u' s ) = with three independent variables. 13. Show that the planes z = ax + by + ('. where a and h are connected by the relation F(a, b) = 0. are complete solutions of the equation F(p. q) = 0. Integrate : (a) pq = 1, (/3) q = p 2 + 1, (7) p 2 + q 2 = m a , ( 5 ) pq = k, (e) k log q + p = 0, ( f) 3 j, 2 - 2 r/ 2 = 4pg, and determine also the singular solutions. 14. Note that a simple change of variable will often reduce an equation to the type of Ex. 13. Thus the equations F(f,|) = 0, F { xp,q) = 0, F(^,fj=0, with z = c c/ , a; = c r \ z = e z ', j = c r ', ?/ = e^', take a simpler form. Integrate and determine the singular solutions: (a) q-z\ P£, (/3) x 2 p 2 + y 2 q 2 = z 2 , (7) z = 7/7. (5) 7 = 2 ,jp 2 . (e) (p - y) 2 + (7 - z) 2 = 1, (f) Z = p m 7 m . 15. What is the obvious complete solution of the extended Clairaut equation z = xp + yq +f(p. 7) ? Discuss the singular solution. Integrate the equations: (a) z = xp + yq + Vp 2 + q 2 + 1, (p) z = xp + yq + (p +_7) 2 , (7) z -xp + yq + pq, (8) z = xp + yq — 2 Vpq. 116. Types of partial differential equations. In addition to the linear equation and the types of Exs. 13-15 above, there are several types which should be mentioned. Of these the first is the general equation of the first order. If F(x, y, z, p, q) = is the given equation and if a second equation (.r, //, z, p, y, «) = 0, which holds simultane- ously with the first and contains an arbitrary constant can be found, the two equations may be solved together for the values of p and 7, and the results may be substituted in the relation dz = pdx + y"7/ to give a total differential equation of which the integral will contain the con- stant a and a second constant of integration b. This integral will then 274 DIFFERENTIAL EQUATIONS be a complete integral of the given equation ; the general integral may then he obtained by (18) of § 115. This is known as Charpit's method. To find a relation $ = differentiate the two equations F(x, y, z, p, q) = 0, <$> (x, y, z, p, q, a) = (19) with respect to x and y and use the relation that dz be exact. , d n , da , dp , dq p dx dx , dp , da K + Ki + K^ + Kf^o, , d i) , d'F' 0. (20) r/ Multiply by the quantities on the right and add. Then Now this is a linear equation for and is equivalent to dp _ dg _ d.r _ dy _ dz K +i Any integral of tins system containing p or q and a will do for <&, and the simplest integral will naturally be chosen. As an example take zp (x + y) + p(q — P) — z 1 = 0. Then Charpit's equa- tions are dp dq dx — zp + p 2 {x + y) zp — 2zq + pq (x + y) 2p — q — z(x + y) dy _ dz — p 2p' 2 — 2pq — pz (x + y) How to combine these so as to get a solution is not very clear. Suppose the sub- stitution 2 = e 2 ', p = e z 'p', q = e z 'q' be made In the equation. Then p' {x + )/) + p' iff - p') — 1 = is the new equation. For this Charpit's simultaneous system is dp' _ dq' dx dy dz p' p' 2 p' — q' — (x + y) — p' 2p" 2 — 2pq — p' (x + y) The first two equations give at once the solution dp' = dq' or q' = p' + a. Solving P' (* + y) + p' ('/ — P') — 1 = and 2 ~" D „ + ■ • • + «•„ -i B*D; ~ X + «» D?) * = R (■>', //)• (22) Methods like those of § 95 may be applied. Factor the equation. a Q (D x - ai D y ) (D x - a. 2 D y ) ■ • • (D x - a n D y ) r: = n (,; y). (22') Then the equation is reduced to a succession of equations Dx z-aD y z = R(x,y), each of which is linear of the first order (and with constant coefficients). Short cuts analogous to those previously given may be developed, but will not be given. If the derivatives are not all of the same order but the polynomial can be factored into linear factors, the same method will apply. For those interested, the several exercises given below will serve as a synopsis for dealing with these types of equation. There is one equation of the second order,* namely 1 d 2 u __ g 2 ?t dht dSt V~ cf- dx 2 ci/' 1 dz 1 * This is one of the important differential equations of physics : other important equa- tions and methods of treating them are discussed in Chap. XX. 276 DIFFERENTIAL EQUATIONS which occurs constantly in the discussion of waves and which has there- fore the name of the wave equation. The solution may be written down by inspection. For try the form u (x, y, z, t) = F(ax + by + cz - 17) + G (ax + by + cz + 17). (24) Substitution in the equation shows that this is a solution if the relation a' 1 + Ir + c 2 = 1 holds, no matter what functions F and G may be. Note that the equation ax + by + cz — T7 = 0, q — 4> («.,) is a first or intermediary integral of the given equation, the general integral of which may lie found by integrating this equation of the firs! order. If the two factors arc distinct, it may happen that the two systems which arise may both be integrated. Then two first integrals w, = $(».,) and r, = ^ (i\.) will be found, and instead of integrating one of these equations it may be better to solve both for p and (y — x) (27) is a first integral. This is linear and may be integrated by dx dy dz T , dx dz ■ x + y x + y x + y$(y — x) — 2z A\ (A", — 2x) — 2z This equation is an ordinary linear equation in z and x. The integration gives 2.r p 2x K^ze A 'i = / e A "i * (K 1 — 2 x) dx + K 2 . _£*_ /- ?.r Hence (x + 2/) zef- + •"— j e A 'i 4>(7v'j — 2 x) dx = Jf 2 = ^ ( A' t ) = ^ (x + ?/) is the general integral of the given equation when Jv'j has been replaced by x + y after integration, — an integration which cannot be performed until is given. The other method of solution would be to use also the second system containing dy + dx = instead of dy — dx = 0. Thus in addition to the iirst integral (27) a second intermediary integral might be sought. The substitution of dy + dx = 0, y + x =C, in (A) gives G', (dp + dq) + 4pdx = 0. This equation is not integrable, because dp + dq is a perfect differential and pdx is not. The combination with dz = pdx + qdy = (p — q)dx docs not improve matters. Hence it, is impossible to determine a second intermediary integral, and the method of completing the solution by integrating (27) is the only available method. Take the equation ps — qr = 0. Here S = p, It = — q, T= V — 0. Then — qdy 2 — pdxdy = or dy = 0, pdx + qdy — and — dx + qdy are not very satisfactory for obtaining an intermediary integral a l = < M«. ) ), although p = ^(z) is an obvious solution of the first set. It is better to use a method adapted to this special equation. Note that and ±(9) = o gives q = f(y). p) p.- c.e\i By (11), p. 124, 1 = - (~) ■ then ~ =-f((l) P x cy L dy and x = - ffdf) dy + * (2) = * (y) + * (z). 278 DIFFERENTIAL EQUATIONS EXERCISES 1. Integrate these equations and discuss the singular solution: (a) p* + q* = 2 x, (/3) (p 2 + q 2 )x = pz, (7) (p + q) {px + qy) = 1, (5) pq = px + qy, (e) p 2 + q 2 = x + y, (f) xp 2 — 2zp + xy = 0, (V) q 2 = z 2 (p - q), (0) q (P 2 z + q 2 ) = l, («) p (1 + (y) = (1 + axB y + \ a 2 x 2 B y + . . •) (y) = (y + ax), 03) ■— — — = e«**>y — = e«*i>y v R (|, y) d| = f*B (£, y + ax-a^)di. D x — cr J^ •/ «/ 10. Prove that if [(A- - ff 1 A,) ra i • ■ • (A - a /J),,)"*] z = 0, then z = * u (?/ + <*!*) + z 12 (2/ + c^z) + ■ • • + x m i- 1 \ mi {y + a l x) -\ + <*>«( t2 (y + ai-x) + ■ ■ ■ + x"'k - 1 km k (y + «&), where them's are all arbitrary functions. This gives the solution of the reduced equa- tion in the simplest case. What terms would correspond to (D x — al) y — /3) m z = 0? 11. Write the solutions of the equations (or equations reduced) of Ex.8. 12. State the rule of Ex. 9 (7) as: Integrate E (x, y — ax) with respect to x and in the result change y to y + ax. Apply this to obtaining particular solutions of Ex. 8 (5). (e). (77) with the aid of any short cuts that are analogous to those of Chap. VIII. 13. Integrate the following equations: (a) (]>-,. - I)':,, + D y - 1) z = c« is (x + 2 y) + e". (/3) x 2 r 2 + 2 xys + yH* = x 2 + y 2 , (7) (If + D. r// + I) y - 1) z = sin (jc + 2y), (5) r-t- 3p + 3q= e* + *», (e) (Dl-2D x Dl + Dl)z = x-2, (f) r-t + p + 3q-2z = e*-y-x 2 y, { n ) ( D 2 - B x B y - 2 D y + 2 D x + 2 B y ) z = e 2 * + »* + sin (2 x + y) + xy. 14. Try Monge's method on these equations of the second older : {a) q-r - 2pqs + pH = 0. (j3) r - aH = 0, (7) r + s = - p. ( 5 ) q(l + q)r-{p + q + 2 pq) s + p (1 + p) « = 0, (e) a; 2 r + 2 .n/.v + ?/'^ = 0, ( t ) (/' + c, l)->' ~ 2 ( b + C 'J) < ( ' + cp) s + (a + cp)H = 0, (77) r + fc« 2 !! = 2 a.s-. If any simpler method is available, state what it is and apply it also. 280 DIFFERENTIAL EQUATIONS 15. Show that an equation of the form Br + Ss + Tt + U (rt — s 2 ) = V neces- sarily arises from the elimination of the arbitrary function from M x > y-> z > Pi v) =/[" 2 ( x ' Vi z i Pi ?)]■ Note that only such an equation can have an intermediary integral. 16. Treat the more general equation of Ex. 15 by the methods of the text and thus show that an intermediary integral may be sought by solving one of the systems Udy + \ Tdx + \ Udp = 0, Udx + \Rdy + \ Udq = 0, Udx + \Jldy + KUdq = 0, Udy + \. 2 Tdx + \.,Udp = 0, dz = pdx + qdy, dz = pdx + qdy, where X t and X„ are roots of the equation \ 2 (RT,+ UV) + \US + U 2 = 0. 17. Solve the equations : (a) s 2 — rt = 0. (/3) s 2 — rt = a 2 , (y) ar + bs + ct + e (rt — s 2 ) = h, (5) xqr + ypt + xy (s 2 — rt) — pq. PAET III. INTEGRAL CALCULUS CHAPTER XI ON SIMPLE INTEGRALS 118. Integrals containing a parameter. Consider <£ 00 = / f( x > a ) dx > (1) a definite integral which contains in the integrand a parameter a. If the indefinite integral is known, as in the ease / cos axdx = - sin ax, a f'""* axdx = - sin ax a it is seen that the indefinite integral is a function of x and a, and that the definite integral is a function of a alone because the variable x disappears on the substitution of the limits. If the limits themselves depend on a, as in the case I cos axdx = - sin ax a = - (sin a 2 — sin 1), a ' the integral is still a function of a. In many instances the indefinite integral in (1) cannot be found explicitly and it then becomes necessary to discuss the conti- nuity, differentiation, and integration of the function (a) defined by the integral with- out having recourse to the actual evaluation of the integral; in fact these discussions may be required in order to effect that evaluation. Let the limits x Q and x 1 be taken ' as constants independent of a. Consider the range of values x Q =§j x =E x x for x, and let a == a == a be the range of values over which the func- tion 4>(a) is to be discussed. The function f(x, a) may be plotted as the surface z =f(x, a) over the rectangle of values for (x, a). The 281 282 INTEGRAL CALCULUS value (a,) of the function when a = a t is then the area of the section of this surface made by the plane a = a-. If the surface f(x, a) is con- tinuous, it is tolerably clear that the area <£(«) will be continuous in a. The function <£(«) is continuous iff(x, a) is continuous 'm the two varia- bles (x, a). To discuss the continuity of 0(a) form the difference (a + Aa) -

    (a + Aa) - 0(a)] = /"Vfo < r + Aa ) -/(*> «)](Zx < f%dx = e(x 1 - x ). It is therefore proved that the function 0(a) is continuous provided /(x. a) is con- tinuous in the two variables (x. a): for e(x L — x ) may be made as small as desired if e may be made as small as desired. As an illustration of a case where the condition for continuity is violated, take /»i cidx x i ^ 4>(a)=l - — r^tan- 1 - - cot-^a if a j± 0, and 0(O) = O. Jo a 2 + x- a o Here the integrand fails to be continuous for (0. 0): it becomes infinite when (x. a) = (0. 0) along any curve that is not tangent to a — 0. The function (a) is defined for all values of a §= 0. is equal to cot _1 a when a ^ 0. and should there- fore be equal To \tt when a = if it is to be continuous, whereas it is equal to 0. The importance of the imposition of the condition that /(x, a) be continuous is clear. It should not be inferred, however, that the function 0(a) will necessarily be discontinuous when/(x. a) fails of continuity. For instance r l dx 1 / /— 1 0(a) = I — = = -(Va + l-V«), 0(0) = -- 170 Va + x l * This function is continuous in a for all values a 2=0; yet the integrand is dis- continuous and indeed becomes infinite at (0. 0). The condition of continuity imposed on /(x, a) in the theorem is sufficient to insure the continuity of 0(a) but by no means necessary; when the condition is not satisfied some closer exami- nation of the problem will sometimes disclose the fact that

    («) = I /(•*■> a ) ilr , % = a = a i> ( 3 ) , a = a x and the curves x = g (ct), x x = g,Ox), and if the functions <7 («) and g-fa) are continuous. For in this case

    (a) I < e| ;/l (/r) - g {a) | + | /(£„ rt + Art) 1 1 Ag, | + |/(£ , a + Aa) \ \ Ag \, where £ and £ t are values of x between g and g + Ag , and g 1 and f/, + Ag v By taking Art small enough, f/,(rt + Art) — f/,( may be made as small as desired. 119. To find the derivative of a function (/') defined by an integral containing a jjarameter, form the quotient A(f> _ t f> Or + Aa) - (a) Aa 1_ Aa An Xf/jC.t + A«) /-»<7, (a) f(x, a + Aa) d;r - / f(x, a)dx .„(n+A,i) Jy n (; a + Aa) Aa J , . Aa dx + p fix, a c/.'7 + Ar/ n + An) la j Aa The transformation is made by (-33), p. 2."). A further reduction may be made in the last two integrals by (05'), p. 25, which is the Theorem of the Mean for integrals, and the integrand of the first integral may be modified by the Theorem of the Mean for derivatives (p. 7, and Ex. 14, p. 10). Then Aa and a, (a) S7„(<0 f' a {x, a + 6Aa) dx -f(£ , a + An) ^-° + f($ v a + Aa) |j d(j> da C'^cf dg, ,/ r/l J , s en ° 7 ' (a) exists and may be obtained by (4) in case f' a exists and is continuous 284 INTEGRAL CALCULUS in (x, a) and g ('(), ffii a ) are differential )le. In the particular case that the limits g and g are constants, (4) reduces to Leibnh's Rule da /.''. which states that the derivative of a function defined by an integral with fixed limits mug he obtained by differentiating under the sign of integration. The additional two terms in (4), when the limits are varia- ble, may be considered as arising from (66), p. 27, and Ex. 11, p. 30. This process of differentiating under the sign of integration is nf fre'/uent use in evaluating the function 4>( a ) m cases where the indefi- nite integral of f(x, a) cannot be found, but the indefinite integral of f' a can be found. For if («) = f f(x, a) dx, then ^ = £ />/,• = $ (a). Now an integration with respect to a will give <£ as a function of a with a constant of integration which may be determined by the usual method of giving a some special value. Thus 4 ( a ) — I 1 ( ' x > T = 1 dx = I x ' Jo ^o S x da J log,- J Hence -f- = — — r x a +1 = — — r , <£ (a) = log (a + 1) + C. da a + 1 | a -f- 1 But (0) = | da = and <£ (0) = log 1 + C. Jo r 1 x a — i Hence d>(a)= | dx = log (a -f- 1). Jo 1(, y * In the way of comment upon this evaluation it may he remarked that the func- tions (x a — l)/logx and x a are continuous functions of (x, a) for all values of x in the interval 0==.e;s=l of integration and all positive values of a less than any assigned value, that is, 0^ a = K. The conditions which permit the differen- tiation under the sign of integration are therefore satisfied. This is not true for negative values of a. When a<0 the derivative ./•« becomes infinite at (0. 0). The method of evaluation cannot therefore be applied without further examination. As a matter of fact — 1, and it would be natural to think that some method could be found to justify the above formal evaluation of the integral when — 1 < a = K (see Chap. XIII). To illustrate the application of the rule for differentiation when the limits are functions of a, let it be required to differentiate 1 (70 /•'" a-" — 1 a a — 1 0(a) = I fix. - = I x"rix + «/a log X (la J a log a log a ON SIMPLE INTEGRALS 285 d da a a + 1 — 1 M \ a- a — a a — a + 1 • a+ll J logorL J This formal result is only good subject to the conditions of continuity. Clearly a must be greater than zero. This, however, is the only restriction. It might seem at first as though the value x = 1 with logx = in the denominator of (x a — l)/logx would cause difficulty ; but when x = 0, this fraction is of the form 0/0 and has a finite value which pieces on continuously with the neighboring values. 120. The next problem would be to find the integral of a function defined by an integral containing a parameter. The attention will be restricted to the case where the limits x and x 1 are constants. Consider the integrals I (a) da =| • I f(x, a) dx ■ da, where a may be any point of the interval a == a =§ a of values over which (f> (a) is treated. Let (a)= f ■ f f(x, a)da-dx. Then &(a) = j '• — j f(x,a)da-dx= f l f(x,a)dx = $(a) J., n Ja Q J:r n by (4'), and by (66), p. 27; and the differentiation is legitimate if f(x, a) be assumed continuous in (x, a). Now integrate with respect to a. Then f $'( cc ) = $(or)-$(o: )= f (a)da. Ja Ja a But &(a^)— 0. Hence, on substitution, (a)da= J ■ j f(x,a)dx-da. (5) Hence 1 appears the rule for integration, namely, integrate under the sign, of integration. The rule lias here been obtained by a trick from the previous rule of differentiation; it could be proved directly by considering tin 1 integral as the limit of a sum. It is interesting to note the interpretation of this integration on the figure, p. 281. As (a) is the area of a section of the surface, the product (f>(a)da is the infinitesimal volume under the surface and included between two neighboring planes. The integral of 4>(a) is therefore the volume * under the surface and boxed in by the four * For the "volume of a solid with parallel buses and variable cross section'' see Ex. 10, p. 10, and § 35 with Exs. 20, 2o thereunder. 286 INTEGRAL CALCULUS planes a = a , a — a, x = x , x = x . The geometric significance of the reversal of the order of integrations, as XX, s*a x ^a x r»x, ■ J f(- r , a) da ■ dx = J • I /(■'', a) dx • da, is in this case merely that the volume may be regarded as generated by a cross section moving parallel to the «a-plane, or by one moving parallel to the .~.r-plane, and that the evaluation of the volume may be made by either method. If the limits x Q and x x depend on a, the integral of <£(/<) cannot be found by the simple rule of integration under the sign of integration. It should be remarked that integration under the sign may serve to evaluate functions defined by integrals. As an illustration of integration under the sign in a ease where the method leads to a function which may be considered as evaluated by the method, consider C 1 , 1 r h , . 7 r b da .6 + 1

    — "■' '■ — e~~ 3* /i re (5) I e _arx sin?rcxdx = — - to show | — dx = tan -1 — — tan -1 — , Jo a 2 + m 2 Jo xcscmx m in n* dx -n- ,. . r n dx r n , ft— cosx (e) I — — = ■ — ; to find I , I log — , Jo a— cosx Va 2 — 1 ° ( sJo where in the second expression the subscripts P and N denote that the integrals are evaluated for the parabola and semicubical parabola. As a change in the order of the limits changes the sign of the integral, the area may be written ydx + i ydx = - / yl? - / ydx, and is the area bounded by the closed curve formed of the portions of the parabola and semicubical parabola from to 1. In considering the area bounded by a closed curve it is convenient to arrange the limits of the different integrals so that they follow the curve in a definite order. Tims if one advances along P from to 1 and re- turns along N from 1 to 0, the entire closed curve has been described in a uniform direction and the inclosed area has been constantly on the right-hand side ; whereas if one advanced along S from to 1 and ON SIMPLE INTEGRALS 289 returned from 1 to along P, the curve would have been described in the opposite direction and the area would have been constantly on the left-hand side. Similar considerations apply to more general closed curves and lead to the definition : If a closed curve which nowhere crosses itself is described in such a direction as to keep the inclosed area always upon the left, the area is considered as positive ; whereas if the description were such as to leave the area on the right, it would be taken as negative. It is clear that to a person standing in the inclosure and watching the description of the boundary, the descrip- tion would appear counterclockwise or positive in the first case (§ 76). In the case above, the area when positive is / ydx + / ydx = - I ydx sJa pJi Jo (6) r where in the last integral the symbol O denotes that the integral is to be evaluated around the closed curve by describing the curve in the positive direction. That the formula holds for the ordinary case of area under a curve may be verified at once. Here the circuit consists of the con- tour ABB' A 1 A. Then A BX [ydx = f ydx + f ydx + f ydx + f ydx. Jo J. i Jn J n> J A' The first integral vanishes because y = 0, the second and fourth vanish because x is constant and dx = 0. Hence — J ydx = — I ydx- = I ydx. It is readily seen that the two new formulas A = f xdy and A = \ I {xdy - ydx) (7) Jo Jo also give the area of the closed curve. The first is proved as (6) was proved and the second arises from the addition of the two. Any one of the three may be used to compute the area of the closed curve ; the last has the advantage of symmetry and is particularly useful in finding the area of a sector, because along the lines issuing from the origin y : ,r = dy : dx and xdy — ydx = : the previous form with the integrand xdy is advantageous when part of the contour consists of lines parallel to the .''-axis so that dy = Q; the first form has similar advantages when parts of the contour are parallel to the //-axis. 290 INTEGRAL CALCULUS The connection of the third formula with the vector expression for the area is noteworthy. For (p. 175) dk = \ Txdt, A = l I Txdr, Jo and if r = xi + y], dr — idx + ]dy, then A= J rxf/r = J k I (xdy — ydx). Jo Jo The unit vector k merely calls attention to the fact that the area lies in the x//-plane perpendicular to the s-axis and is described so as to appear positive. These formulas for the area as a curvilinear integral taken around the boundary have been derived from a simple figure whose contour was cut in only two points by a line parallel to the axes. The exten- sion to more complicated contours is easy. In the first place note that if two closed areas are contiguous over a part of their contours, the inte- gral around the total area following both contours, but omitting the part in common, is equal to the sum of the integrals. For J PR.SP JPQRP JPR J lisr J PQR J RP J QRSP p[ since the first and last integrals of the four are in oppo- site directions along the same line and must cancel. But the total area is also the sum of the individual areas and hence the integral around the contour PQRSP must be the total area. The for- mulas for determining the area of a closed curve are therefore applicable to such areas as may be composed of a finite number of areas each bounded by an oval curve. If the contour bounding an area be expressed in parametric form as x—f(t), y —

    0(o/ / (O at = if[f(t) vd %,•]■ If, when n becomes infinite so that A.r and Ay each approaches as a limit, the sum a- approaches a definite limit independent of how the individual increments A.r- and Ay ( approach 0, and of how the point (£,, rji) is chosen in its segment of the curve, then this limit is defined as the line integral r (ii) (x,y) AX; X lini o- = f [P (a?, y) dx + Q (x, y) dy-]. cJa, h (12) It should be noted that, as in the case of the line integral which gives the area, any line integral which is to be evaluated along two curves which have in common a portion described in opposite directions may be replaced by the integral along so much of the curves as not repeated ; for the elements of ///, F*dr — Xdx + Ydy, Feos Ods = / (Xdx + Ydy), b Ju. h where X and 3' are the components of the force along the axes. It is readily seen that any line integral may be given this same inter- pretation. If 1=1 Pdc + Qdij, form F = V'i + Q). J (I, h X.i. y /-» .; , y Pdx -\-Qdy = I Fcos 6ds. . h J ii. h To the principles of momentum and moment of momentum ($80) may now be added the principle of work and energy for mechanics. Consider d-r d-x m = F and in — . dx = F>dx = d IT. dl- di- rt /l t7r dx\ 1 d-x r?r \ dx d-x d-x dx Then — (- • -)= \- = - , dt\2dt dlj 2cu- dl 2dl dl- dl- dt d ( l v-\ = ~['dl and d ( - miA = dlj Hence 2 / dt- 1,1, r r -mv 2 mv- = I F-dx = ir 2 2 " Jr.. In words: The change of the kinetic energy \ mv- of a particle moving under the action of the resultant force F is equal to the workdone hy the force, that is, to the line integral of the force along the path. If there were several mutually interacting particles in motion, the results for the energy and work would merely be added as ■2 I mv- — 2 I mi - Q — 2 W. and the total change in kinetic energy is the total work done by all tliG forces. The result gains its significance chiefly by the consideration of what forces may be disregarded in evaluating the work. As dlF=F«dr, the work done will be zero if dx is zero or if F and dx are perpendicular. Hence in evaluating IF, forces whose point of application does not move may be omitted (for example, forces of support at pivots), and so may forces whose point of appli- cation moves normal to the force (for example, the normal reactions of smooth curves or surfaces). When more than one particle is concerned, the work done by the mutual actions and reactions may be evaluated as follows. Let r x . r., be the vectors to the particles and r t — r., the vector joining them. The forces of action and re- action may be written as ± c (r x — r 2 ), as they are equal and opposite and in the line joining the particles. Hence d W - d W\ + d »F 2 = c (r x - r 2 ).dr a - c (r x - r 2 ).dr 2 = c (x x - x.,).d (r t - r 2 ) = \ cd [(r, - r 2 ).(rj - r 2 )] = * cdr\ 2 , where /y, is the distance between the particles. Now dW vanishes whe.i and only when dr v , vanishes, that is, when and only when the distance between tie particles 294 INTEGRAL CALCULUS remains constant. Hence when a system of particles is in motion the change in the total kinetic energy in passing from one position to another is equal to the work done by the forces, where, in evaluating the work, forces acting at fixed points or normal to the line of motion of their points of application, and forces due to actions and reactions of particles rigidly connected, may be disregarded. Another important application is in the theory of thermodynamics. If U, p. v are the energy, pressure, volume of a gas inclosed in any receptacle, and if dl' and dv are the increments of energy and volume when the amount dll of heat is added to the gas, then „ dlf — dU + pdv, and hence 11= I dU + pdv is the total amount of heat added. By taking p and v as the independent variables, 11 = jV-\- dp + (j~+ p) dv \ = f U'(P, ") dp + U I P- <-') dv] . The amount of heat absorbed by the system will therefore not depend merely on the initial and final values of (p, v) but on the sequence of these values between those two points, that is. upon the path of integration in the pu-plane. 123. Let there be given a simply connected region (p. 89) bounded by a closed curve of the type allowed for line integrals, and let P(x, y) and Q(x, y) be continuous functions of (./•, y) over this region. Then if the line integrals from (a, li) to (x, y\ along two paths Xx, y /-» x, >/ Pdx + Qdy = I Pdx + Qdy b rJti.h are equal, the line integral taken around the combined path f ' + f = f ' Pdx + Qdy = vanishes. This is a corollary of the fact that if the order of description of a curve is reversed, the signs of A.'-,- and A//,- and hence of the line integral are also reversed. Also, conversely, if the in- tegral around the closed circuit is zero, the integrals from any point (a, h) of the circuit to any other point (x, //) are equal when evaluated along the two different parts of tht 1 circuit leading from (>/, //) to (,r, >/). The chief value of these observations arises in their application to the case where P and (J happen to lie such functions that the line inte- gral around any and every closed path lying in the region is zero. In this case if (", />) be a fixed point and (./'. //) be any point of the region. the line integral from (". />) to (x, y) along any two paths lying within the region will he the same: for the two paths may lie considered as forming one (dosed path, and the integral around that is zero by hy- pothesis. The value of the integral will therefore not depend at all on OX SIMPLE INTEGRALS 295 the path of integration but only on the final point (x, y) to which the integration is extended. Hence the integral x: \_P (x, y) dx + Q (x, y) dy] = F(x, y), (14) extended from a fixed lower limit (a, />) to a variable upper limit (x, y), must be a function of (x, y). This result may be stated as the theorem : The necessary and suffi- cient condition that the line integral L [P(x,y)dx + Q(x,y)dy] define a single valued function of (x, y) over a simply connected region is that the circuit integral taken around any and every closed curve in the region shall he zero. This theorem, and in fact all the theorems on line integrals, may lie immediately extended to the case of line integrals in space, [P (x, y, z) dx + Q (,; y, z) dy + R (x, y, z) dz]. (15) L If the integral about every closed path is zero so that the integral from a fixed lower limit to a variable upper limit F(x, V)= f ' P(*> V)dx + Q(x, y)dy J a, b defines a function F(x, y), that function has continuous first partial derivatives and hence a total differential, namely, CF CF , — = P. x- = Q, dF = Pdx + Qdii. (16) Cx Cy s \ / To prove this statement apply the definition of a derivative. Xx+ Aj\ y px, y Pdx + Qdy - | Pdx + Qdy . b J a, b dF _. A/' ■7— = lim = lim OX Ax = A./' Aj=0 Xow as the integral is independent of the path, the integral to (x + A./-, //) may follow the same path as that to (x, y), except for the passage from (>. //) to (x + A./', //) which may be taken along the straight line joining them. Then A// = and A.7 A - = A I P (x, y) dx = -r-p(i, V) &x = p (*, y), J .C, 1/ 296 INTEGRAL CALCULUS by the Theorem of the Mean of (Qo 1 ), p. 25. Now when A.r = 0, the value £ intermediate between x and x + A.? will approach x and P ($, y) will approach the limit P(x, y) by virtue of its continuity. Hence AF/Ax- approaches a limit and that limit is P(x, y) = cF/cx. The other derivative is treated in the same way. If the Integrand Pdx + Qdy of a line integral is the total differential dj-' of a single valued function F(x, y), then the integral about any closed circuit is zero ami f Pdx + Qdy = f ' dF = F(x, y) - F(a, b). (17) J a, h J a, b If equation (17) holds, it is clear that the integral around a closed path will be zero provided F(x, y) is single valued; for F(x, y) must come back to the value F(a, b) when (x, y) returns to (a, b). If the function were not single valued, the conclusion might not hold. To prove the relation (17), note that by definition jdF = jPdx + Qdy = lim^ [P(fe, m )Axi + Q(fc, vd^fi] and Ap = P(£,-, w) Ax t + Q(£, V i) Ay,- + e x AXi + e. 2 Ayi, where e t and e., are quantities which by the assumptions of continuity for P and Q may be made uniformly (§ 25) less than e for all points of the curve provided Ax; and Ayi are taken small enough. Then ^ ( PiAxi + QiAyt ) - ^ AFi I < ejj ( | An | + | Ay, |) ; = F(x, y) — F(a. b), the sum 2P,Ar,- + Q t Ayi appr< limV [PiAxt + QiAyt] = f '""pj.r + Qdy = F(x, y) - F(a. b). ^^ J a . b and since SAP- = F(x,y) — F(a. b), the sum 2P,Ar,-f Q,A?A approaches a limit, and that limit is EXERCISES 1. Find the area of the loop of the strophoid as indicated above. 2. Find, from (6), (7), the three expressions for the integrand of the line inte- grals which give the area of a closed curve in polar coordinates. 3. (liven the equation of the ellipse x = a m>t. y = bsin t. Find the total area. the area of a segment from the end of the major axis to a line parallel to the minor axis and cutting the ellipse at a point whose parameter is t. also the area of a sector. 4. Find the area of a segment and of a sector for the hyperbola in its parametric form x — a cosh t. y = b sinht. 5. Express the folium x z + y 3 = Saxy in parametric form and find the area of the loop. 6. What area is given by the curvilinear integral around the perimeter of the closed curve r = a sin 3 J

    I , .10 ^ 1 7 1 - I , i(Zu +i . , \dv \ 8. Compute these line integrals along the paths assigned: >i, l x-ydx + y ? 'dy, y- = x or y - x or y 3 = x 2 , 0, o /•l.l (/3) I (x 2 + ■ subtended by the curve at r — 0. Hence inter that rdlogr rdlogr rdlogr I ds = 2 7r or I ds = or I ds = Jo dn Jo dn Jo dn according as the point r = is within the curve or outside the curve or upon the curve at a point where the tangents in the two directions are inclined at the angle 9 (usually it). Note that the formula may be applied at any point (£, -q) if r 2 = (£ — x)' 2 + (17 — y)' 1 where (1. y) is a point of the curve. What would the inte- gral give if applied to a space curve? 17. Are the line integrals of Ex. 10 of the same type I P(x, y)dx + Q(x, y)dy as those in the text, or are they more intimately associated with the curve ? Cf . § 155. J-.0, 1 /i0,l (x — y)ds, (j3) I xyds along a right line, along a quad- 1,0 J— 1,0 rant, along the axes. 124. Independency of the path. It has been seen that in case the integral around every closed path is zero or in case the integrand Pdx + Qdy is a total differential, the integral is independent of the path, and conversely. Hence if /"*■''•." £/,' ftp F(x,y)=\ Pdx + Qdy, then - = P, - = Q, ' a, h C.r Gil c-F _ cQ c-F _ cP cP _ cQ dxdy C.r dljGX ClJ CiJ c.r provided the partial derivatives P' y and Q' x are continuous functions.* It remains to prove the converse, namely, that: If the two ptarthd derivatives P' y end Q' x are continuous and equal, the integral I Pdx + Qdy with P' y = Q' x (18) J a, h is independent of the path, is zero around a closed path, and the quant it y Pdx + Qdy is a total differentia/. To show that the integral of Pdx + Qdy around a closed path is zero if P' = Q' x , consider first a region II such that any point (./-, //) of it may * See § 52. In particular observe the comments there made relative to differentials which are or which arc not exact. This difference corresponds to integrals which are and which arc not independent of the path. OX SIMPLE INTEGRALS 299 be reached from (ft, b) by following the lines y = b and x = x. Then define the function F(x, //) as P (*, V)=Jp (•'•, V) dx +j J Q O, y) dy (19) for all points of that region 11. Now dx P(r,b) + But Q (■<; //) rfy = dx C.r "'( 1' 1 Q(x. h J = P (.•'■,//) This results from Leibniz's rule (4') of § 119, which may be applied since Q' x is by hypothesis continuous, and from the assumption Q' x = P'y Then g^ P(.r, J) + /*(*> y)-P{x, b) = P(x,y). Hence it follows that, Avithin the region specified, Pdx + Qdy is the total differential of the function F(x, //) defined by (19). Hence along any closed circuit within that region Jl the integral of Pdx + Qdy is the integral of dF and vanishes. It remains to remove the restriction on the type of region within which the integral around a closed path vanishes. Consider any closed path C which lies within the region over which P' and Q' x are equal continuous functions of (x, y). As the path lies wholly within R it is possible to rule 1! so finely that any little rectangle which contains a portion of the path shall lie wholly within R. The reader may construct his own figure, possibly with reference to that of § 128, where a finer ruling would be needed. The path C may thus be surrounded by a zigzag line which lies within R. Each of the small rectangles within the zigzag line is a region of the type above considered and. by the proof above given, the integral around any closed curve within the small rectangle must be zero. Now the circuit C may be replaced by the totality of small circuits consisting either of the perim- eters of small rectangles lying wholly within C or of portions of the curve C and portions of the perimeters of such rectangles as contain parts of C. And if C be so replaced, the integral around C is resolved into the sum of a large number of inte- grals about these small circuits; for the integrals along such parts of the small circuits as are portions of the perimeters of the rectangles occur in pairs with oppo- site signs.* Hence the integral around C is zero, where C is any circuit within R. Hence the integral of Pdx + Qdy from (a, b) to (x, ?/) is independent of the path and defines a function F(x, y) of which Pdx -f Qdy is the total differential. As this function is continuous, its value for points on the boundary of J! may be defined as the limit of F(r. y) as (,r, y) approaches a point of the boundary, and it may thereby be seen that the line integral of (18) around the boundary is also without any fur- ther restriction than that P' and Q' c be equal and continuous within the boundary. * See Ex. 10 above. It is well, in connection with §§ 123-125, to read carefully the work of §§ 44-45 dealing with varieties of regions, reducibility of circuits, etc. 300 INTEGRAL CALCULUS It should be noticed that the line integral f Pdx + Qdy = f P(r, b)dx + C' Q(x, y)dy, J a, b J a Jb (19) when Pdx + Qdy is an exact differential, that is, when P' = Q' x , way be evaluated by the rale given for integrating an exact differential (p. 209), provided the path along y — b and x = x does not go outside the region. If that path should cut out of R, some other method of evaluation would be required. It should, however, be borne in mind that Pdx + Qdy is best integrated by inspection whenever the function F, of which Pdx + Qdy is the differential, can be recognized ; if F is multiple valued, the consideration of the path may be required to pick out the par- ticular value which is needed. It may be added that the work may be extended to line integrals in space without any material modifications. It was seen (§ 73) that the conditions that the complex function F ( x > y) = x (*> y) + /r 0''> y)> * = « + iy, be a function of the complex variable z are X;=-Y' x and x:=y;. (20) If these conditions be applied to the expression (13), Cf(x, //) = f Xdx - Ydy + I I Ydx + Xdy, J Ja. h J-:, b for the line integral of such a function, it is seen that they are pre- cisely the conditions (18) that each of the line integrals entering into the complex line integral shall be independent of the path. Hence the integral of a function of a complex variable is independent of the path- of integration in the complex plane, and the integral around a closed path vanishes. This applies of course only to simply connected regions of the plane throughout which the derivatives in (20) are equal and continuous. If the notations of vectors in three dimensions be adopted, \ Xdx + Ydy + Zdz = jF.t/r, where F = A'i + Fj + Zk, dx = idx + yly + kdz. In the particular case where the integrand is an exact differential and the integral around a closed path is zero, Xdx + ydy + Zdz = F.tfr = dU = dr.\U, OX SIMPLE INTEGRALS 301 where U is the function defined by the integral (for VU see p. 172). When F is interpreted as a force, the function V = — U such that F = — V V or A = — - — > 1 = — -7T- 3 Z = 7— ex o// on is called the potential function of the force F. The negative of the slope of the potential function is the force F and the negatives of the partial derivatives are the component forces along the axes. If the forces are .such that they are thus derivable from a potential function, they are said to be conservative. In fact if i — .dr =- di.VV ' = - dV, dt 2 d 2 x 7H : dt 2 = F = -VI '' r*\ d 2 x m — •< Jr *» 7r — ( i *r - 2 -<) = = l r o - m (It dr I n -1" and 2 dt at m ., 1 x or - i>f + I ! = - n - + I Thus the sum of the kinetic energy $mv 2 and the potential energy V is the same at all times or positions. This is the principle of the conservation of energy for the simple case of the motion of a particle when the force is conservative. In case the force is not conservative the integration may still be performed as (v?-v 2 )=f Tl F.dT = W, where W stands for the work done by the force F during the motion. The result is that the change in kinetic energy is equal to the work done by the force ; but dW is then not an exact differential and the work must not be regarded as a function of (.r. y. z). — it depends on the path. The generalization to any number of particles as in § 123 is immediate. 125. The conditions that P' v and Q' T be continuous and equal, which insures independence of the path for the line integral of Pdx + Qdy, need to be examined more closely. Consider two examples : First | Pdx + Qdy = I ' - dx -\ dy , J J x- + y- x- + y- where + //" x- + y- cP y 2 — x- dQ y' 2 — x- cy (x 2 + y 2 ) 2 ' ex (x 2 + y 2 ) 2 It appears formally that P' — Q' r . If the integral be calculated around a square of side 2 a surrounding the origin, the result is X + « + adx r + " ady r - a — adx r - a — ady _ r + " adx -a x 2 + a 2 J -a a 2 + y 2 J+u x 2 + a 2 «/ + « a 2 + y 2 J- a x 2 + a 2 „ C + a ady . r + " adl- . ir „ . + 2 I — = 4 j — = 4- = 27T5^0. J -a a 2 + y 2 J -a £- + a- 2 302 INTEGRAL CALCULUS The integral fails to vanish around the closed path. The reason is not far to seek, the derivatives P' y and Q' x are not defined for (0, 0), and cannot be so defined as to be continuous functions of (x, y) near the origin. As a matter of fact r ■•'• y — ydx xdy r x > v ,y I — 1 — = I dtan- 1 - = tan- 1 J a, b X- + 1J- X 2 + y 2 Ja,b X and tan _1 (y/x) is not a single valued function ; it takes on the increment 2 it when one traces a patli surrounding the origin (§45). Another illustration may be found in the integral dz r dx + idy _ r xdx + ydy . r — ydx + xdy z J x+ iy J x- + y' 2 J x 2 + y' 2 taken along a path in the complex plane. At the origin z = the integrand \/z becomes infinite and so do the partial derivatives of its real and imaginary parts. If the integral be evaluated around a path passing once about the origin, the result is /i n-, r i i,~i x. v (21) r dz T 1 , „ vl x > v ( ™ - \t log ( X a + y 2 ) + i tan -1 ^ =2; Jq Z \_2 X_]a,b In this case, as in the previous, the integral would necessarily be zero about any closed path which did not include the origin ; for then the con- ditions for absolute independence of the path would be satisfied. Moreover the integrals around two different paths each encircling the origin once would be equal ; for the paths may be considered as one single closed circuit by joining them with a line as in the device (§ 44) for making a multiply connected region simply con- nected, the integral around the complete circuit is zero, the parts due to the description of the line in the two directions cancel, and the integrals around the two given circuits taken in opposite directions are therefore equal and opposite. (Compare this work with the multiple valued nature of log z, p. 161.) Suppose in general that P(oc, //) and Q(x, y) are single valued func- tions which have the first partial derivatives P' and Q' x continuous and equal over a region R except at certain points A, ]>, ■ ■■. Surround these points with small circuits. The remaining portion of 7t is such that P' y and Q' x are everywhere equal and continuous; but the region is not simply connected, that is, it is possible to draw in the region circuits which cannot be shrunk down to a point, owing to the fact that the circuit may surround one or more of the regions which have been cut out. If a circuit can be shrunk down to a point, that is, if it is not inextricably wound about one or more of the deleted portions, the integral around the circuit will vanish; for the previous reasoning will apply. But if the circuit coils about one or more of the deleted regions so that the attempt to shrink it down leads to a circuit which consists of the contours of these regions and of lines joining them, the integral need not vanish ; it reduces to the sum of a number of integrals ON SIMPLE INTEGRALS 303 taken around the contours of the deleted portions. If one circuit can be shrunk into another, the integrals around the two circuits are equal if the direction of description is the same ; for a line connecting the two circuits will give a combined circuit which can be shrunk down to a point. The inference from these various observations is that in a multiply connected region the integral around a circuit need not be zero and the integral from a fixed lower limit (a, l>) to a variable upper limit (,/', ?/) may not be absolutely independent of the path, but may be dif- ferent along two paths which are so situated relatively to the excluded regions that the circuit formed of the two paths from (a, b) to (x, y) cannot be shrunk down to a point. Hence F (x, y) = r V pdx + Qdy, 1>; = Q' x (generally), the function defined by the integral, is not necessarily single valued. Nevertheless, any two values of F(x, y) for the same end point will differ only by a sum of the form F*(x, y) - F~i(x, y) = m x l x + »'i 2 + • ■ where I 1} L 2 , . . . are the values of the integral taken around the con- tours of the excluded regions and where m w w 2 , . . . are positive or negative integers which represent the number of times the combined circuit formed from the two paths will coil around the deleted regions in one direction or the other. 126. Suppose that f(z) = X(x, y) + iY(x, y) is a single valued func- tion of z over a region it surrounding the origin (see figure above), and that over this region the derivative /'(«) is continuous, that is, the relations X' t = — Y' x and X'. = Y' y are fulfilled at every point so that no points of 11 need be cut out. Consider the integral J'— — dz = j r— (dx + id//) (22) over paths lying within R. The function f(z)/z will have a contin- uous derivative at all points of 11 except at the origin z = 0, where the denominator vanishes. If then a small circuit, say a circle, be drawn about the origin, the function f(z)/z will satisfy the requisite condi- tions over the region which remains, and the integral (22) taken around a circuit which does not contain the origin will vanish. The integral (22) taken around a circuit which coils once and only once about the origin will be equal to the integral taken around the 304 INTEGRAL CALCULUS small circle about the origin. Now for the circle, Jq Jo Jo " Jo where the assumed continuity of f(z) makes \r/(z)\ < e provided the circle about the origin is taken sufficiently small. Hence by (21) C£& dz = C f -& dz = 2 irif(O) + £ Jo Jo 1*1- f|*| s f|i|w a .f Jo Jo J(, with ■ 2 7T d6 Hence the difference between (22) and 2 irif(0) can be made as small as desired, and as (22) is a certain constant, the result is I ^-dz = 27rlf(0). (23) A function f(z) which has a continuous derivative /'(*) at every point of a region is said to be analytic over that region. Hence if the region includes the origin, the value of the analytic function at the origin is given by the formula / ^ = 2^ f^**' < 23 ') Jo where the integral is extended over any circuit lying in the region and passing just once about the origin. It follows likewise that if z = a is any point within the region, then Jo where the circuit extends once around the point a and lies wholly within the region. Tins important result is due to Cauchy. A more convenient form of (24) is obtained by letting t = z repre- sent the value of z along the circuit of integration and then writing a = z and regarding z as variable. Hence Cauchy's Integral : f(*) = 2^if£z L J t - ( 25 > Jo This states that if any run/ it be drawn in the region over which f(z) is analytic , the value of f(z) at all points within that circuit may be ob- tained by evaluating Cauchy's Integral (25). Thus f(z) may be regarded ON SIMPLE INTEGRALS 305 as defined by an integral containing a parameter z ; for many pur- poses this is convenient. It may be remarked that when the values of f(z) are given along any circuit, the integral may be regarded as defining /'(.ti) for all points within that circuit. To find the successive derivatives of f(z), it is merely necessary to differentiate with respect to z under the sign of integration. The condi- tions of continuity which are required to justify the differentiation are satisfied for all points z actually within the circuit and not upon it. Then At //i>f a \ / w- (fy •»- >. ^'V. i ex I / i /'(*) a. *r As the differentiations may be performed, these formulas show that an analytic function has continuous derivatives of all orders. The definition of the function only required a continuous first derivative. Let a be any particular value of z (see figure). Then 1 I 1 1 (t — a) — (z — a) t — A z - a (z 1 + + *r- t — a (t af + + (t-ay-^ (z — a) n (t — a)" fit) (t- a y '2ttJ q [ > (t-ay 2™ Jo (*-«)" with ,. 1 MJ^M, t — a t — a Now £ is the variable of integration and z — a is a constant with respect to the integration. Hence /(*)=/(«) + (* - ")/'(«) + ^"97^ /» + ••• + (* - «y (26) (,-l)!^"» +7 - This is Taylor's Formula for a function of a complex variable. 806 INTEGRAL CALCULUS EXERCISES 1. If V' y = Q' x , Q' z ■= R' v , li' x = P' z and if these derivatives are continuous, show that Pdx + Qdy + Rdz is a total differential. 2. Show that | P{x, V, cc)dx + Q(x, y, a)dy, where C is a given curve, C J a, h defines a continuous function of or, the derivative of which may be found by differ- entiating under the sign. What assumptions as to the continuity of P, Q, P' a , Q' a do you make ? • x < 'J xdx + ydy . r *> v — ydx + xdy ,o x' z + y 2 J i,o x' 2 + y' 1 definition of log z, draw paths which make logQ + \ V— 3) = \iri, 2 \ m, — 1| irl „ /-*az r x ■» — y ) and (r, <£) are the component forces resolved along the radius vector and perpendicular to the radius, show that dW = lldr + i^ed

    be conservative. 11. Show that if a particle is acted on by a force R = —/(»') directed toward the origin and a function of the distance from the origin, the force is conservative. 12. If a force follows the Law of Nature, that is, acts toward a point and varies inversely as the square r- of the distance from the point, show that the potential is — k/r. 13. From the results F=-V7 or V = — f F.Jr = i Xdx + Ydij + Zdz show that if V x is the potential of F, and V., of F 2 then V = J\ + V„ will be the potential of F = F t + F 2 , that is, show that for conservative forces the addition of potentials is equivalent to the parallelogram law for adding forces. 14. If a particle is acted on by a retarding force — kv proportional to the velocity, show that R = \ kt- is a function such that cR . cR cR - - = — kv x , — = — kv w , — = — kv z , CV X CVy CIV dW= — kv>th = — k(i\,d.r + r/li/ + v z dz). Here R is called the dissipative function ; show the force is not conservative. 15. Tick out the integrals independent of the path and integrate: (a) J i/zdx + xzdy + xydz, (£) j ydx/z + xdy/z - xydz/z*, (7) J xyz {dx + dy + dz), (5) J li >g {xy) dx + xdy + ydz. 16. Obtain logarithmic forms for the inverse trigonometric functions, analogous to those for the inverse hyperbolic functions, either algebraically or by considering the inverse trigonometric functions as defined by integrals as dz . , r z dz tan r • az . . r z a - J 2 = | — , sin- 1 * = I JO 1 +2" Jo Vl 17. Integrate these functions of the complex variable directly according to the rules of integration for reals and determine the values of the integrals by substitution : (a) f 1+ 'ze**°-dz, (j8) f 2l cos3zdz, (y) f ~' + '(1 + z^dz, Jo Jo J\ (5) ( -=' ( £ ) . - 7=' (f) f -!==■ t/o vi — ^ Ji z\z- — \ " x v 1 + z- In the case of multiple valued functions mark two different paths and give two values. 18. Can the algorism of integration by parts be applied to the definite (or indefi- nite) integral of a function of a complex variable, it being understood that the integral must be a line integral in the complex plane ? Consider the proof of Taylor's Formula by integration by parts, p. 57. to ascertain whether the proof is valid for the complex plane and what the remainder means. 308 INTEGRAL CALCULUS 19. Suppose that in a plane at r = there is a particle of mass in which attracts according to the law F=m/r. Show that the potential is 1"= mdogr, so that F = — VF. The induction or Jiiu of the force F outward across the element ds of a curve in the plane is by definition — Fcos(F, n)ds. By reference to Ex. 16, p. 297, show that the total induction or flux of F across the curve is the line integral (along the curve) — I Fcos(F, n)ds = m | — -^— ds = I — ds; J J dn J dn l r „ ,~ . , L C — ! 7r Jq dn and m = j F cos (F, n) ds = — | — ds, 2 tv Jq 2 7r Jo dn where the circuit extends around the point r = 0, is a formula for obtaining the mass m within the circuit from the field of force F which is set up by the mass. 20. Suppose a number of masses //(, . »*.,. • ■ • . attracting as in Ex. 1!). are situated at points (£ r ij x ), (£._,, rj 2 ), • • ■ in the plane. Let F = F x + F 2 + • ■ • , V=V l + V a + ~-, Vi = rrn log [(& - x)* + ( Vi - y)*]l he the force and potential at (x, y) due to the masses. Show that -i rFcos(F,n)d, = - 1 -V rlId.= s V'ni| = Jf ) 2 77 Jq 2 7r ^-1 Jq dn ^—t where S extends over all the masses and 2' over all the masses within the circuit (none being on the circuit), gives the total mass M within the circuit. 127. Some critical comments. In the discussion of line integrals and in the future discussion of double integrals it is necessary to speak frequently of curves. For the usual problem the intuitive conception of a curve suffices. A curve as ordinarily conceived is continuous, lias a continuously turning tangent line except perhaps at a finite number of angular points, and is cut by a line parallel to any given direction in only a finite number of points, except as a portion of the curve may coincide with such a line. The ideas of length and area are also appli- cable. For those, however, who arc interested in more than the intuitive presentation of the idea of a curve and sonic of the matters therewith connected, the following sections are offered. If 4>(t) and 0(0 are two single valued real functions of the real variable t defined for all values in the interval t = t =§ t v the pair of equations a; = 0(0, V = f(t), t Q ^t^t v (27) will be said to define a curve, if and are continuous functions of t, the curve will be called continuous. If 0(0) = {t), ^ = ^ = t x , be continuous. Let P (x, y) he a continuous function of (x. //). Form the sum ]T P (f ,- . ,.-) AiX = 2} P (fc , 17,-) A,*" - J P (& ■ Vd A/*', (28) where A x x, A £, ■ • • are the increments corresponding to A x t, AJ. ■ ■ ■ . where (£,-. 77,) is the point on the curve which corresponds to some value of t in A,/, where x is assumed to be of limited variation, and where x" and x' are two continuous increas- ing functions whose difference is x. As x" (or x') is a continuous and constantly increasing function of t. it is true inversely (Ex. 10, p. 45) that t is a continuous and constantly increasing function of x" (or x'). As P(x. y) is continuous in (x, y). it is continuous in t and also in x" and x' . Xow let A;t = : then A;x" = and A,.r' = 0. Also imVP,A;/'= C Jl Pdx" and lim V P,A ; x' = C^Pdx'. The limits exist and are integrals simply because P is continuous in x" or in x' . Hence the sunt on the left of (28) has a limit and limV PA ; x = f'Pdx = f ] Pdx" - f'^Pdx' OX SIMPLE INTEGRALS 311 may be defined as the line integral of P along the curve C of limited variation in x. The assumption that y is of limited variation and that Q(x, y) is continuous would lead to a corresponding line integral. The assumption that both x and y arc of limited variation, that is, that the curve is rectifiable, and that P and Q are continuous v:ould lead to the existence of the line integral f' v,, P{x, y)dx + Q(x, y)dy. A considerable theory of line integrals over general rectifiable curves may be con- structed. The subject will not be carried further at this point. 128. The question of the area of a curve requires careful consideration. In the first place note that the intuitive closed plane curve which does cut itself is intui- tively believed to divide the plane into two regions, one interior, one exterior to the curve ; and these regions have the property that any two points of the same region may be connected by a continuous curve which does not cut the given curve, whereas any continuous curve which connects any point of one region to a point of the other must cut the given curve. The first question which arises with regard to the general closed simple curve of page 308 is : Does such a curve divide the plane into just two regions with the properties indicated, that is, is there an interior and exterior to the curve ? The answer is affirmative, but the proof is somewhat difficult — not because the statement of the problem is involved or the proof replete with advanced mathematics, but rather because the statement is so simple and elemen- tary that there is little to work with and the proof therefore requires the keenest and most tedious logical analysis. The theorem that a closed simple plane curve has an interior and an exterior will therefore be assumed. As the functions x(t), y(t) which define the curve are continuous, they are lim- ited, and it is possible to draw a rectangle with sides x = a, x = b. y = r, y = d so as entirely to surround the curve. This rectangle may next lie ruled with a num- ber of lines parallel to its sides, and thus be divided into smaller rectangles. These little rec- tangles may be divided into three categories, those outside the curve, those inside the curve, and those upon the curve. By one upon the curve is meant one which has so much as a single point of its perimeter or interior upon the curve. Let .1. A;. A u , A e denote the area of the large rec- tangle, the sum of the areas of the small rectan- gles, which are interior to the curve, the sum of the areas of those upon the curve, ami the sum of those exterior to it. Of course A=A { + A u + A e . Now if all methods of ruling be considered, the quantities Ai will have an upper frontier L;. the quantities A e will have an upper frontier L e . and the quantities A u will have a lower frontier l„. If to any method of ruling new rulings be added, the quantities A; and A e become A\ and A' e with the conditions A' ; j== Ai, A' e === A,., and hence A' =§ A u . From this it follows that A = L; + l u + L, . For let there be three modes of ruling which for the respective cases .1,-. .1,,. A u make these three quantities differ from their frontiers J.,. L e , l„ by less than J e. Then the superposition of the three systems of rulings gives rise to a ruling fur which A' ; , A' A' must differ from the frontier values by less than 312 INTEGRAL CALCULUS S re, and hence the sum Li + l u + L e , which is constant, differs from the constant A hy less than e, and must therefore be equal to it. It is now possible to define as the (qualified) areas of the rune Li = inner area, l u = area on the curve, i, + l u = total area. In the case of curves of the sort intuitively familiar, the limit l u is zero and Li = A — L e becomes merely the (unqualified) area bounded by the curve. The question arises : Does the same hold for the general curve here under discussion ? This time the answer is negative ; for there are curves which, though closed and simple, are still so sinuous and meandering that a finite area l u lies upon the curve, that is, there is a finite area so bestudded with points of the curve that no part of it is free from points of the curve. This fact again will be left as a statement with- out proof. Two further facts may be mentioned. In the first place there is applicable a theorem like Theorem 21, p. 51, namely: It is possible to find a number 5 so small that, when the intervals between the rulings (both sets) are less than 5, the sums A„. A(, A c differ from their frontiers by less than 2e. For there is, as seen above, some method of ruling such' that these sums differ from their frontiers by less than e. Moreover, the adding of a single new ruling cannot change the sums by more than A/A where A is the largest inter- val and D the largest dimension of the rectangle. Hence if the total number of intervals (both sets) for the given method is X and if 5 he taken less than e/XAI). the ruling obtained by superposing the given ruling upon a ruling where the inter- vals are less than 5 will be such that the sums differ from the given ones by less than €, and hence the ruling with intervals less than 5 can only give rise to sums which differ from their frontiers by less than 2 e. In the second place it should be observed that the limits i,-, l u have been obtained by means of all possible modes of ruling where the rules were parallel to the x- and y-axes, and that there is no a priori assurance that these same limits would have been obtained by rulings parallel to two other lines of the plane or by covering the plane with a network of triangles or hexagons or other figures. In any thorough treatment of the subject of area such matters would have to be discussed. That the discussion is not given here is due entirely to the fact that these critical com- ments are given not so much witli the desire to establish certain theorems as with the aim of showing the reader the sort of questions which come up for considera- tion in the rigorous treatment of such elementary matters as "the area of a plane curve,'' which he may have thought he "knew all about." It is a common intuitive conviction that if a region like that formed by a square be divided into two regions by a continuous curve which runs across the square from one point of the boundary to another, the area of the square and the sum of the areas of the two parts into which it is divided are equal, that is. the curve (counted twice) and the two portions of the perimeter of the square form two simple closed curves, and it is expected that the sum of the areas of the curves is the area of the square. Now in case the curve is such that the frontiers /,, and l' u formed for the two curves are not zero, it is clear that the sum L, + L' ( for the two curves will not give the area of the square but a smaller area, whereas the sum (L{ + l u ) -\- (L\ + /,',) will give a greater area. Moreover in this case, it is not easy to formulate a general definition of area applicable to each of the regions and such that the sum of the areas shall be equal to the area of the combined region. Hut if /„ and l' u both vanish, then the sum L, + L ( ' does give the combined area. OX SIMPLE INTEGRALS 313 It is therefore customary to restrict the apj)lication of the term "area" to such simple closed curves as have l u = 0, and to say that the quadrature of such curves is possible, but that the quadrature of curves for which l^ -^ is impossible. It may be proved that : If a curve is rectifiable or even if one of the functions x (t). or y{t) is of limited variation, the limit l u is zero and the quadrature of the curve is possible. For let the interval t === t =§ < x be divided into intervals A t £, A 2 t, • ■ • in which the oscillations of x and y are e 15 e Q . • • • , ij 1? ij 2 , • • • . Then the portion of the curve due to the interval Ait may be inscribed in a rectangle e,i7,-, and that portion of the curve will lie wholly within a rectangle 2e i -277 i - concentric with this one. In this way may be obtained a set of rectangles which entirely contain the curve. The total area of these rectangles must exceed l u . For if all the sides of all the rectangles be produced so as to rule the plane, the rectangles which go to make up A u for this ruling must be contained within the original rectangles, and as A u >l u , the total area of the original rectangles is greater than l u . Next suppose x(l) is of limited variation and is written as x + n (t) — X(t), the differ- ence of two nondecreasing functions. Then 2e,- =j= II (tA + N(t x ), that is, the sum of the oscillations of x cannot exceed the total variation of x. On the other hand as y(t) is continuous, the divisions A,-£ could have been taken so small that 77, < 77. Hence l u < A u =§ ]>} 2 e ; ■2, ; <4^e,s4, [n^) + JV(*i)]- The quantity may be made as small as desired, since it is the product of a finite quantity by 77. Hence /„ = and the quadrature is possible. It may be observed that if x (t) or y (t) or both are of limited variation, one or all of the three curvilinear integrals -Jydx, Jxdy, Ijxdy-ydx may be defined, and that it should be expected that in this case the value of the integral or integrals would give the area of the curve. In fact if one desired to deal only with rectifiable curves, it would be possible to take one or all of these integrals as the definition of area, and thus to obviate the discussions of the pres- ent article. It seems, however, advisable at least to point out the problem of quadrature in all its generality, especially as the treatment of the problem is very similar to that usually adopted for double integrals (§ 132). From the present viewpoint, therefore, it would be a proposition for demonstration that the curvi- linear integrals in the cases where they are applicable do give the value of the area as here defined, but the demonstration will not be undertaken. EXERCISES 1. For the continuous curve (27) prove the following properties: (a) Lines x = a, x = b may be drawn such that the curve lies entirely between them, has at least one point on each line, and cuts every line x = £, a < £ < b, in at least one point ; similarly for y. (/3) From p — x cos a + y sin a. the normal equation of a line, prove the prop- ositions like those of (a) for lines parallel to any direction. (7) If (£, 77) is any point of the jy-plane, show that the distance of (£, 7?) from the curve has a minimum and a maximum value. 314 INTEGRAL CALCULUS (5) If /«(£, 7)) and -V(£, 77) are the minimum and maximum distances of (£, 77) from the curve, the functions m(£ , 77) and 3/(£, 77) are continuous functions of (£, 77). Are the coordinates x(£, 77), y (£. 77) of the points on the curve which are at mini- mum (or maximum) distance from (£, 17) continuous functions of (£, 77) ? ( e ) If £', £", • • • , t^'K ■ ■ ■ are an infinite set of values of t in the interval i = i = i t and if i° is a point of condensation of the set, then x° = ^>(t°), y° — \j/ (t°) is a point of condensation of the set of points (x% y'), (x", y"), •••, (j;W. y^), ••• corre- sponding to the set of values t\ t" • • • , £(*), ■ • • . (f ) Conversely to (e) show that if (x', y'), (x", ?/'), • • • , (/W, y a >), • • • are an infinite set of points on the curve and have a point of condensation (x°, t/°), then the point (x°, ]f>) is also on the curve. (77) From (f) show that if a line x = £ cuts the curve in a set of points if, y", ■ ■ ■ , then this suite of y's contains its upper and lower frontiers and has a maximum or minimum. 2. Define and discuss rectifiable curves in space. 3. Are y = x 2 sin - and y = Vx sin - rectifiable between x = 0, x — 1 ? x x 4. If x(t) in (27) is of total variation II (^ + N(C 1 ), show that PV(/, y)dx | Ax| and 5 > | Ay j may be made as small as desired by taking At sufficiently small and where it is assumed that in ^ 0. 6. From Ex.5 infer that #(£, 77. t) is of limited variation when t describes the interval t = t =2 t x defining the curve. Show that #(£, 77, t) is continuous in (£, 77) through any region for which m > 0. 7. Let the parameter t vary from t to ^ and suppose the curve (27) is closed so that (x, y) returns to its initial value. Show that the initial and final values of 6 (£, 77, t) differ by an integral multiple of 2 tt. Hence infer that this difference is constant over any region for which m > 0. In particular show that the constant is over all distant regions of the plan:'. It may be remarked that, by the study of this change of 6 as t describes the curve, a proof may be given of the theorem that the closed continuous curve divides the plane into two regions, one interior, one exterior. 8. Extend the last theorem of § 123 to rectifiable curves. CHAPTER XII ON MULTIPLE INTEGRALS 129. Double sums and double integrals. Suppose that a body of matter is so thin and fiat that it can be considered to lie in a plane. If any small portion of the body surrounding a given point P (>. //) be considered, and if its mass be denoted by A/» and its area by A.l. the average (surface) density of the portion is the quotient A/// /A.l, and the, actual density at the point P is defined as the limit of this quotient when A.l = 0, that is, . D{x,y) = \\m — -• The density may vary from point to point. Now conversely suppose that the density D(x, >/) of the body is a known function of (x, >j) and that it be required to find the total mass of the body. Let the body be considered as divided up into a large number of pieces each of which is small in eri'rij dirertlon, and let A.l ( - be the area of any piece. If (£,-. 77,) be any point in A.4,-, the density at that point is D(£ i} rji) and the amount of matter in the piece is approxi- mately D(i h ^/iA.-I,- provided the density he regarded as continuous, that is, as not varying much over so small an area. Then the sum D($ v Vl )±A 1 + D(i 2 , rj.^A 2 + ••• + D(i n , Vn ) A.!„ = ^ />(£, t^A.I,, extended over all the pieces, is an approximation to the total mass, and may be sufficient for practical purposes if the pieces be taken tolerably small. The process of dividing a body up into a large number of small pieces of which it is regarded as the sum is a device often resorted to : for tin; properties of the small pieces may be known approximately, so that the corresponding property for the whole body can be obtained approx- imately by summation. Thus by definition the moment of inertia of a small particle of matter relative to an axis is //>/■-. where m is the mass of the particle and /• its distance from the axis. If therefore the moment of inertia of a plane body with respect to an axis perpendicular 315 316 INTEGRAL CALCULUS to its plane were required, the body would be divided into a large number of small portions as above. The mass of each portion would be approximately D(^ u ■qAAA i and the distance of the portion from the axis might be considered as approximately the distance i\ from the point where the axis cut the plane to the point (£, 77,-) in the por- tion. The moment of inertia would be D<&, Vl )r^A x + ... + D(i n , Va )r^A n = ^D($ i} rjAr^A,, or nearly this, where the sum is extended over all the pieces. These sums may be called double sums because they extend over two dimensions. To pass from the approximate to the actual values of the mass or moment of inertia or whatever else might be desired, the underlying idea of a division into parts and a subsequent summation is kept, but there is added to this the idea of passing to a limit. Com- pare §§16-17. Thus n Jl^oX D ^vd^ and J^Z^rir?^ would be taken as the total mass or inertia, where the sum over n divisions is replaced by the limit of that sum as the number of divisions becomes infinite and each becomes small in every direction. The limits are indicated by a sign of integration, as lim^ 2>(£, V! )AA { =Cd(x, y)dA, lim^ D(£ i} rj^rfAA, = C DMA. The use of the limit is of course dependent on the fact that the limit is actually approached, and for practical purposes it is further depend- ent on the invention of some way of evaluating the limit. Both these questions have been treated when the sum is a simple sum (§§ 16-17, 28~30, 35) ; they must now be treated for the case of a double sum like those above. 130. Consider again the problem of finding the mass and let D { be used briefly for I>(i h 97,). Let M t be the maximum value of the density in the piece A.l } - and let m { be the minimum value. Then W;AAi^ Di&Ai^Mi&Ai. In this way any approximate expression 1\AA t for the mass is shut in between two values, of which one is surely not greater than the true mass and the other surely not less. Form the sums s = 2} w^A/C-sV l) i AA i sT MA A i = S extended over all the elements Ayl,. Now if the sums s and S approach the same limit when A.l i: = 0, the sum 'S,D i AA i which is constantly ON MULTIPLE INTEGRALS 3U included between s and S must also approach that limit independently of how the points (£,-, 77,) are chosen in the areas A.1,. That s and S do approach a common limit in the usual case of a continuous function D(x, //) may be shown strikingly if the surface z = J) (x, }j) be drawn. The term D { AA { is then repre- sented by the volume of a small cylinder upon the base AA t and with an altitude equal to the height of the surface z = D (x, y) above some point of AA { . The sum 2/>,A-'/ of all these cylinders will be ap- proximately the volume under the surface z = D(x,y) and over the total area A = 2A.1-. The term M { AA . is represented by the volume of a small cylin- der upon the base AA t and cir- cumscribed about the surface ; the term m^A^ by a cylinder inscribed in the surface. When the number of elements A.l,- is increased without limit so that each becomes indefinitely small, the three sums s, S, and %D { AA t all approach as their limit the volume under the surface and over the area A. Thus the notion of volume does for the double sum the same service as the notion of area for a simple sum. Let the notion of the integral he applied to find the formula for the center of gravity of a plane lamina. Assume that the rectangular coordinates of the center of gravity are (x, y). Consider the body as divided into small areas AA{. If (£,-, ?;,•) is any point in the area A. 1 ,-. the approximate moment of the approximate mass D,-A/l ,• in that area with respect to the line x = x is the product (& — x)_D,A^L of the mass by its distance from the line. The total exact moment would therefore be limV (fc - x)D { AAi = f(x - x)D(x, y)dA = 0, and must vanish if the center of gravity lies on the line x — x as assumed. Then CxD (x, y) <1A - Cxi) (x, y) dA - or CxDdA =x C D (x, y) dA . These formal operations presuppose the facts that the difference of two integrals is the integral of the difference and that the integral of a constant x times a function I) 318 INTEG UAL CALCULUS is the product of the constant by the integral of the function. It should be imme- diately apparent that as these rules arc applicable to sums, they must be applicable to the limits of the sums. The equation may now be solved for x. Then CxDdA Cxdm CyBdA C ydr I DdA f DdA (1) where m stands for the mass of the body and dm for DdA. just as Aw?j might replace DjAAj ; the result for y may be written down from symmetry. As another example let the kindle energy of a lamina moving in its plane be cal- culated. The use of vectors is advantageous. Let r be the vector from a fixed origin to a point which is fixed in the body, and let r! be the vector from this point to any other point of the body so that dr.- dr n dr, ; r, = r + ri ,-, — = — + — - or v,- = v + v u . dt dt dt The kinetic energy is 2 i- vjXntj or better the integral of i c 2 dm. Now vj = v/.v,- = Vo«Vo + v 1( -.v 1( - + 2v .Vii = r 2 + r{;(o- + 2v .Vi,-. That Vi,«Vi, = /'j",-w-. where /• li = : r ll - and w is the angular velocity of the body about the point r , follows from the fact that r 1( - is a vector of constant length r u and hence din = >'nd6. where d6 is the angle that i\i turns through, and conse- quently w = d6/dt. Next integrate over the body. I }, r-dm — I I i\;dm + I i r^u-dm + ( v «Vi(Z«i = I r,;.1/ + .Vo;' 2 Crfdm + V - fvidm ; (2) for r 2 and u- are constants relative to the integration over the body. Note that v u . | v x dm = if v = or if iv^lm — j - x x dm = - Cr x dm = 0. But v = holds only when the point r is at rest, and I x^lm = is the condition that r„ be the center of gravity. Jn the last case T = fu, ■ U 2 M + I frfrn As / is the integral which has been called the moment of inertia relative to an axis through the point r„ perpendicular to the plane of the body, the kinetic energy is seen to be the sum of I Mr-, which would be the kinetic energy if all the mass were concentrated at the center of gravity, and of \ lu' 2 , which is the kinetic energy of rotation about the center of gravity; in case r indicated a point at rest (even if only instantaneously as in $:-}!») the whole kinetic energy would reduce to the kinetic energy of rotation J lu-. In case r (l indicated neither the center of gravity nor a point at rest, the third term in (2) would not vanish and the expression for the kinetic energy would be more complicated owing to the presence of this term. ON MULTIPLE INTEGRALS 319 131. To evaluate the double integral in case the region is a rectangle parallel to the axes of coordinates, let the division be made into small rectangles by drawing lines parallel to the axes. Let there be in equal divisions on one y m columns i=i,2... ,m side and n on the other. There will then be ' • th p:i".t"+:: mn small pieces. It will be convenient to in- rowf't troduce a double index and denote by AJ, ; the ,, -"•-"t;t"-t- './ y \ 1 ' ' ' area of the rectangle in the tth column and./'th -^ — ^r row. Let (ifj, 77,/) be any point, say the mid- dle point in the area A.l, 7 — A.<-,A//-. Then the sum may be written 2 D(jt U} V!j ) a.i v . = /'n^A.'h + AA^A^ + ■■■ + i>„,A->;,A>h X.\ X + AAAZA. + £>,oA.r 2 A>/o + /<„,A/,„A//o + D lll \.r 1 \a a + D. 2n Xr. 2 \t/ ll H h D m ,A.r tl Al/n- Now the terms in the first row are the sum of the contributions to 2,- j of the rectangles in the first row, and so on. But (AA'i + D iA*-i + • • • + /^A'»A/// = A ///2 />tt: >- ni)^ and A / / / ^/^ ; .^)A/ i = D(r, r,j)d.r + £. \'lr That is to say, by taking m sufficiently large so that the individual increments A./', are sufficiently small, the sum can be made to differ from the integral by as little as desired because the integral is by definition the limit of the sum. In fact Cl^2l^- W y A. ', == «(/, if e be the maximum variation of D(x, ;,) over one of the little rectangles. After thus summing up according to rows, sum up the rows. Then 2 y> A-'.:/ = f 1 D(X, Vl )d.r\ ;/l + f 'jhr. V .)d.r\,,„ + ••• + j l D(.r, nj'lx\ lln + A, \X\ = [C^ ! / l + ^!/, f ■■■+L\'I II ^ ^>-./' )^Ay = e(./--;r )(^- Z / ) If j D(x, ij)/), then J AA 1;/ = (ji(rj 1 )\i/ 1 + (r].,)A>/, H h f',;„)A//„ + A r , = I (y)d// + k -\- X. k. A small. 320 IN TE( J K A L C ALC U LU S Hence * lim 2) ZV^l;/= f^-l= f ' f l D(x,y)dxdy. i.i J «/'/n J*n (3) It is seen that the double integral is equal to the result obtained by first integrating with respect to x, regarding y as a parameter, and then, after substituting the limits, integrating with respect to y. If the sum- mation had been first according to columns and second according to rows, then by symmetry J Del A =j l £ l D(x, y)dxdy =J 'J ' D(x, y)dydx. (3' This is really nothing but an integration under the sign (§ 120). If the region over which the summation is extended is not a reef angle parallel to the axes, the method could still be applied. But after summing or rather integrating according to rows, the limits would not be constants as x () and x p but would be those func- tions x = (y) and x — $ x {y) of y which represent the left-hand and right-hand curves which bound the region. Thus 0\ D,IA= J, J D ( x >y) d *< h J (3") And if the summation or integration had been first with respect to columns, the limits would not have but the functions esent the lower and upper bounding curves of the region. Thus o, '' /» + r V b 1 — (x 2 + y 2 )dxdy J-b J -^Vb--!/- = I> f '" f\" % ^—^ (-'■- + y 2 )dxdy. Either of these forms might be evaluated, but the moment of inertia of the whole ellipse is clearly four times that of a quadrant, and hence the simpler results J-b r , V 6- - ir f (x 2 + y-)dxdy a " x / (( , _ r o = 4 Z> f C b (x 2 + ij-) dydx = ~ Lab (a- + b 2 ). J'i Jo 4 It is highly advisable to make use of symmetry, wherever possible, to reduce the region over which the integration is extended. 132. With regard to the more careful consideration of the limits involved In the definition of a double integral a few observations will be sufficient. Consider the sums S and s and let .U,A.l,- be any term of the first and ?«,A.t,- the corresponding term of the second. Suppose the area AA{ divided into two parts A-4i. and AA 2 {, and let 3/],-, 3/-> t - be the maxima in the parts and mi,-, m- 2 i the minima. Then since the maximum in the whole area A.l,- cannot be less than that in either part, and the minimum in the whole cannot be greater than that in either part, it follows that mn == mi, nio. s m.-, Mu == Mi, Mm === Mi, and m.-A-l; ^ mi.-A.lu + m 2i AA. 2i , M U AA U + M. 2i AA. 2i ^ M t AA;. Hence when oik; of the pieces AA{ is subdivided the sum S cannot increase nor the sum s decrease. Then continued inequalities may be written as mA s Vm.-A.l,- sVfl(£ ( , Vi )AAi S Vjl/.-A^l,- S MA. If then the original divisions A.-1,- be subdivided indefinitely, both N and s will approach limits (§§ 21-22) ; and if those limits are the same, the sum Z-D.A-1; will approach that common limit as its limit independently of how the points (&, 77.) are chosen in the areas AA ,-. 322 INTEGKAL CALCULUS It lias not been shown, however, that the limits of S and s are independent of the method of division and subdivision of the whole area. Consider therefore not only the sums S and s due to some particular mode of subdivision, but consider all such sums due to all possible modes of subdivision. As the sums S are limited below by mA they must have a lower frontier L, and as the sums s are limited above by MA they must have an upper frontier I. It must lie shown that I === J.. To see this consider any pair of sums S and s corresponding to one division and any other pair of sums 6" and ,s' corresponding to another method of division ; also the sums S" and s" corresponding to the division obtained by combining, that is, by superposing the two methods. Now S' s S" s s" s s , »S s S" s , s " s , s ', S s L. S' S L, s ^ ?, s' s /. It therefore is seen that any S is greater than any s, whether these sums correspond to the same or to different methods of subdivision. Now if L < I, some S would have to be less than some s ; for as L is the frontier for the sums >', there must be some such sums which differ by as little as desired from L ; and in like manner there must be some sums s which differ by as little as desired from I. Hence as no S can be less than any s, the supposition L < I is untrue and L == /. Now if for any method of division the limit of the difference lim (S - s) = lim V (M; - m t ) AA,- = lim V 0,-A.L = of the two sums corresponding to that method is zero, the frontiers L and / must be the same and both S and s approach that common value as their limit; and if the difference S—s approaches zero for ever}' method of division, the sums S and s will approach the same limit J, = / for all methods of division, and the sum -I),AA; will approach that limit independently of the method of division as well as independently of the selection of (',■. 77/). This result follows from the fact that L — I == S — s, S — L ^ »S — s, !-sg N — s, and hence if the limit of S — * is zero, then L = l and S and s must approach the limit L = I. One ease, which covers those arising in practice, in which these results are true is that in which l)(.r. 1/) is continuous over the area .1 except perhaps upon a Unite number of curves, each of which maybe inclosed in a strip of area as small as desired and upon which J)(.r. y) remains finite though it be discontinuous. For let the curves over which D(x, y) is discontinuous be inclosed in strips of total area a. The con- tribution of these areas to the difference S — s cannot exceed (M — m)(i. Apart from these areas, the function J)(r. y) is continuous, and it is possible to take the divisions A-l,- so small that the oscillation of the function over any one of them is less than an assigned number e. Hence the contribution to N — .x is less than e(A — a) for the remaining undeleted regions. The total value of $ — ,s is there- fore less than (M — m) a + e(.\ — a) and can certainly be made as small as desired. The proof of the existence and uniqueness of the limit of Z7>,A.l,- is therefore obtained in case 1) is continuous over the region .1 except for points along a finite number of curves where it may be discontinuous provided it remains finite. Throughout the discussion the term " area '* has been applied : this is justified by the previous work (§128). Instead of dividing the area A into elements A.l. one may rule the area with lines parallel to the axes, as done in § 128. and consider the sums Z-VAxAy, linXrXij. -I)\.r\ii. where the first sum is extended over all the rectan- gles which lie within or upon the curve, when* the second sum is extended over all the rectangles within the curve, and where the last extends over all rectangles ON MULTIPLE INTEGRALS 323 within the curve and over an arbitrary number of those upon it. In a certain sense this method is simpler, in that the area then falls out as the integral of the special function which reduces to 1 within the curve and to outside the curve, and to either upon the curve. The reader who desires to follow this method through may do so for himself. It is not within the range of this book to do more in the way of rigorous analysis than to treat the simpler questions and to indicate the need of corresponding treatment for other questions. The justification for the method of evaluating a definite double integral as given above offers some difficulties in case the function D(x, y) is discontinuous. The proof of the rule may be obtained by a careful consideration of the integration of a function defined by an integral containing a parameter. Consider

    (y) is a continuous function of y if D(x, y) is a con- tinuous function of (x, y). Suppose that L){x, y) were discontinuous, but remained finite, on a finite number of curves each of which is cut by a line parallel to the j-axis in only a finite number of points. Form A<£ as before. Cut out the short intervals in which discontinuities may occur. As the number of such intervals is finite and as each can be taken as short as desired, their total contribution to (y) might not be defined and might have a discontinuity for the value y = ft. But there can be only a finite num- ber of such values if I)(x. y) satisfies the conditions imposed upon it in considering the double integral above. Hence cp (y) would still be integrable from y to y v Hence / ' / l D(x, y)dxdy exists m (x t - x )(y x - 2/ ) = J J '^(A y)dxdy § M(x x - x )(y 1 - y ) and Add and under the conditions imposed for the double integral. Now let the rectangle x = x = x v y = y = y x be divided up as before. Then jriijAXjAyj =§= / / D(x, y)dxdy =s M,;A;xAjy. ^ mfjAXiAyj ^ ^ f" '"J" V 1) (x. y) dxdy ^ ]T MyApA/y V. f" " V ' f * 1; 'l>(x. y)dxdy = f '"' f'^Dix. y)dxdy. *"* Jy Jx «/2/[ /^X 1 p I/, fX= l (y) r> I D (x, y) dxdy = / I D(x, y) dxdy = D(x, y) dA . The rule for evaluating the double integral by repeated integration is therefore proved. EXERCISES 1. The sum of the moments of inertia of a plane lamina about two perpendicular lines in its plane is equal to the moment of inertia about an axis perpendicular to the plane and passing through their point of intersection. 2. The moment of inertia of a plane lamina about any point is equal to the sum of the moment of inertia about the center of gravity and the product of the total mass by the square of the distance of the point from the center of gravity. 3. If upon every line issuing from a point of a lamina there is laid off a dis- tance OP such that OP is inversely proportional to the square root of the moment of inertia of the lamina about the line OP, the locus of P is an ellipse with center at 0. 4. Find the moments of inertia of these uniform laminas: (a) segment of a circle about the center of the circle, (/3) rectangle about the center and about either side, (7) parabolic segment bounded by the latus rectum about the vertex or diameter, (5) right triangle about the right-angled vertex and about the hypotenuse. 5. Find by double integration the following areas: (a) quadrantal segment of the ellipse, (/3) between y' 2 — x 3 and y = x, (7) between 3 y' 2 = 25x and 5x 2 = 9y, (5 ) between x 2 + y' 2 — 2 x = 0, x 2 + y 2 — 2 y = 0, (e) between y 2 — A ax + 4a 2 , y' 2 = — 46x + 4// 2 , (f ) within (y - x - 2) 2 = 4 — x 2 , (??) between x 2 = 4 ay, y(x' 2 + 4 a 2 ) = 811 s , ( 9 ) y 2 = ax, x' 2 + y' 2 — 2 ax = 0. 6. Find the center of gravity of the areas in Ex. 5 (a), (/3), (7), (5), and 222 (a) quadrant of a 4 // 2 = a 2 x 4 — x c , (p) quadrant of x» + yi = as, (7) between x'i = y 2 + a^, x + y = a, (5) segment of a circle. 7. Find the volumes under the surfaces and over the areas given : (a) sphere z = w — x 2 — y' 2 and square inscribed in x 2 + y' 2 = a 2 , (j8) sphere z = Va 2 — x 2 — y' 2 and circle x 2 + y 2 — ax = 0, (7) cylinder z = V4a 2 — y 2 and circle x 2 + y' 2 — 2 ax = 0, (5) paraboloid z = kxy and rectangle =§ x =§ a, =2 y ^ b, ( e ) paraboloid z — kxy and circle x 2 + y 2 — 2 ax — 2 ay = 0, (f) plane x/a + y/b + z/c = 1 and triangle xy(x/a + y/b — 1) = 0, (17) paraboloid z = \ — x 2 /4 — y' 2 /9 above the plane z — 0, (6) paraboloid z = (x -f y)' 2 and circle x' 2 + y' 2 = a 2 . ON MULTIPLE INTEGRALS 325 8. Instead of choosing (ft, -17/) as particular points, namely the middle points, of the rectangles and evaluating 2Z>(ft, ny) AXiAijj subject to errors A, k which vanish in the limit, assume the function D(x, y) continuous and resolve the double integral into a double sum by repeated use of the Theorem of the Mean, as 4>(y) = ( D(x, y)dx = ^J /•>(£,-, y)Ax,, £'s properly chosen, ■'0 ,- f % (V) dy = 2 4> (Vj) A/// = ^ [ 2J J) (&, Vj) Axjl Ay,- =J>D (ft, v ) A,l (> '' /o ./' ./' i »'iJ 9. Consider the generalization of Osgood's Theorem (§ 35) to apply to double integrals and sums, namely: It' a,j are infinitesimals such that a u = D(&, r U )AA ii + tvAAq, where f y - is uniformly an infinitesimal, then lim \ a;j = fl)(x, y)dA = f '"' f X *D(x, y)dxdy. Discuss the statement and the result in detail in view of §34. 10. Mark the region of the zy-plane over which the integration extends:* (a) f a f X mydx, (p) f 2 f'"l)d>/dj; (y) f* f % ' Ddxdy, Jo Jo J i J .,■ Jo J ir /*V2 /»V3 — a- 2 ,- - f% a "V 2 cos 2 pi(i n\co&~ w~ (5) ,- Ddydx, (e) \ J l)dnl vdm. or the difference between the moments of momentum about P and Q is the moment about P of the total momentum considered as applied at Q. 14. Show that the formulas (1) for the center of gravity reduce to | xyDdx f \yyDdx ( ' 'x(y l — y )Ddx - Jo Jo - Jx„ x = — , y = - or x — — i 1 , j yBdx ( ylhlx J \i/i — i/ )l)dx J 'i(//i + !/ u )(Vi- Vo) DtU J {y 1 -y )Bdx * Exercises involving polar coordinates may be postponed until § ir>4 is reached, unless the student is already somewhat familiar with the subject. 326 IXTEGKAL CALCULUS when -D(x, y) reduces to a function D(x), it being understood that for the first two the area is bounded by x = 0. x = a, y =/(■/■), y = 0, and for the second two by x = x , x = x v y x = f l (x). y = ./;,(/). 15. A rectangular hole is cut through a sphere, the axis of the hole being a diameter of the sphere. Find the volume cut out. Discuss the problem by double integration and also as a solid with parallel bases. 16. Show that the moment of momentum of a plane lamina about a fixed point or about the instantaneous center is Iw, where o> is the angular velocity and / the moment of inertia. Is this true for the center of gravity (not necessarily fixed)? Is it true for other points of the lamina ? f 1 f V:i '/ f 2 VA 17. Invert the order of integration in Ex. 10 and in / I Ddydx. J-i ^Vw 18. In these integrals cut down the region over which the integral must be extended to the smallest, possible by using symmetry, and evaluate if possible: (a) the integral of Ex. 17 with 1) — y ?j — 2 .<■'-;/, (p) the integral of Ex. 17 with l) = (x- 2V3) 2 ^ or D = (x - 2 V$)y 2 , (y) the integral of Ex. 10(e) with I) = r(l + cos0) or 1) = sin

    . 19. The curve y = f(x) between x = a and x = b is constantly increasing. Express the volume obtained by revolving the curve about the X-axis as 7r[f((()J 2 (b — a) plus a double integral, in rectangular and in polar coordinates. 20. Express the area of the cardioid r = a(l — cos<£) by means of double inte- gration in rectangular coordinates with the limits for both orders of integration. 133. Triple integrals and change of variable. In the extension from double to triple and higher integrals there is little to cause difficulty. For the discussion of the triple integral the same foundation of mass and density may be made fundamental. If T)(.r, y, ,*■) is the density of a body at any point, the mass of a small volume of the body surround- ing the point (£,-, rj ; , £,•) will be approximately />(£,-, 77,, £,-)AJ r ,-, and will surely lie between the limits J/,Ar ; and m^AE,, where .1/,- and ///,- are the maximum and minimum values of the density in the element of volume AT;. The total mass of the body would be taken as lim V />(£, Vi , Q AV { = f ']>(..; y, x)dV, (5) where the sum is extended over the whole body. That the limit of the sum exists and is independent of the method of choice of the points (£;, rj;. fj) and of the method of division of the total volume into elements AT,, provided /)(,r, //. ::) is continuous and the elements Al', : approach zero in such a manner that they become small in every direction, is tolerably apparent. ON MULTIPLE INTEGRALS 327 The evaluation of the triple integral by repeated or iterated integra- tion is the immediate generalization of the method used for the double integral. If the region over which the integration takes place is a rec- tangular parallelepiped with its edges parallel to the axes, the integral is D (x, y, z) d V = j j D (x, y, z) dxdydz. (5') The integration with respect to x adds up the mass of the elements in the column upon the base dydz, the integration with respect to y then adds these columns together into a lamina of thickness dz, and the integration with respect to z finally adds together the laminas and obtains the mass in the entire parallelepiped. This could be done in other orders ; in fact the inte- gration might be performed first with re- gard to any of the three variables, second with either of the others, and finally with the last. There are, therefore, six equiva- lent methods of integration. If the region over which the integration is desired is not a rectangular parallele- piped, the only modification which must be introduced is to adjust the limits in the successive integrations so as to cover the entire region. Thus if the first integration is with respect to x and the region is bounded by a surface ./■ = ^ (//, z) on the side nearer the //-.'-plane and by a surface x = ij/ l (y, z) on the remoter side, the integration D (x, //, z) dxdydz = O (//, z) dydz ^ 0/.O will add up the mass in elements of the column which has the cross section dydz and is intercepted between the two surfaces. The problem of adding up the columns is merely one in double integration over the region of the //,~-plane upon which they stand; this region is the pro- jection of the given volume upon the //.v-plane. The value of the integral is then ' 1 (z) pz x p^Cz) pip 1 (x,!/) L /pz x py = ^z) pz x p$ x (z) p^ x (x,y) DdV= I f O dydz =1 j / Ddxdydz. Jz a «/v = <|, CO Jz J {U- + z 2 ) d T, I v = l)f (x- + z 2 ) dV, 7, = I)f(x 2 + y 2 ) d V. The consideration of how the figure looks shows that the limits for z are z = and z = (x 2 + y 2 )/a if the first integration be with respect toz ; then the double integral in x and y has to be evaluated over a semi- circle, and the first integration is more simple if made with respect to y with limits y = and // = v2kx — x-. and final limits x = and x — 2a for x. If the attempt were made to integrate first with respect to y, there would be difficulty because a line parallel to the y-axis will give different limits according as it cuts both the paraboloid and cylinder or the xz-plane and cylinder ; the total integral would be the sum of two integrals. There would be a similar difficulty with respecc to an initial integration by x. The order of integration should therefore 1-8 z, y, x. X=2Cl Wia ' ' X- + I/' 2 a lydx [x - a (1— eos#) V : = I xdzdydx = I) / t/ .. J II — •J z = «/ ./• = J u = = - - / x 3 \ 2 ax — x- + ■ x (2 ax — x 2 )'i dx = 7nt 4 l). " Jo L 3 J Hence x = 4 a/3. The computation of the other integrals may be left as an exercise. 134. Sometimes the region over which a multiple integral is to be evaluated is such that the evaluation is relatively simple in one kind of coordinates but entirely impracticable in another kind. In addition to the rectangular coordinates the most useful systems are polar coor- dinates in the plane (for double integrals) and polar and cylindrical coordinates in space 1 for triple integrals 1. It has been seen 1 •: 40) that the clenient of area or of volume in these cases is d.l = rdrd, ,1V = rn\nddrd6dcf>. dV = rdrd^dr;, , 7i OX MULTIPLE INTEGRALS 329 except for infinitesimals of higher order. These quantities may be substituted in the double or triple integral and the evaluation may be made by successive integration. The proof that the substitution can be made is entirely similar to that given in §§ 34-35. The proof that the integral may still be evaluated by successive integration, with a proper choice of the limits so as to cover the region, is contained in the statement that the formal work of evaluating a multiple integral by repeated integration is independent of what the coordinates actually represent, for the reason that they could be interpreted if desired as representing rectangular coordinates. Find the area of the part of one loop of the lemni.scate r' 2 = 2 a' 2 cos 2 <£ which is exterior to the circle r = a ; also the center of gravity and the moment of inertia rela- tive to the origin under the assumption of constant density. Here the integrals are A CdA, Ax = CxdA, Ay = fydA, 1 = 1) f 'r-dA, m = DA, The integrations may be performed first with respect to r so as to add up the elements in the little radial sectors, and then with regard to

    /• 77 /I — 77, 6 J rdrd

    - a 2 ) d

    2 r b/ r~ o s Ax = 21' r cos

    d = 2 tt. Then l = u 330 INTEGRAL CALCULUS p 2 77 n - r>itl Cos e m = I I " I fcr • r- sin 0drd0d, J^=o Je=o Jr=0 J"2jt /> ^- •.r = 2acos(J / I At ■ r cos (9 • r' 2 ah\0drd0d = f * * - fra 4 d«£ = ^ rfca * , Ja=o J = — — The center of gravity is therefore z = 8 «/7. Sometimes it is necessary to make a change of variable x — (u, r), !/ = ^(>f, v) or x = 4>( lt > ''j "')> V ~ ^( u ) v > "')' z ~ M ("> ''■> "') (8) in a double or a triple integral. The element of area or of volume has been seen to be (§ 63, and Ex. 7, p. 135) dA = V «> / \u, V, i ~) dudvdw. (8') \H—) \ \U, VI dudv (8") V", v, ;) dudvdw. It should be noted that the Jacobian may be either positive or negative but should not vanish; the difference between the ease of positive and the case of negative values is of the same nature as the difference between an area or volume and the reflection of the area or volume. As the elements of area or volume are considered as positive when the increments of the variables are positive, the absolute value of the Jacobian is taken. EXERCISES 1. Show that ((i) arc the formulas for the center of gravity of a solid body. 2. Show that /,. - C (if 2 + z-)dm, T u = f (./'- + z-)dm, I z = f (x 2 + y-)dm are the formulas for the moment of inertia of a solid about the axes. 3. Prove that the difference between the moments of inertia of a solid about any line and about a parallel line through the center of gravity is the product of the mass of the body by the square of the perpendicular distance between the lines. 4. Find the moment of inertia of a body about a line through the origin in the direction determined by the cosines I, m, ?(, and show that if a distance OP be laid off along this line inversely proportional to the square root, of the moment of inertia, the locus of P is an ellipsoid with as center. ON MULTIPLE INTEGRALS 331 5. Find the moments of inertia of these solids of uniform density: (a) rectangular parallelepiped abc, about the edge a, (/3) ellipsoid x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1, about the z-axis, (7) circular cylinder, about a perpendicular bisector of its axis, (5) wedge cut from the cylinder x 2 + y 2 = r- by z = ± nix, about its edge. 6. Find the volume of the solids of Ex. 5 (/3), (5), and of the : (a) trirectangular tetrahedron between xyz = and x/a + y/b + z/c = 1, (jS) solid bounded by the surfaces y 2 + z 2 = 4 ax, y 2 = ax, x = 3 a, (7) solid common to the two equal perpendicular cylinders x 2 + y 2 = a 2 , x 2 + z 2 = ). (7) r = a sin 2 , (5) r = asin 3 J (small loop), (e) circular sector of angle 2 nr. 11. Find the moments of inertia of the areas in Ex. 10 (a), (/3), (7) about the initial line. 12. If the density of a sphere decreases uniformly from I) at the center to D x at the surface, find the mass and the moment of inertia about a diameter. 13. Find the total volume of : (a) (x 2 + y 2 + z 2 ) 2 = axyz, (/3) (x 2 + y 2 + z 2 )* = 27 a*xyz. 14. A spherical sector is bounded by a cone of revolution; find the center of gravity and the moment of inertia about the axis of revolution if the density varies as the nth power of the distance from the center. 15. If a cylinder of liquid rotates about the axis, the shape of the surface is a paraboloid of revolution. Find the kinetic energy. 16. Compute J l'-' V ), j( X,V,Z \, J (^'J) and hence verify (7). 1 \r, ) \r, Jo Jv = \1 + U 1 + «/ (1 + ») 2 l + « r (y) or = f n f f dudv - f' a f a ~ f dudv. KU Jo Ju = (1 + M) 2 Ja Ju = l (1 + U) 2 8-32 INTEGRAL CALCULUS 18. Find the volume of the cylinder r = 2 a cos between the cone z — r and the plane z — 0. 19. Same as Ex.18 for cylinder /- 2 = 2a 2 cos20; and find the moment of inertia about r = if the density varies as the distance from r = 0. 20. Assuming the law of the inverse square of the distance, show that the attraction of a homogeneous sphere at a point outside the sphere is as though all the mass were concentrated at the center. 21. Find the attraction of a right circular cone for a particle at the vertex. 22. Find the attraction of (a) a solid cylinder, (/3) a cylindrical shell upon a point on its axis ; assume homogeneity. 23. Find the potentials, along the axes only, in Ex. 22. The potential may be defined as Hr- 1 dm or as the integral of the force. 24. Obtain the formulas for the center of gravity of a sectorial area as 1 r *t 1 ] /■•* I 1 x = — I ~r s cos', y)dA the conception of den- sity and mass of a lamina was made fundamental : as was pointed out, it is possible to go into three dimensions and plot the surface ." —f(.r, y) ),1j OX MULTIPLE INTEGRALS 080 and interpret the integral as a volume. In the treatment of the triple integral J f(x, //, z)dV the density and mass of a body in space were made fundamental; here it would not be possible to plot u = f(x, //, s) as there are only three dimensions available for plotting. Another important interpretation of an integral is found in the con- ception of avcvage value. If q v q o} ■ ■ ■, q n are n numbers, the average of the numbers is the quotient of their sum by n. . y = V ' + ^ + --' + V " = ^- (9) n n If a set of numbers is formed of ir numbers q , and vj numbers q , ••■, and w n numbers q n , so that the total number of the numbers is w x + iv 2 + • • • + ?'"„, the average is 1 >'\ + v: 2 + ■■■ + tr„ ' Sw* ' The coefficients tr, ?'".„•• -,'>''„, or any set or numbers which are pro- portional to them, are called the weights of q , q , •••, q n . These defi- nitions of average will not apply to hading the average of an infinite number of numbers because the denominator n would not be an arith- metical number. Hence it would not be possible to apply the definition to finding the average of a function /(.r) in an interval x ?=x = o\. A slight change in the point of view will, however, lead to a defi- nition for the average value of a function. Suppose that the interval x = x = x is divided into a number of intervals Ax j} and that it be imagined that the number of values of y = f(x) in the interval Ar, is proportional to the length of the interval. Then the quantities Ax { would be taken as the weights of the values _/"(£■) and the average would be r ,-^ I \f(x)dx _ SAav/(£.) . ,, _ J, n ... // = — ■--- j or better u = -— — (10) I ax by passing to the limit as the A.>y's approach zero. Then !J = — — or f Xl f(x)dx=(x 1 -x )j/. (10') In like manner if z = f(x, if) be a function of two variables or a =f(.r. ?/. ■:) a function of three variables, the averages over an area 334 INTEGRAL CALCULUS or volume would be defined by the integrals ff(x,y)dA Cf(x,y,z)dV » = *- and u = ^ (10") CdA = A CdV = V It should be particularly noticed that the value of the average is de- fined with reference to the variables of which the function averaged is a function ; a change of variable will in general bring about a change in the value of tit e ace rage. For if y = /(,■), Ji7) = -A- pfix) dx ; ' 1 ' Jx a but if y=/(*(0), W) = t^j fV(*(*))*; and there is no reason for assuming that these very different expres- sions have the same numerical value. Thus let y = ./•", =j= x == 1, x = sin t, = t == \ it, _ 1 r 1 , 1 1 /*£ , 1 y( .r) = j j o ,,/,• = -, y(*) = — j Bin'ftfc = 2' The average values of x and ?/ over a plane area are x = — I xdA, T J=\~) V dA > when the weights are taken proportional to the elements of area; but if the area be occupied by a lamina and the weights be assigned as proportional to the elements of mass, then — I xdm, y = — / udm, m J J m J J and the average values of x and y are the coordinates of the center of gravity. These two averages cannot be expected to be equal unless the density is constant. The first would be called an area-average of x and a ; the second, a mass-average of x and y. The mass average of the square of the distance from a point to the different points of a lamina would be -, p ?=k* = —j AI»i=I/M, (11) and is defined as the radius of gyration of the lamina about that point; it is the quotient of the moment of inertia by the mass. ON MULTIPLE INTEGRALS 335 As a problem in averages consider the determination of the average.value of a proper fraction ; also the average value of a proper fraction subject to the condi- tion that it be one of two proper fractions of which the sum shall be less than or equal to 1. Let x be the proper fraction. Then in the first case x = - I xdx = - 1 Jo 2 In the second case let y be the other fraction so that x + y =§ 1. Now if (x, y) be taken as coordinates in a plane, the range is over a triangle, the number of points (x, y) in the element dxdy would naturally be taken as proportional to the area of the element, and the average of x over the region would be fxdA ^J l ~"xdxdy f\l-2y + y*)dy 1 CdA f l C l ~"dxdy 2f\l-y)dy Now if x were one of four proper fractions whose sum was not greater than 1, the problem would be to average x over all sets of values (x, y. z, u) subject to the relation x-fj + .'-f iigl. From the analogy with the above problems, the result would be -./•A.rAyAzA lim XI /% 1 — U rt \ — H — Z p \ — ! = 0«'C = J» = «b=0 xdxdydzdu ZAxAyAzAu r 1 r - «r - — /• *- " ~ ' ~ '^y^ J u = o J z = " »/ = Jx = The evaluation of the quadruple integral gives x = 1/5. 136. The foregoing problem and other problems which may arise lead to the consideration of integrals of greater multiplicity than three. It will be sufficient to mention the case of a quadruple integral. In the first place let the four variables be • T = ■'-• = ''V Vo^!/=!/v .~o = * = *l> "0=" = "!* ( 12 ) included in intervals with constant limits. This is analogous to the case of a rectangle or rectangular parallelepiped for double or triple integrals. The range of values of x, ?/, z, u in (12) may be spoken of as a rectangular volume in four dimensions, if it be desired to use geo- metrical as well as analytical analogy. Then the product Ax i A.y i A.z i &n < t would be an element of the region. If x t === ii = x { + A.r., • • -, ?/. s; 0j ^ a i + A",-, the point (£,-, 77,-, £,-, 0,-) would be said to lie in the element of the region. The formation of a quadruple sum ;/(£,., Vi , £» e^AxAyAzAui X- could be carried out in a manner similar to that of double and triple sums, and the sum could readily be shown to have a limit when 336 INTEGEAL CALCULUS A./',-, Ay { , -Az i} A?/,- approach zero, provided f is continuous. The limit of this sum could be evaluated by iterated integration lim]£./;-A.r l .A#A2 1 -A" I . = f If f{x,y,z,u)dudzdydx ■A) J'Ja "J" ^"0 where the order of the integrations is immaterial. It is possible to define regions other than by means of inequalities such as arose above. Consider F(x, y, z, v.) = and F(x, >/, z, u) S 0, where it may be assumed that when three of the four variables are given the solution of F=0 gives not more than two values for the fourth. The values of x, y, z, u which make F< are separated from those which make F> by the values which make F = 0. If the sign of F is so chosen that large values of x, y, z, u make F positive, the values which give F > will be said to be outside the region and those which give F < will be said to be inside the region. The value of the integral of /'(•'', y, z, )') over the region Fg could be found as If / I /(■'•, V, z, «) dudzdydx, where u = w 1 (a*, y, z) and u = w (x, y, z) are the two solutions of F = for u in terms of .r, y, z, and where the triple integral remaining after the first integration must be evaluated over the range of all possible values for (./•, y. ::). By first solving for one of the other variables, the integrations could be arranged in another order with properly changed limits. If a change of variable is effected such as x = (fi(x',y',z',a'), y = ^{x',y',z',u'), z = x(x',y\ *', u'), u = w(x',y%z',u') (13) the integrals in the new and old variables are related by fffff(*, V, z, u)dxdydzdu =fffff(4>, *, X, «)|, ^, x, u)\J [dx'dy'dz' , and each side may be integrated with respect to u. The rule therefore holds in this case. It remains therefore merely to show that any transformation (13) may be resolved into the succession of two such as (13'), (13"). Let x l = x', y x =-y', z l -z', u l = u{x',y',z',u')- u(x v y 1 ,z v u'). Solve the equation u l = «(x t , y v z v »') for u' — co, (x,, y x , z v u v ) and write x = 0(x t , y v z v w t ), y = f (x v y v z v w,), z = x (x v y v z v w x ), u = u v Now by virtue of the value of w v this is of the type (13"), and the substitution of x v y v z v itj in it gives the original transformation. EXERCISES 1. Determine the average values of these functions over the intervals: (a) x 2 , OSj^lfl, (p) sin x, ^ x s i tt, (7) x", s x ^ n, (5) eos"x, g x § \ir. 2. Determine the average values as indicated : (cr) ordinate in a semicircle ./- + y' 2 = a 2 , y > 0, with x as variable, (£) ordinate in a semicircle, with the arc as variable, (7) ordinate in semiellipse x = acos<£, y = bm\(p, with

    y to the plane cuts the surface in only one point. / Y Over any element <1A of the projection there will / 'I3dA he a small portion of the surface. If this small portion were plane and if its normal made an angle y with the s-axis, the area of the surface (p. 1G7) would be to its projection as 1 is to OX MULTIPLE INTEGRALS 339 cos y and would be sec yd A. The value of cos y may be read from (9) on page 96. Tins suggests that the quantity •-/->- -/jfh® + £) dxdy (15) be taken as the definition of the area of the surface, Avhere the double integral is extended over the projection of the surface ; and this defi- nition will be adopted. This definition is really dependent on the particular plane upon which the surface is projected ; that the value of the area of the surface would turn out to be the same no matter what plane was used for projection is tolerably apparent, but will be proved later. Let the area cut out of a hemisphere by a cylinder upon the radius of the hemisphere as diameter be evaluated. Here (or by geometry directly) x 2 + y- + z 2 cy s =/h^g J —x:.x v«:.-^ dydx. "\ "- — x* This integral may be evaluated directly, but it is better to transform it to polar coordinates in the plane. Then >U \w I ...; ._.. rdrd = 2 J a 2 (1 - sin sin # + + cos# nsin# + cos# . . ., cos 7 = — — = — = cos 7 (cos & + ]t sm 6). Vi + p- + q- sec y Hence the new form of the area is the integral of sec y'dA' and equals the old form. The integrand dS = sec yd A is called the element of surface. There are other forms such as dS = sec (>; ri) r 2 sin 9:Wd, where (/•, u) is the angle between the radius vector and the normal : but they are used comparatively little. The possession of an expression for the element of surface affords a means of computing averages orer surfaces. For if u = u(x, y, z) be any function of (x, y, z), and z =f(x } y) any surface, the integral Tt = \j " (■'"' v> z ) ds = sjj " (:, '> !h - f) Vi+y 2 + '/-w.77/ v (i g) will be the average of u over the surface S. Thus the average height of a hemisphere is (for the surface average) 1 f 7 ■ 1 CC " 7 7 1 2 • 1 . z = ; I zdS = — ; I I z ■ - dxdy = — — — , • ira- = - ; 2 ira" J lira- J J z 'lira- 2 whereas the average height over the diametral plane would be 2/3. This illustrates again the fact that the value of an average depends on the assumption made as to the weights. 138. If a surface z =f(x. y) be divided into elements AS { , and the function u (.r, //, z) be formed for any point ('£,-, rj : , £,) of the element, and the sum Hu i \S i be extended over all the elements, the limit of the sum as the elements become small in every direction is defined as the surface Integral of the function over the surface and may be evaluated as lim^J »/(£., y,., &)AS,'= ju(.r y, z)dS That the sum approaches a limit independently of how (t,. 77,, £/) is chosen in AX- and how A.S',- a])proaches zero follows from the fact that the element u($ i} 77,-, £,)A.s'. of the sum differs uniformly from the integrand of the double integral by an infinitesimal of higher order, provided u(.r. //, z) be assumed continuous in (./', //. z) for points near the surface and VI +/','" +,/', J hl> continuous in (.r. y) over the surface. For many purposes it is more convenient to take as the normal form of t lie integrand of a surface integral, instead of in/S, the ON MULTIPLE INTEGRALS 841 product R cos yds of a function R (x, y, z) by the cosine of the in- clination of the surface to the .--axis by the element dS of the surface. Then the integral may be evaluated over either side of the surface ; for R (x, y, z) has a definite value on the surface, dS is a positive quantity, but cos y is positive or negative according as the normal is drawn on the upper or lower side of the surface. The value of the integral over the surface will be j R (:>; y, «) <-° s r ,s = if nfady according as the evaluation is made over the upper or lower side. If the function R (x, y, z) is continuous over the surface, these integrands will be finite even when the surface becomes perpendicular to the ,r//-plane, which might not be the case with an integrand of the form u (x, y, z)dS. An integral of this sort may be evaluated over a closed surface. Let it be assumed that the surface is cut by a line parallel to the .-.--axis in a finite number of points, and for convenience let that number be two. Let the normal to the surface be taken con- stantly as the exterior normal (some take the interior normal with a resulting change of sign in some formulas), so that for the upper part of the surface cosy > and for the lower part cos y < 0. Let z =f i (,r, ?/) and z =f(x, y) be the upper and lower values of z on the surface. Then the exterior integral over the closed surface will have the form Jr cos ydS =JJr [.r, ]h f x (,; yftdxdy -jjli [x, y,f Q (x, y)\dxdy, (18' "where the double integrals are extended over the area of the projection of the surface on the r//-plane. From this form of the surface integral over a closed surface it appears that a surface integral over a closed surface may be ex- pressed as a volume integral over the volume inclosed by the surface.* * Certain restrictions upon the functions and derivatives, as regards their hecoming infinite and the like, must hold upon and within the surface. It will be quite sufficient if the functions and derivatives remain Unite and continuous, but such extreme conditions an- by no means necessary. 342 INTEGRAL CALCULUS For by the rule for integration, dxdy. 2 =f (x, v) (19) Hence J R cos ydS = I — f/U or 1 1 Rdxdy = I I 2L dxdydz if the symbol O be used to designate a closed surface, and if the double integral on the left of (19) be understood to stand for either side of the equality (18'). In a similar manner f P cos adS = If Pdydz = f f f y dxdydz = f y dV, jQ cos pdS =11 Qdxdz = f f f y dydxdz = f y dV. Then / (P cos a + Q cos (3 + R cos y )dS= f (y + y + y \ dV Jo J \ x y -V ^ or II (Pdydz + Qdzdx + Rdxdy) = f f f ly + y + y ) tfa%dz follows immediately by merely adding the three equalities. Any one of these equalities (19), (20) is sometimes called Gauss's Formula, some- times Green's Lemma, sometimes the divergence formula owing to the interpretation below. The interpretation of Gauss's Formula (20) by vectors is important. From the viewpoint of vectors the element of surface is a vector dS z have been replaced by the elements dydz, dxdz, dxdy, which would be used to evaluate, the integrals in rectangular coordinates, ON MULTIPLE INTEGRALS 343 without at all implying that the projections dS x , dS y , dS z are actually rectangular. The combination of partial derivatives ^ + ^+^ = divF = v . F> (21) CX CtJ cz y / where V.F is the symbolic scalar product of V and F (Ex. 9 below), is called the divergence of F. Hence (20) becomes CdivFdV= fv.F^F= f F.tfS. (20') Now the function F(./\ y, z) is such that at each point (x, y, z) of space a vector is defined. Such a function is seen in the velocity in a moving fluid such as air or water. The picture of a scalar function u (x, ?/, z) was by means of the surfaces u = const.; the picture of a vector function F(x, y. z) may be found in the system of curves tangent to the vector, the stream lines in the fluid 7 ~ if F be the velocity. For the immediate purposes it is better to consider the function F(x, y, z) as the flux Dv, the prod- uct of the density in the fluid by the velocity. With this interpretation the rate at which the fluid flows through an element of surface dS is Jw«dS = F-dS. For in the time dt the fluid will advance along a stream line by the amount vdt and the volume of the cylindrical volume of fluid which advances through the surface will be V'dSdt. Hence ~7Jv-dS will be the rate of diminution of the amount of fluid within the closed surface. As the amount of fluid in an element of volume dV is JMV, the rate of diminution of the fluid in the element of volume is — cl)/ct where cD/dt is the rate of increase of tin' density J) at a point within the element. The total rate of diminution of the amount of fluid within the whole volume is therefore — ~2cD/itdY. Hence, by virtue of the principle of the indestructibility of matter, f F-dS = f Dv.dS = -f — dV. (20") Xow if r.,., r,,. v.. be the components of v so that P = Dv x , Q = Dv v , B = Dv z are the components of F. a comparison of (21), (20'). (20") shows that the integrals of — cD/ct and div F are always equal, and hence the integrands, cD cP cQ cR cDfr cl)v„ dDv z ct ex cy cz ex cy cz are equal ; that is. the sum P' x + Q^ + B' z represents the rate of diminution of density when i/' + jQ + kZ? is the flux vector; this combination is called the divergence of the vector, no matter what the vector F really represents. 139. Xot only may a surface integral be stepped up to a volume integral, but a line integral around a closed curve may be stepped up into a surface integral over a surface which spans the curve. To begin 344 INTEGRAL CALCULUS with the simple case of a line integral in a plane, note that by the same reasoning as above f Pdx = If — ~ dxdy, I Qdy = ( I ~ dxdy, C[P{x, y)dx + Q(x, y)dy] =ff(^ ~ ^) dxd ^ (22) This is sometimes called Green's Lemma for the plane in distinction to the general Green's Lemma for space. The oppo- site signs must be taken to preserve the direction of the line integral about the contour. This result may be used to establish the rule for transforming a double integral by the change of variable x = (u, v), y = $(11, v). For I* Qy X = I xdy = ± I a Jo Jo — (III + X 77- dV CU CO ( x d A- CU \ C( c / Oy ~| X 7T" cc\ CU dudv J J \CU CO CO CU) J\dudo. (The double signs have to lie introduced at first to allow for the case where / is negative.) The element of area dA —\J\dudv is therefore established. To obtain the formula for the conversion of a line integral in space to a surface integral, let P(.r, y, z) be given and let z =f(x, //) be a surface spanning the closed curve O. Then by virtue of , z =f(x, y), the function J i (.r, y, z) = P x (x, y) and where O' denotes the projection of O on the ./-//-plane. Xow the final double integral may be transformed by the introduction of the cosines of the normal direction to z =f(x, y). cos (3 : cos y = — a .- 1, dxdy = cos ydS, qdxdy = — cos fidS = — dxdz. OX MULTIPLE INTEGRALS 345 If this result and those obtained by permuting the letters be added, If (23) f(Pih- + Qdy + Rdz) >cR cQ\ , , (cP cR\ . 7 led dP\ , cy e\-.-/ ' \OZ CXJ \CX Clj) This is known as Stokes's Formula and is of especial importance in hydromechanics and the theory of electromagnetism. Note that the line integral is carried around the rim of the surface in the direction which appears positive to one standing upon that side of the surface over which the surface integral is extended. Again the vector interpretation of the result is valuable. Let F (x, y, z) = iP(.r, y, z) + jQ (x, y, z) + kR (x, y, z), , „ .('cR cQ\ JcP cR\ , ie(l cP\ , nt% rari F = ' W " &) + ] ( T, ~ si) + k (c, - Ji) ■ < 24 > Then Cf^It = fcurl F.rfS = fvxF.tfS, (23') where Y*F is the symbolic vector product of V and F (Ex. 9, below), is the form of Stokes r s Formula ; that is, the line integral of a vector around a closed curve is equal to the surface integral of the curl of the vector, as defined by (24), around any surface which spans the curve. If the line integral is zero about every closed curve, the surface inte- gral must vanish over every surface. It follows that curl F = 0. For if the vector curl F failed to vanish at any point, a small plane sur- face dS perpendicular to the vector might be taken at that point and the integral over the surface would be approximately [curl F\dS and would fail to vanish, — thus contradicting the hypothesis. Now the vanishing of the vector curl F requires the vanishing ir y - Q' t = o, j>: - r' x = 0, q' x - p; = of each of its components. Thus may be derived the condition that Pdx + Qdy + Rdz be an exact differential. If F be interpreted as the velocity v in a fluid, the integral | V'dr = ( v-,dx + Vydy + v z dz of the component of the velocity along a curve, whether open or closed, is called the circulation of the fluid along the curve ; it might be more natural to define 346 INTEGRAL CALCULUS the integral of the flux Dv along the curve as the circulation, but this is not the convention. Now if the velocity be that due to rotation with the angular veloc- ity a about a line through the origin, the circulation in a closed curve is readily computed. For v = axr. f v-di — Taxr.rZr = ( a.rxdr = a. ( ixdr = 2a.A. The circulation is therefore the product of twice the angular velocity and the area of the surface inclosed by the curve. If the circuit be taken indefinitely small, the integral is 2 a«dS and a comparison with (23') shows that curl v = 2 a; that is. the curl of the velocity due to rotation about an axis is twice the angular velocity and is constant in magnitude and direction all over space. The general motion of a fluid is not one of uniform rotation about any axis; in fact if a small element of fluid be considered and an interval of time St be allowed to elapse, the element will have moved into a new position, will have been somewhat deformed owing to the motion of the fluid, and will have been somewhat rotated. The vector curl v. as defined in (24), may be shown to give twice the instantaneous angular velocity of the element at each point of space. EXERCISES 1. Find the areas of the following surfaces : (a) cylinder x- + y- — ax = included by the sphere x' 2 + y- + z- — a' 2 . (/3) x/a + y/b -f z/c = 1 in first octant. (7) x' 2 + y 2 + z- = a- above r — a cosn0, (5) sphere x' 2 + y' 2 + z' 2 = a' 2 above a square \x\=*b,\y\^b,b< Jv2«. ( e ) z — xy over x' 2 + y' 2 = a' 2 . ( f) 2 az = x 2 — y 2 over r 2 — a' 2 cos , (r)) z' 2 + (x cos a + y sin a)' 2 = a 2 in first octant, (6) z = xy over r 2 = cos 2 z ) wcydxdy = fff{x, (£? V, f), z = / 3 (£, v- f) is a transformation of space which transforms the above surface into a new surface £ = ^(u, v), y — ^ 2 ( u > v )i f = ^(u, y), then Vm, v/ \£, 77/ \W, 17 \tj. f/ \«, ly \f, £/ \«, V Show 4° that the surface integral of the second type becomes where the integration is now in terms of the new variables £, 77, f in place of x, y, z. Show 5° that when Ii = z the double integral above may be transformed by Green's Lemma in such a manner as to establish the formula for change of variables in triple integrals. 6. Show that for vector surface integrals j UdS — j VJJdV. 7. Solid angle as a surface integral. The area cut out from the unit sphere by a cone with its vertex at the center of the sphere is called the solid angle w subtended at the vertex of the cone. The solid angle may also be defined as the ratio of the area cut out upon any sphere concentric with the vertex of the cone, to the square of the radius of the sphere (compare the definition of the angle between two lines 348 INTEGRAL CALCULUS in radians). Show geometrically (compare Ex. 16, p. 297) that the infinitesimal solid angle dw of the cone which joins the origin r = to the periphery of the element dS of a surface is do = cos(r, n)dS/f 2 , where (r, n) is the angle between the radius produced and the outward normal to the surface. Hence show cos(r, n) „ c _ r r-dS _ r 1 dr . a _ r d 1 , Jv __ /• „ 1 /. cos ( r,n) = /-Mb = r i or dS = _ /•_* 1 = _ r J r 2 J r 8 J f 1 dn J d)i r J dS-V where the integrals extend over a surface, is the solid angle subtended at the origin by that surface. Infer further that -C±ldS = irt or _f-*! d S = or _ f A 1 dS = tf do dn t Jq dn r d Q dn r according as the point, r = is within the closed surface or outside it or upon it at a point where the tangent planes envelop a cone of solid angle 6 (usually 2ir). Note that the formula may be applied at any point (ft 77, f) if where (x, ?/, z) is a point of the surface. 8. Gauss's Integral. Suppose that at r = there is a particle of mass m which attracts according to the Newtonian Law F=m/r' 2 . Show that the potential is V =— m/r so that F=— VI'. The induction or flux (see Ex. 19, p. 308) of the force F outward across the element dS of a surface is by definition — Feos(F, n)d*S = F*dS. Show that the total induction or flux of F across a surface is the surface integral f F-dS = - fdS.VF = - f — dS = m fdS-V l ; J J J dn J r and m=- ] f F-dS = ± f dS-V V = - ] f * m dS, 4 7r do 4 7r do 4 tv «'c dn r where the surface integral extends over a surface surrounding a point r = 0, is the formula for obtaining the mass m within the surface from the field of force F which is set up by the mass. If there are several masses »i,, m„, • • ■ situated at points (ft, Vl , j-j), (ft, 77,. f 2 ), ■ • •, let F = F 1 + F, + .... r= r, + )', + ..., Vi = - m [(ft - a-,) 2 + ( v; - mf + (ft - z,-) 8 ]- 1 be the force and potential at (j, ?/, z) due to the masses. Show that - 1 fF.dS= L fdS.Vr=- 1 V f * 1 d.s-V' W; = 3f, (25) 4 7T do 4 7r d c 4 77 jLi Jj dn n £4 where 2 extends over all the masses and 2' over all the masses within the surface (none being on it), gives the total mass M within the surface. The integral (25) which gives the mass within a surface as a surface integral is known as Gauss's Integral. If the force were repulsive (as in electricity and magnetism) instead of attracting (as in gravitation), the results would be V = m/r and - 1 - f F .dS = ^ fdS.VV=-W f ' J ^d.S=y / m l = 3f. (25') 4 7rd 4tt Jj iv^/Jodnn ^4 OX MULTIPLE INTEGRALS 349 9. If V = i f-j (-k — be the operator defined on page 172, show ex cy cz ex cy cz \cy cz] \cz ex J \cx dy J by formal operation on F = Pi + Qj + Hk. Show further that V*V U = 0, V-VxF = 0, (V-V) (*) = (~ + ~ + -4^ (*), \cx- cy' 1 cz' 1 ! Vx(VxF) = V (V«F) - (V'V) F (write the Cartesian form). Show that (V*V) U = V«(VJ7). If u is a constant unit vector, show cF cF (F dF (u»V)F = — cos a H cos/3 -\ cos 7 = — ex cy cz ds is the directional derivative of F in the direction u. Show (tZr-V) F = <1F. 10. Green's Formula (space). Let F(x, y, z) and G (x, y, z) be two functions so that VF and V(? become two vector functions and FVG and GV F two other vector functions. Show V.(FVG) = VF-VG + FV.VG*, V.(GYF) = VF.VG + GV.VF, or A^ G ) + I(f" ; UA(^) ex \ ex / cy \ cy I cz\ cz ) cF cG cF cG cFcG r ll 2 G c' 2 G c 2 G\ ex ex dy cy cz dz \cx 2 cy 1 cz 2 J and the similar expressions which are the Cartesian equivalents of the above vector forms. Apply Green's Lemma or Gauss's Formula to show CFVG'dS = CvF'VGdV + fFV.VGdV, (26) f GVF'dS = CvF'VGdV + fGV.VFdV, (26') C (FVG - GVF)'dS = f{FV.VG - GV.VF) dV, (26") r^dG 1L , r/cF c;G cF cG dFbG\ jrT /• r ,/e 2 G c°-G c 2 G\ ,_ Jo aw ^ \c.c cx c?/ c?/ cz cz/ J \cx 2 cy 2 - cz 2 1 r( F dG G dF \ ds _ C\f(°* g — b * G \ G ( £2F — — Yldv Jo\ cZ'i dn/ ^ L \?^ 2 cy 2 Sz 2 / \cx 2 cy' 2 cz 2 ) \ The formulas (26), (26'), (26") are known as Green's Formulas; in particular the first two are asymmetric and the third symmetric. The ordinary Cartesian forms of (26) and (26") are given. The expression e 2 F/cx 2 + c 2 F/cy' 2 + c 2 F/cz 2 is often written as AF for brevity ; the vector form is V'VF. 11. From the fact that the integral of F.cZr has opposite values when the curve is traced in opposite directions, show that the integral of VxF over a closed surface vanishes and that the integral of V'VxF over a volume vanishes. Infer that V.VxF = 0. 350 INTEGRAL CALCULUS 12. Reduce the integral of VxVU over any (open) surface to the difference in the values of Z7 at two same points of the bounding curve. Hence infer VxVJ7 = 0. 13. Comment on the remark that the line integral of a vector, integral of F«dr. is around a curve and along it, whereas the surface integral of a vector, integral of F»cZS, is over a surface but through it. Compare Ex. 7 with Ex. 16 of p. 297. In particular give vector forms of the integrals in Ex. 1(5. p. 297. analogous to those of Ex. 7 by using as the element of the curve a normal dn equal in length to dr, instead of dr. 14. If in F = Pi + Qj + Pk, the functions P. Q depend only on x. y and the function E = 0, apply Gauss's Formula to a cylinder of unit height upon the a;?/-plane to show that Cv.FdV = f F-dS becomes ff(~ + °^) dxdy = f F-dn, where dn has the meaning given in Ex. 13. Show that numerically F-dn and Fxdr are equal, and thus obtain Green's Lemma for the plane (22) as a special case of (20). Derive Green's Formula (Ex. 10) for the plane. 15. If fF-dr = fG'dS, show that f(G- YxF).dS = 0. Hence infer that if these relations hold for every surface and its bounding curve, then G = VxF. Ampere's Law states that the integral of the magnetic force H about any circuit is equal to \tt times the flux of the electric current C through the circuit, that is, through any surface spanning the circuit. Faraday's Law states that the integral of the electromotive force E around any circuit is the negative of the time rate of flux of the magnetic induction B through the circuit. Phrase these laws as integrals and convert into the form 4 ttC = curl H, - B = curl E. 16. By formal expansion prove V-(ExH) = H-VxE - E'VxH. Assume VxE = — H and VxH = E and establish Poynting's Theorem that f(ExH).(/S = - C - Tl(E.E+ H.H)dT T 17. The " equation of continuity " for fluid motion is cD cT)v T cDv„ cDv z . dU T /cv r cv„ cv,\ 1 1 -] 1 + 1 = or — + J) I — + -' + — = 0, ct ex, cy cz dt \ ex cy cz / where I) is the density, v = ir T + jr,, + kr s is the velocity. cD/ct is the rate of change of the density at a point, and dD/dt is the rate of change of density as one moves with the fluid (Ex. 14. p. 101). Explain the meaning of the equation in view of the work of the text. Show that for fluids of constant density V-v = 0. 18. If f denotes the acceleration of the particles of a fluid, and if F is the external force acting per unit mass upon the elements of fluid, and if p denotes the pressure in the fluid, show that the equation of motion for the fluid within any surface may be written as V iDdV = VFPWr - V pdS or CiDdY = fFDdV- f pdS, ON MULTIPLE INTEGRALS 351 where the summations or integrations extend over the volume or its bounding sur- face and the pressures (except those acting on the bounding surface inward) may be disregarded. (See the first half of § 80.) 19. By the aid of Ex. 6 transform the surface integral in Ex. 18 and find CntdV = f(IW - Vp) dV or — = F - - Vp J J dt 2 D as the equations of motion for a fluid, where r is the vector to any particle. Prove . , d 2 r dv (V , cv 1. dt 2 dt it ct 2 , * d , 7 , dv , dr , d' 2 r 1 , , (P) - (dr.v) = dr. - + d .v = dr. _ + - d v-v . dt at dt dt 2 2 20. If F is derivable from a potential, so that F = — VL", and if the density is a function of the pressure, so that dp/I) = dP, show that the equations of motion are £Z_ V xvxv=- V (u + P + lvA, or -(y.di)=-d(u+P--v 2 \ after multiplication by dr. The first form is Helmholtz's, the second is Kelvin's. Show X- r , y,z d dp x < 'J< z r l.d - T > y> z r — (v.dr) = — I v-dr = — U + P r 2 and | v-dr = const. .., b, c dt dtJa.h.c 2 _]n,b,c Jq In particular explain that as the differentiation d/dt follows the particles in their motion (in contrast to l/ct. winch is executed at a single point of space), the integral must do so if the order of differentiation and integration is to be inter- changeable. Interpret the final equation as stating that the circulation in a curve which moves with the fluid is constant. 21. T ,c 2 r c"-l l 2 l n , rVjcl \ 2 (cl\ 2 /cU\*l.- f 7 .dr If — + — + —j- =0, show / — ) + — +( — \dV= I L —dS. ex- cy- cz- J \_\cx/ \cy J \cz I J Jq dn 22. Show that, apart from the proper restrictions as to continuity and differen- tiability, the necessary and sufficient condition that the surface integral CfPdydz + Qdzdx + Hdxdy = C pdx + qdy + rdz depends only on the curve bounding the surface is that P' r + Q' + B' z — 0. Show further that in this case the surface integral reduces to the line integral given above, provided p, q, r are such functions that r' — q' z = P. p' z — r' x = Q, q' x — p' — R. Show finally that these differential equations forp, q. r may be satisfied by p=C Qdz -fli (x. y. z ) dy. q = - f *Pdz, r = ; and determine by inspection alternative values of p, q. r. CHAPTER XIII ON INFINITE INTEGRALS 140. Convergence and divergence. The definite integral, and hence for theoretical purposes the indefinite integral, has been defined, f f(x)dx, F(x)= f f(x)dx, when the function f(x) is limited in the interval a to b, or a to x ; the proofs of various propositions have depended essentially on the fact that the integrand remained- finite over the finite interval of integration (§§ 16-17, 28-30). Nevertheless problems which call for the determina- tion of the area between a curve and its asymptote, say the area under the witch or cissoid, I ' Sa 3 dx x 1 + 4 w = 4 rr tan" 2 a = 4 ira' 1 . Jr- a xMx V2 a — of. 3 ira~, have arisen and have been treated as a matter of course.* The inte- grals of this sort require some special attention. When the integrand of a definite integral becomes infinite within or at tlte extremities of the interval of integration, or when one or both of the limits of integration become infinite, the integral is called, an infinite integral and is defined, not as the limit of a sum, but as the limit of an integral with a, variable limit, that is, as the limit of a function. Thus f f(x)dx = lim f(x)dx — lim \x)=Tf{x)dx U a '(.>■)= £f(x)dx infinite upper limit, integrand f(U) = co. These definitions may be illustrated by figures which show the connec- tion with the idea of area between a curve and its asymptote. Similar definitions would be given if the lower limit were — cc or if the inte- grand became infinite at x = a. If the integrand were infinite at some intermediate point of the interval, the interval would be subdivided into two intervals and the definition would be applied to each part. * Here and beluw the construction of figures is left to the reader. 352 OX INFINITE INTEGRALS 853 Now the behavior of F(x) as x approaches a definite value or becomes infinite may be of three distinct sorts ; for F(a") may approach a definite finite quantity, or it may become infinite, or it may oscillate without approaching any finite quantity or becoming definitely infinite. The examples r x Sa 3 dx ,. f* 8a?dx I ~5 : — ' ; = bin I — : — 7, = = 4 a 2 tan -1 — 9 i £ dx ■ — = lim dx — = log x X I cos xdx = lim P Jo osxd.r = sin x a limit, becomes infinite, no limit, oscillates, no limit, illustrate the three modes of behavior in the case of an infinite upper limit. In the first case, where the limit exists, the infinite integral is said to converge; in the other two cases, where the limit does not exist, the integral is said to diverge. If the indefinite integral can be found as above, the question of the convergence or divergence of an infinite integral may be determined and the value of the integral may be obtained in the case of convergence. If the indefinite integral cannot be found, it is of prime importance to know whether the definite infinite integral converges or diverges ; for there is little use trying to compute the value of the integral if it does not converge. As the infinite limits or the points where the integrand becomes infinite are the essentials in the discussion of infinite integrals, the integrals will be written with only one limit, as / f(x)dx, f f{ " )dx, f(x) dx. To discuss a more complica :d combination, one would write d.r J VV'log.r Jo J € Ji J^ Vrlogx and treat all four of the infinite integrals Jr e~ x dx r 1 e~ x dx r e~ x dx f r K, where m and 31 are the minimum and maximum values of E (x) between K and x. Xow let x become infinite. As the integrands are positive, the integrals must increase Avith x. Hence (p. 35) if I f(x) dx converges, I f(x) E (x) dx < 31 I fix) dx converges, Jk Jk Jk if / /(•'•) E (,r) dx converges, I f(x) dx < - / /< x ) E ( ./■ ) dx converges ; Jk '" Jk ' and divergence may be treated in the same way. Hence the integrals (1) converge or diverge together. The same treatment could be given for the case the integrand became infinite and for all the variety of hypotheses which could arise under the theorem. This theorem is one of the most useful and most easily applied for determining the convergence or divergence of an infinite integral with an integrand which does not change sign. Tims consider the case 3 3 C* xdx ft-V x- 1 2 dx r,, , r p r*> dx 1 x J ( C(X + X 2 )2 L".C + X-J X- L"-'' + .'-J J X- X Here a simple rearrangement of the integrand throws it into the product of a func- tion K(x). which approaches the limit 1 as x becomes infinite, and a function !/.;■-. the integration of which is possible. Hence by the theorem the original integral converges. This could have been seen by integrating the original integral : but the integration is not altogether short. Another case, in which the integration is not possible, is r 1 dx r 1 1 dx J v T — x* J Vl + x 2 Vl + x V 1 — x 1 r l dx i 1 1 E(x)= - =, f ,_ = -2VTT7' . Vl + x- VI + x J VI — x ON INFINITE INTEGRALS 355 Here -E'(l) = \. The integral is again convergent. A case of divergence would be r dx r 1 dx _ . , 1 r dx 2 1 j ={ , E(x) — , | — = . Jo (2x-x^'i Jo (2-x)iA (2-x)i Jo z? Vxlo 141. The interpretation of a definite integral as an area will suggest another form of test for convergence or divergence in case the inte- grand does not change sign. Consider two functions f(x) and ^(x) both of which are, say, positive for large values of x or in the neigh- borhood of a value of ./• for which they become infinite. If the curve y = \\j(x) remains above y = f(x), the integral off(x) must converge if the integral of \\i(x) converges, and the integral of \ty(x') must diverge if the integral of f(x) diverges. This may be proved from the definition. For /(./•) < tf/(x) and | f(x)dx g\j/, and the result above may be applied to show that the integral of f(x) diverges if that of \p (x) does. For an infinite upper limit a direct integration shows that /" dx _ -1 _1_ x k ~ 1; - 1 ar*- or lo< converges if k > 1, ._. diverges if 1c ^ 1. ( "^ Now if the test function {x) be chosen as l/x k = x~ k , the ratio f(x)/(x) becomes x K 'f(x), and if the limit of the product x k f(x) exists 356 INTEGRAL CALCULUS and may be shown to be finite {or zero) as x becomes infinite for any choice of k greater than 1, the integral off(x) to infinity will converge ; but if the product approaches a finite limit (not zero) or becomes infinite for any choice of k less titan or equal to 1, the integral diverges. This may be stated as : The integral of f(x) to infinity will converge if f(x) is an infinitesimal of order higher than the first relative to 1/x as x becomes infinite, but will diverge if f(x) is an infinitesimal of the first or lower order. In like manner f da (b-x) k k — 1 (b-xf- \og(b-x) converges if k 0, the integrand never becomes infinite, and the only integral to examine is that to infinity ; but if n < the integral from has also to be consid- ered. Now the function c- x for large values of x is an infinitesimal of infinite order, that is, the limit of x k + n e~ x is zero for any value of k and n. Hence the integrand x"e~ x is an infinitesimal of order higher than the first and the integral to infinity converges under all circumstances. Forx = 0, the function e~ x is finite and equal to 1 ; the order of the infinite x n e~ x will therefore be precisely the order n. Hence the integral from converges when n > — 1 and diverges when n =g — 1. Hence the function T(a)=f x a - 1 e- x dx, a > 0, Jo defined by the integral containing the parameter a, will be defined for all positive values of the parameter, but not for negative values nor for 0. Thus far tests have been established only for integrals in which the integrand does not change sign. There is a general test, not particularly useful for practical purposes, but highly useful in obtaining theoretical results. It will be treated merely for the ease of an infinite limit. Let ■V)= P. U x' *•(*)= J /(•'•) )=\ f(x)dx } x',x">K. (4) Now (Ex. 3, ]>. 44) the necessary and sufficient condition that F(x) approach a limit as x becomes infinite is that F(x") — F(x') shall approach the limit when x' and x", regarded as independent varia- bles, become infinite ; by the definition, then, this is the necessary and sufficient condition that the integral of f(x) to infinity shall converge. Furthermore ON INFINITE INTEGRALS 357 if f f(x) | dx converges , then j f(a )dx (5) must converge and is said to be absolutely convergent. The proof of this important theorem is contained in the above and in f f(x)dx*f'\f(x)\dx. To see whether an integral is absolutely convergent, the tests estab- lished for the convergence of an integral with a positive integrand may be applied to the integral of the absolute value, or some obvious direct method of comparison may be employed; for example, /cos xdx C '' 1 dx . . . —, : =§ | — : winch a- + ,-- J a 2 + x 2 converges, and it therefore appears that the integral on the left converges abso- lutely. When the convergence is not absolute, the question of con- vergence may sometimes be settled by integration by parts. For suppose that the integral may be written as f x f(x)dx=r f(x)dx = (x)t(x)dx 4>(x)f$(x)dx -C '(x)jij i (x)dx 2 by separating the integrand into two factors and integrating by parts. Xow if, when x becomes infinite, each of the right-hand terms approaches a limit, then I f(x)d> lim >/ (•'0 $(x)d* and the integral of f(x) to infinity converges. x x cos xdx As an example consider the convergence of i I cf> '(./■) I if/(x)dxdx, 00 x I cos x I dx r x x cos xdx TX r — -. Here I J a- + x 2 J + X- J a- + x- does not appear to be convergent ; for, apart from the f actor | cos j | which oscillates between and 1, the integrand is an infinitesimal of only the first order and the integral of such an integrand does not converge ; the original integral is therefore apparently not absolutely convergent. However, an integration by parts gives /■<•' x c a 2 lim x cos xdx x sin x + x 2 a- + x- a 2 + x : \ x r x x- — a- - — | — - —cos - J (x 2 + a 2 ) 2 xdx, r x x- — a 2 , r x dx I cos xdx < — J {x- + dx, (f) r (Zj , (,) r._^ == J Jo Vox-x 2 J i xVx 2 - 1 «/o 1 — .f 4 *^°(1 rif t/ u 1 — X (X) f 1 . ^ =. A < 1, A: = l t (m) f 1 J l ~ k2x * dx ,k <1. Ju \ (_1 -x 2 )(l -/c-' 2 x 2 ) Ju \ 1-/- 2. Point out the peculiarities which make these integrals infinite integrals, and test the integrals for convergence or divergence: r 1 1 1\" r l lo"-x (a) (log-) dx, conv. if n > — 1, div. if n S — 1, (/3) f -dx, Jo \ x/ Jo 1 — X (7) I (— logx)"dx, (5) j 2 logsinxdx, (e) f xlogsinxdx, Ju Jo Jo v ^J \ x/l + x 2 v " Jo (sinx + cos x)* Jo \ x/ , . r r e~ x dx , , r r - - 1 , , r 1 , ?rc , (0 I - , (k) x x dx, (\) I loir x tan — dx, Ju Vxlog(x + l) Ju Ju - «oo X ar-l ^ + =o -00 x a ~ 1 dX M) I — - ;dx, (") I e— dx, (0) I — -, Jo 1+X J- x Jv (1+X)- ^»sin 2 x fUogxdx , /•« -(*-"Y (tt) — — dx, (p) _ , ( xsinx , .„. r x , , . . r* cosx , (tj) I dx, {6) I e-^cosbxdx, (i) — — dx, J o x 2 + A; 2 Jo Jo -y/j. . , /•* a ,,. /"" sin x cos ax , (a:) / a- ir - 1 t— :reos Pcos(xsin/3)dx, (X) / — dx, Jo Jo X C ^ c "° /x 2 ct^ \ z* 00 sin^-'x^ (m) I cos x 2 cos 2 axdx, (v) I sin) 1 )dx, (o) { '- dx. Jo Jo \2 2x 2 / Jo x m 5. If /^(x) and / 2 (x) are two limited functions integrable (in the sense of §§ 28-30) over the integral a == x =§= b, show that their product /(x) =/ 1 (x)/ 2 (x) is integrable over the interval. Note that in any interval 5,-, the relations m u m 2 i s im s J/- ^ M u M 2 i and J/i^,- — m u m 2 i = M u M 2 i — M u m2i + M\im-2i — mnm-2i = MnO-a + vuiOn hold. Show further that £f x {x)f 2 {x) dx = lilll ^ / 1 (ft)/ 2 (f/)«i = liin^fr) [ i £ 3 "' + 1 / 2 (x)tZx-£" + 1 {/ 2 (f l ) -/ 2 (x)dx}l or f b f(x)dx = lim^/ 1 (f,-)jr ar ' +1 / 2 WdJ: = lim ^ /j (f e ) r c V 2 (x) czx - r " / 2 (x) dx] , or £f{x)dx = f^ l )jj„{x)dx + lim^ [/,,(£,-) -/„(&_!)] £ux)dx. 6. TAe Second Theorem of the Mean. If /(x) and 0(x) are two limited functions integrable in the interval a =s x =§ b, and if (x) is positive, nondecreasing, and less than K, then f V (x)/(x) dx = K f f(x) dx, a S £ ^ 6. And, more generally, if (x) satisfies — cc < & =§ (x) =s K < oo and is either nondecreasing or nonincreasing throughout the interval, then f (x)/(x) dx = k f f(x) dx +■ Jl f /(x) dx, a S £ g h. Iii the first case the proof follows from Ex. 5 by noting that the integral of (x)f(x) may be regarded as the limit of the sum (* 2 ) - &) + • ■ • + (£„) - (€„ -i) + A' - (£„)] = ^if if [x be a properly chosen mean value of the integrals which multiply these coeffi- cients : as the integrals are of the form | f(x) dx where £ = a, x,. • • • , x n , it follows 160 INTEGRAL CALCULU S tliat fj. must be of the same form where a = £ = b. The second form of the theorem follows by considering the function

    d path in the z-plane. It is known that if the points where F(z) = X(x, y) + iY(x, y) ceases to have a derivative F'(::). that is, where X(x, y) and Y(x, //) cease to have continuous first par- tial derivatives satisfying the relations A','. = Y'„ and X' y = —Y' x , are cut out of the plane, the integral of F(z) around any closed path which does not include any of the excised points is zero (§ 124). It is some- times possible to select such a function F(z) and such a path of integration that part of the integral of the complex function reduces to the given infinite integral while the rest of the integral of the complex function may be computed. Thus there arises an equation which determines the value of the infinite integral. -A+iB A + iB dz=+dx dz=+ idy dz= idy , >-n dz=dx -A (J Consider the integral J X which is known to conver; r *A^dx = r e ~ ^dx = r°>^_r<°e Z i* Jo ./• Jo 2 ix Jo 2 Ix Jo 2 ix Now dx suggests at onrc that the function e'-/z be examined. This function lias a definite derivative at every point except z = 0, and the origin is therefore the only point ON INFINITE INTEGRALS 361 which has to be cut out of the plane. The integral of e iz /z around any path such as that marked in the figure * is therefore zero. Then if a is small and A is large, p e iz pA t ix pli e iA-y p-A e ix-B 0=1 — dz = I — dx + \ idy + j dx Jq z Ja x Jo A + iy J a x + IB p e - !A -y p-a e ix p + a e iz + ; idy + —dx+ — dz. Jjl — A + IV J - A X J-a Z + iy A e-*dx _ /• + «**_ r + a l + v p-a e x p-A e ix p A e - ixdx p + a e iz p~ But I — dx = — / — dx = — I and I — dz=\ J- A X J-a X Ja X J-a Z J- a the first by the ordinary rules of integration and the second by Maclaurin's Formula. Hence Xpiz p A (Ax g— ix p + a (Jg — dz = | - + I h four other integrals. J Z J a X ■ J- a Z It will now be shown that by taking the rectangle sufficiently large and the semicircle about the origin sufficiently small each of the four integrals may be made as small as desired. The method is to replace each integral by a larger one which may be evaluated. I pB e iA-y | pB \ e iA\ e -y pB\ ]] I -7— —idy == | ' . . \i\dy< I -e-vdy<-. \Jo A + iy Jo \A + iy \ Jo A A These changes involve the facts that the integral of the absolute value. is as great as the absolute value of the integral and that e iA - 'J = e' A e~ '-'. \ e iA \ = 1, | A -f iy I > A . e~y <1. For the relations [ e iA | = 1 and \A -\- iy\>A, the interpretation of the quantities as vectors suffices (§§ 71-74) ; that the integral (if the absolute value is as great as the absolute value of the integral follows from the same fact for a sum (p. 154). The absolute value of a fraction is enlarged if that of its numerator is enlarged or that of its denominator diminished. In a similar manner | A (Ax-F, pA ( .-H [ I p0 e -iA-y I ]J I dx < | dx = 2c-< ; — , ! I idy\<—. \Ja x + iB J-a J! B Jn-A + iy \ A X+ a y. p + a dz\ r n ~ dz == | 1 7) J - — = I | r] | d = \ 7r. Hence I r w c Ui '" !> Ric''hl(h\ p* Re-x*h , p\ Bc- R8in , I - s / dd> = 2 j A dd>. | Jo £ 2 e«* + Jfc a | Jo R 2 -k 2 Jo R 2 -k 2 Now by Ex. 28, p. 11, sin > 2 /ir. Hence the integral may be further increased. e iii< J4, Rie\ , pi Re~ K * d Jr-n e iR, • lUeit'd\ p o IV 1 v 2 '"'/' + k 2 " Jo + k 2 | Jo R 2 - /.■- /.'- - k- (c-R-1). -. p c' z dz p <'~ dz p (e~ k \ dz Moreover. = | = / h v) - ~ > Jaa'a Z 2 + k 2 Jaa'a Z + ik Z - ik Jaa'a \2ki ) Z - ik where 77 is uniformly infinitesimal with the radius of the small circle. But p dz , . p < !z dz 2wc- k - — — 2m, and (- t, Jaa'a Z-ik Jaa'a Z 2 + k 2 2 k wliere |f| == 2 ire if e is the largest value of \t][. Hence finally ON INFINITE INTEGRALS 363 J" A> eos.r o x- + k~ - e~ k + f+ i^T^ (e ^- 1 >- By taking the small circle small enough and the large circle large enough, the last two terms may be made as near zero as desired. Hence j: o x' 2 + k 2 dx ■2k {') dz is exact It may be noted that, by the work of § 12(5. J ' Jaa'a Z + kl Z — ki 2 lii and not merely approximate, and remains exact for any closed curve about z — kl which does not include z = — ki. That it is approximate in the small circle follows immediately from the continuity of e iz /{z + ki) = c~ k /2kl + 17 and a direct inte- gration about the circle. c x x a ~~ * As a third example of the method let / dx be evaluated. This integral Jo 1 + x will converge if < a < 1, because the infinity at the origin is then of order less than the first and the integrand is an infinitesi- mal of order higher than the first for large values of x. The function z a ~ V(l + z) becomes infinite at z — and z =— 1, and these points must be excluded. The path marked in the figure is a closed path which does not contain them. Xow here the integral back and forth along the line aA cannot be neglected; for the function has a fractional or irrational power z a ~ x in the nu- merator and is therefore not single valued. In fact, when z is given, the function z a ~ 1 is deter- mined as far as its absolute value is concerned, but its angle may take on any addition of the form 2 irk (a — 1) with k integral. Whatever value of the function is assumed at one point of the path, the values at the other points must be such as to piece on continuously when the path is followed. Thus the values along the line aA outward will differ by 2 7r(« — 1) from those along A a inward because the turn has been made about the origin and the angle of z has increased by 2tt. The double line be and cb. however, may be disregarded because no turn about the origin is made in describing cdc. Hence, remembering that e 7 " =—1, /i ?a — 1 n fa — lp(a — l)d>i n A ya — 1 /*1tt Aapat&i 0= / dz= d(re*')= / -dr+ ~id

    Z a ~ l e 27 "'dr + | dz + - dz. A 1 + re 2w( Jabba 1 + 2 Jcdc 1 + Z Jr. A yCl — 1 pfl ylX — 1 p'2 TTCli r* A y(Z — 1 dr + I ■ dr = / (1 - &" ai )dr, a 1 + >• J A 1 + r J a 1 + r I r' 27T A a e a < . , I r 2rr A a I ,.| , 2irA a I Jo 1 + Aefi I Jo A-l\ | A-l J' <•> yi f2-rv (l a 2 7T(l a dz\=\ I — — id i ^ _ d

    (x)2dx = / /(/<■) dx + i $ (x) dx when one of the limits is infinite or one of the functions becomes infinite in the interval. For, the fact that the integral on the left converges is no guarantee that either integral upon the right will converge; all that can be stated is that If one of the integrals on the right converges, the other will, and the equation will be true. The same remark applies to integration by parts, 1 f(x)V{x)dx = /(*)*(*) f'(x)(x)dx. If in the process of taking the limit which is required in the defi- nition of infinite integrals, two of the three terms in the enaction approach limits, the third will approach a limit, and the equation will be true for the infinite integrals. The formula for the change of variable is I = *(0 Ji where it is assumed that the derivative '(t) is continuous and does not vanish in the interval from t to T (although either of these con- ditions may be violated at the extremities of the interval). As these two quantities are equal, they will approach equal limits, provided they approach limits at all, when the limit f(x)dx = I fl4»(t)^'(t)dt required in the definition of an infinite integral is taken, where one of the four limits a, b, t , t x is infinite or one of the integrands becomes 366 INTEGRAL CALCULUS infinite at the extremity of the interval. The formula for the change of variable is therefore applicable to infinite integrals. It should be noted that the proof applies only to infinite limits and infinite values of the integrand at the extremities of the interval of integration : in ease the integrand becomes infinite within the interval, the change of variable should be examined in each subinterval just as the question of convergence was examined. p * sin x it As an example of the change of variable consider f dx = and take x = ax'. J o x 2 Jr>x=v sin ax' , , r + °° sin ax' , , r~ r sin ax' , , r x '= x sinax' , , dx — | dx or = | ax = — \ ax. x=o x' J.r' = o x' «/.)' = o x' Jx'=a x' according as a is positive or negative. Hence the results r x sin ax . , it ., n , tt .. . ,,_ s I -dx=+- if a>0 and if a < 0. (10) Jo x 2 2 Sometimes changes of variable or integrations by parts will lead back to a given integral in such a way that its value may be found. For instance take 1 = f 2 log sin xdx -— - I log cos ydij = f " log cos ydy, y = - — x. t/0 «';r <-') 2 Then 2 7= f " (log sin x + 1< »g cos x) dx = f " log tllLJI dx \ P~ TT f ■> 7T = - | loirsinxdx lo<*2 = | "log sin xdx log 2. 2 Jo 2 Jo 2 I = f ' log sin xdx = - - log 2. (11) Hence Here the first change was // = .- — — x. The new integral and the original one were then added together (the variable indicated under the sign of a definite inte- gral is immaterial, p. 20). and the sum led back to the original integral by virtue of the substitution y — 2x and the fact that the curve y = log sin.? is symmetrical with respect to x = \ it. This gave an equation which could be solved for I. EXERCISES 1 T . Z( ''~ <• r ,- , C' r - XSinX , 7T 1. Integrate • as for the case of (<)• to show I dx = - e~ 1 '. z- + k 2 Jo x 2 + k 2 2 2. By direct integration show that ( e-( a - hr * z dz converges to (« — bi)~ l , when a > and the integral is extended along the line y = 0. Thus prove the relations I e - "•'' cos bxdx = - — , I e- ar sin6xdx = , a > 0. Jo a- -f h- Jo a 2 + b' 1 Along what lines issuing from the origin would the iriven integral eonvenre '.' ON INFINITE INTEGRALS 367 = — To integrate about z = — 1 use the binomial o (1 + x) 2 sin air expansion z*- 1 = [- 1 + 1 + z]"- 1 = (- l)"- 1 ^ + (1 - a) (1 + z) + t?(1 + z)], t; small. 4. Integrate e~ z "~ around a circular sector with vertex at z = and bounded by the real axis and a line inclined to it at an angle of iw. Hence show "dx = — 2 e* 7 " ( (cos r 2 — i sin r 2 ) dr = I c,- x "% Jo Jo ( cos x 2 dx = | sin x 2 dx = - a /— • Jo Jo 2 \ 2 5. Integrate e-^ 2 around a rectangle y = 0, y = B, x = ± A, and show | e~ •'" cos 2 axdx = I Vttc- a ~, I e~ j3 sin 2 axdx = 0. Jo J- 00 6. Integrate z a - 1 e~ z , < a, along a sector of angle q < \ ir to show secag j x a - 1 e- XC0B i cos(xsinq)dx Jo — cscaq ( x a ~ 1 e~ XC0B i sin (x sin q)dx = I x a - x e~ x dx. Jo Jo 7. Establish the following results by the proper change of variable : , r™ cos ax , -Ke- ak . , . r™ x a ~\lx -n-p a - 1 a) | dx = , a > 0, (p) I - - = -f , /3 > 0, 'Jo i 2 + F 2 A: ,3 + x sm cnr 7) I e- ah: *dx = — Vtt, 'Jo 2 a — JiL e) I e - a "- ) " cos bxdx = , a > 0, Jo 2 a . r™ cosx , z* 00 sinx , V 77) I — —dx= I ax = A?— , Jo Vx Jo Vx >- 8. By integration by parts or other devices show the following : xlogsinxdx = 7r' 2 log2, (p) I dx = — , 2 Jo x' 2 2 /•« sinx cos ax 7r 7r ., , 1 7) / — — dx = - if — 1 < a < 1, or - if a = ± 1, or if a > 1, Jo x 2 4 „ , /• " „ , , , Vi . . r °° , , , , 3 Vw- S) x 2 e- 0, converges. Then if p > , q > 0, Ja x Jo J* aioL"" 31 •'/'a £ ■'9a £ J snow r- /(i»)-/(v') dx = lim r a m *5 =/(0) log « . Jo X « = t/pa X p /•» sinpx — singx ,.x e -,,x_ e - 7 x f/ Hence (a) / — dx = 0, (/3) — dx = log - , Jo x Jo x j) /.lj.ji-1 _ jfl-l n n* c os,/ - — COS "X , (7) I dx — log - , (0) j — dx = \oza. Jo loii'X 7) Jo X . 1 j- v - 1 — j-7 - 1 q , ^» * COS X — COS ax dx — log - >gx p 10. If/(x) and/'(x) are continuous, show by integration by parts that f 1 ' r" sin /,'./■ 7r lim I /(x) sin fcxdx = 0. Hence prove lim I /(x) — — dx = -/(0). *- = x Jn ' A- = x Jo ' x 2 r„. • r"^-/ .sinfcc, ..,,, fsinA-x , , r a f(x) — /(0) . , , 1 Write f(x) dx — /(0) I — dx + — — : smfrxdx. Jo x Jo x Jo x J Apply Kx. G, pj. 359, to prove these formulas under general hypotheses. 11. Show that lim f f(x) '- -dx — if b > a > 0. Hence note that k — x o'rt X /> ^ sin A*x /° ^ sin K'r lim lim I /(x) ' -d~^lim lim j /(x) '— — dx, unless /(0) = 0. k = x a = J« X a = t = x J n X 144. Functions defined by infinite integrals. If the integrand of an integral contains a parameter (§ 118), the integral defines a function of the parameter for every value of the parameter for which it converges. The continuity and the differentiability and integrability of the func- tion have to be treated. Consider first the ease of a:i infinite limit / /(.,-, a)dx = j f(x, a)(lr + /,*(>, a), 11 = I f{:r, n)dx. If this integral is to converge for a given value a = a , it is necessary that the remainder 11 (./•. a ) can be made as small as desired by taking x large enough, and shall remain so for all larger values of x. In like manner if the integrand becomes infinite for the value x = b, the condition that / f(x, a) dx = j /(./', n)dx + 11 (x, a), R== f /(•'">'* dx converge is that /?(.'", n\ can be made as small as desired by taking x near enough to b. and shall remain so for nearer values. Now for different values of a, the least values of ./• which will make //!.'■. a) == e. when e is assigned, will probably differ. The infinite inte- grals are said to converge uniformly for a range of values of a such as ON INFINITE INTEGRALS 309 n Q gaga;, when it is possible to take x so large (or x so near b) that R (x, a) | < e holds (and continues to hold for all larger values, or values nearer b) simultaneously tor all values of a in the range a Q =§ a =E a . The most useful test for uniform convergence is contained in the theorem : If a jtositive function (x) can be found such that f cj> (./•) dx converges and <£ (#) == \f(x, a) I for all large values ofx and for all values of a in the interval a '== a =j= a , the integral of f(x, a) to infinity converges uniform I y (and absolute!//) for the range of values in a. The proof is contained in the relation j f(x, a)dx\^ f (x)dx < c, which holds for all values of a in the range. There is clearly a similar theorem for the case of an infinite integrand. See also Ex. 18 below. Fundamental theorems are: * Over any interval a == a == a where an infinite integral converges uniformly the integral defines a con- tinuous function of a. This function maybe integrated over any finite interval where the convergence is uniform by integrating with respect to a under the sign of integration with respect toa\ The function may be differentiated at any point a% of the interval «„ = « =g a ± by differ- entiating with respect to a: under the sign of integration with respect to x provided the integral obtained by this differentiation converges uniformly for values of a in the neighborhood of a$. Proofs of these theorems are given immediately below, t To prove that the function is continuous if the convergence is uniform let yp (a) = j f(x, a)dx = ( f(x, a)dx + R(x, a), a S a ^ a v f{a + Aa) — j f(x, a + Aa)dx + II (x, a + Aa), J a |A^|^ I [f(x, a + Acx)-f(x, a)]dx + I! (x, a + Aa) \ + ]Ii(x, a)'. * It is (if course assumed that /O, d) is continuous in (.<•, or) for all values of x and a under consideration, and in the theorem on differentiation it is further assumed that f„(x, or) is continuous. t It should be noticed, however, that although the conditions which have been imposed are sufficient to establish the theorems, they are not necessary; that is. it may happen that the function will be continuous and that its derivative and integral may be obtained by operating under the sign although the convergence is not uniform. In this case a special investigation would have to be undertaken : and if no process for justifying the continuity, integration, or differentiation could be devised, it might be necessary in the case of an integral occurring in some application to assume that the formal work led to the right result if the result looked reasonable from the point of view of the problem under discussion, — the chance of getting an erroneous result would be tolerably small. 370 INTEGRAL CALCULUS Now let x be taken so large that | i? | X ] fX f(x, a)dx is continuous in a and hence / [/(£, a + Aa) —f(x, a)]dx can be made less than e by taking Aa small enough. Hence | A^|<3e; that is, by taking Aa small enough the quantity | A^ [ may be made less than any assigned number 3e. The continuity is therefore proved. To prove the integrability under the sign a like use is made of the condition of uniformity and of the earlier proof for a finite integral (§ 120). f 1 \p{a)da= f | f(x, a)dxda + f l Rdx — I f '/(x, a)dadx + f. Ja Ja Q Now let x become infinite. The quantity f can approach no other limit than ; for by taking x large enough R < e and | f | < e (a 1 — a ) independently of a. Hence as x becomes infinite, the integral converges to the constant expression on the left and /•«/•«, | \f/ (a) da = I i f(x, a) dadx. Moreover if the integration be to a variable limit for a, then ¥(a) = f j/(a)da = f f f(x, a)dadx = f F(x, a)dx. J' p-r. \ p-Jl pa I pa p-r- \ F(x, a)dx = I / f(x, a) dadx \—\ I I f(x, a)dxda < e(a — a ). , x I vx *J a Q I «^'(a0. Consider I f' a (x, a)dx = w(a). J a J a As the infinite integral is assumed to converge uniformly by the statement of the theorem, it is possible to integrate with respect to a under the sign. Then C a u>{a)da= C~ f a f' a (x,a)dadx= f X [f(x,a)-f(x.a^)]df = 4>(a)-'(a) and hence a>(af) = (p'(ag). The theorem is therefore proved. This theorem and the two above could be proved in analogous ways in the case of an infinite integral due to the fact that the integrand f(x. a) became infinite at the ends of (or within) the interval of integration with respect to x ; the proofs need not be given here. 145. The method of integrating or differentiating under the sign of integration may be applied to evaluate infinite integrals when the condi- tions of uniformity are properly satisfied, in precisely the same manner as the method was previously applied to the case of finite integrals where OX INFINITE INTEGRALS 371 the question of the uniformity of convergence did not arise (§§ 119-120). The examples given below will serve to illustrate how the method works and in particular to show how readily the test for uniformity may be applied in some cases. Some of the examples are purposely chosen iden- tical with some which have previously been treated by other methods. Consider first an integral which may be found by direct integration, namely, | er ax cos bxdx = Compare | e~ ax dx = -. Jo a 2 + b 2 Jo a The integrand e~ ax is a positive quantity greater than or equal to e _ax cos6x for all values of 6. Hence, by the general test, the first integral regarded as a function of b converges uniformly for all values of 6, defines a continuous func- tion, and may be integrated between any limits, say from to b. Then I I e~ ax cos bxdxdb = | | e~ ax cosbxdbdx Jo Jo Jo Jo sin to , r h adb , b — = tan- 1 -. b 2 a r' r - sin to , r" auu , u ■ | e~ ax dx — I — — — tan- 1 -. Jo x Jo a' 2 C ' r - c ' sin ox p Integrate again. / I e—" x dbdx = | J o J o x Jo sin bx „ , c' r 1 — cos 6a; , •'' dx b a = b tan- 1 log (a- + b 2 ). a 2 cos bx , , r x 1 — cos bx n C* ,,- l — cos ,JX 7 i r Compare I e~ "■< — dx and I Jo x 2 Jo dx. Now as the second integral has a positive integrand which is never less than the inte- grand of the first for any positive value of a, the first integral converges uniformly for all positive values of a including 0, is a continuous function of a, and the value of the integral for a = may be found by setting a equal to in the integrand. Then r * 1 — cosbx , ,. T ,6 «, , , ,, 1 / — dx = Inn b tan- 1 log (a 2 + b 2 ) Jo x 2 « = oL a 2 J in 5 The change of the variable to x' = \ x and an integration by parts give respectively J" cc sin 2 6x , 7r , , , r^sinbx, it tt , , dx = — \b\, I dx = + - or . as b > or b < 0. ox 2 2 Jo x 2 2 This last result might be obtained formally by taking the limit .. r r - sin bx , /^ x sin J Inn e~ ax dx = I «=o Jo x Jo x sin bx , r x sin 6x , ,6 tt dx = tan -1 - = -l """ 2 after the first integration ; but such a process would be unjustifiable without first showing that the integral was a continuous function of a for small positive values of a and for 0. In this case x - 1 e - ax sin bx | =§ | x _1 sin x |, but as the integral of | x _1 sin bx \ does not converge, the test for uniformity fails to apply. Hence the limit would not be justified without special investigation. Here the limit does give the right result, but a simple case where the integral of the limit is not the limit of the integral is lii r x ninbx 7 .. / 7r\ tt r r - ,. sin bx , r x , „ n / dx = hm I ± — J = ± — t^ I Inn ux I -dx = 0. ^0^0 X 6 = \ 2/ 2 «/o 6 = x Jo x 372 INTEGRAL CALCULUS As a second example consider the evaluation of J e~\~x) dx. Differentiate. Jo 0-(a) = A r e -( x -tf dx = 2 rv(-9V»-^i(b da Jo «/o \ x/ x -.Xv(-9 , (x- 5 )* To justify the differentiation this last integral must be shown to converge uni- formly. In the first place note that the integrand does not become infinite at the origin, although one of its factors does. Hence the integral is infinite only by vir- tue of its infinite limit. Suppose a s ; then for large values of x e \ X ■•' (1 — -\ ^ e-'"t'-* 2 and j e~^dx converges (§ 143). Hence the convergence is uniform when a == 0. and the differentiation is justified. But, by the change of variable x' = — a/x, when a > 0, Jo x 2 Jo Jo Hence the derivative above found is zero ; (l>) — I c~ a ~ x ~ coabxdx. Now Jo d

    , (p = Ce ia\ 0(0) = | e-fx-dx = db 2 a' 1 Jo 2 a Hence 0(&) = (b(0)e~iu : - = f Jo : bxdx 2 a In determining the constant. ('. the function 0(7>) is assumed continuous, as the integral for

    //) becomes infinite at any point of the area. The definition of OX INFINITE INTEGKALS 373 convergence is analogous to that given before in the case of infinite simple integrals. If the area A is infinite, it is replaced by a finite area A' which is allowed to expand so as to cover more and more of the area A. If the function f(x, y) becomes infinite at a point or along a line in the area A, the area A is replaced by an area A' from which the singularities of f(x, y) are excluded, and again the area A ' is allowed to expand and approach coincidence with A. If then the double integral extended over A' approaches a definite limit which is independent of how A' approaches .4, the double integral is said to converge. As ffA*, 9 ), if,) dudv, where x = (f>(u, v), y = *]s(", v), is the rule for the change of variable and is applicable to A ', it is clear that if either side of the equality approaches a limit which is independent of how A' approaches A, the other side must approach the same limit. The theory of infinite double integrals presents numerous difficulties, the solution of which is beyond the scope of this work. It will be suffi- cient to point out in a simple case the questions that arise, and then state without proof a theorem which covers the cases which arise in practice. Suppose the region of integration is a complete quadrant so that the limits for x and y are and cc. The first question is, If the double integral converges, may it be evaluated by successive integra- tion as ff(x, y)dA = f f /(■?, V)dydx = f f /(,-, y)dxdy? And conversely, if one of the iterated integrals converges so that it may be evaluated, does the other one, and does the double integral, converge to the same value ? A part of this question also arises in the case of a function defined by an infinite integral. For let (■'-')= f f(*,y)dy and f (x)dx=f f f(x,y)dydx, it being assumed that (.<•) converges except possibly for certain values of x, and that the integral of (x) from to co converges. The question arises, May the integral of (x) be evaluated by integration under the sign ? The proofs given in § 144 for uniformly convergent integrals inte- grated over a finite region do not apply to this case of an infinite inte- gral. In any particular given integral special methods may possibly be devised to justify for that case the desired transformations. But most cases are covered by a theorem due to de la YalleV Poussin : If the 374 INTEGRAL CALCULUS function fix, y) does not change sign and is continuous except over a finite number of lines parallel to the axes of x and y, then the three integrals Cf(x,y)dA, f f X f{x,y)dydx, f f* f(x,y)dxdy, (12) J U x—0 cA/ = Jy — O t/r=0 cannot lead to different determinate results : that is, if any tiro of them lead to definite results, those results are equal* The chief use of the theorem is to establish the equality of the two iterated integrals when each is known to converge ; the application requires no test for uni- formity and is very simple. As an example of the vise of the theorem consider the evaluation of 1=1 e~ x 'dx = J ac - a ~ J ~ tic . Jo Jo Multiply by e~ a " and integrate from to oo with respect to a. Ie- a "' = | ae- a -( l + x2 \lx, if e~ a - da = I 2 = f f ae- a '( 1 + x ') dxda. Jo Jo Jo Jo Now the integrand of the iterated integral is positive and the integral, being equal to I 2 , has a definite value. If the order of integrations is changed, the integral r x c" ■>/■, . -\ , , c r i fa i 1 ■"■ I | ae- a ~v+ x ~>dadx = I = - tan -1 x = -- Jo Jo Jo 1 + x- -1 2 4 is seen also to lead to a definite value. Hence the values I- and \ ir are equal. EXERCISES 1. Note that the two integrands are continuous functions of (x. a) in the whole region = a < x . s= x < x and that for each value of a the integrals converge. Establish the forms given to the remainders and from them show that it is not pos- sible to take x so large that for all values of a the relation | A' (x. a) | < e is satisfied, but may be satisfied for all a'ssuch that < a = a. Hence infer that the conver- gence is nonuniform about a = 0. but uniform elsewhere. Note that the functions defined are not continuous at a = 0. but are continuous for all other values. (a) I ae- ax dx, R(x,a) = ( ae~ ax dx = e~ ax — 1, Jo J a , , r x sin ax , „ , p x sin ax , r^s'mx , (£) | dx, R(x.a) = | d.r = dx. Jo X Jr X J ax X 2. Repeat in detail the proofs relative to continuity, integration, and differ- entiation in case the integral is infinite owing to an infinite integrand at x = b. * The theorem may be generalized by allowing f(x, y) to be discontinuous over a finite number of curves each of which is cut in only a finite limited number of points by lines parallel to the axis. Moreover, the function may clearly he allowed to change sign to a certain extent, as in tl ase where / > when ./• > a, and / < when < .? < a, etc.. where the integral over the whole region may he resolved into the sum of a finite number of integrals. Finally, if the integrals are absolutely convergent and the integrals of /(.i\ in lead to definite results, so will the integrals of /(.;•, y). OX INFINITE INTEGRALS 375 3. Show that differentiation under the sign is allowable in the following cases, and hence derive the results that are given : r K -7 ! /ir r, r x , ■> , Vtt i ■ 3 • • • (2 n — i) I e- ax -dx = -^r_, a>0, | x 2 "e - ax ~ dx = — ', Jo 2 \ a Jo 2 2"a" + 2 J" x o /* " 1 • 2 ■ ■ • n xe- ax ~dx = — , a > 0, | z 2 »+ 1 e- a *'dx = , a 2 a Jo 2a" +1 r*> dx 7r 1 , „ r x dx ir 1 • 3- • • (2n — 1) = = , & >0, | = : ' , Jo x 2 + k 2 V/t ^° (x 2 + k) n + 1 2 2"?i !/,:" + 2 Jr. 1 J s, 1 W j I x"dx = , n > — 1, I x"( — log x)»'dx = , U 71 + 1 Jo (?l + l) m + 1 r x x a ~ l it , / ,I 'x' 1 - 1 10£X 7T 2 C0Sa7T I dx = > < a < 1. I '— dx = — ; Jo 1 + x sin air Jo 1 + x cos' 2 cnr — 1 4. Establish the right to integrate and hence evaluate these : p x p r. (,— ax — g— ftx /j I e-^cZx, < a = a. | dx = log - , b, a === a , Jo Jo x a r 1 ■, r x x a — x h . . a + 1 . I x^dx, — 1 < a ft < a, | ax = log , b. a == a„. Jo ° Jo log X ^ 6+ 1 _ "' r x , __ r=° e-"*— e~te 1 /, 2 + m 2 I e~ <"' cos mxdx, < a === a. | cos ?7ixdx = - log — Jo Jo x 2 « 2 + »< 2 e - ax _ e - to _ Jj /> x p x g— ax g— ox ij f/ I e-^sinrnxdx, 0< a === a. | sin?nxdx = tan -1 tan- 1 —, Jo Jo x mm r x ■> o Vtt r r - -- -- /- | e~ *"*" dx = , < a === a. \ e x- — e x 2 dx = (6 — a) V 7r. Jo 2 a Jo /^ x sin (3x /3 5. Evaluate: (a) \ e~ ax - — — dx=tan-i-, Jo x a r x 1 — cos ax , . /- , s /• x , sin 2 ax , j e~ x dx = log VI + a 2 , (7) | e -1 " ax, Jo x Jo x /•* _(** + £) Vir _ A /x /*« log(l + a 2 x 2 ) , e V x 2 'dx = e- 2 «. a 2= 0. (e) f — — — dx. Jo 2 ' Jo 1 + ^-x 2 /* x 7j- e-^'dx = - a — and iustify the relations: 2 \ r J<"°sinr 2 r b p x „ 2 / >x /* 6 o — — dr = ^^ 1 / e~ ra ~sin ?-dxdr = — zz I I e~ ™"sin rdrdx a -\ r -^ — J a J u -y/^j. J Ja 2 T . /-> x e- ax 'x' 2 dx . , /» x e- hx "x 2 dx — — - sin a sin 6 I VttL Jo 1 + x 4 Jo 1 + x 4 /- x g- ax^aj. ^ x e - 5x^ x -] + cos a cos b . Jo 1 + x 4 Jo l + x 4 J r r sin r w 2 f . /• x e~ rx "'x 2 dx r x e~ rx2 dx~\ — rr dr = \ = sin r \- cos r | ^o % /7 \2 A7r L Jo 1+x 4 Jo l + x 4 J 876 INTEGRAL CALCULUS , . ., , r r coar , /tt 2f /• x e~ ' x2 x 2 dx . /*°° e- rx2 dxl Similarlv, I — — dr = a ' cos r sin r . Jo Vr V2 7T |_ Jo 1 + x* Jo 1 + x 4 J J* °° sin r , c °° cos r , nr r °° . w „ , /* x 7r ., , 1 _— dr = — — dr = i - , | sin - r-dr = I cos - r 2 dr = - . o ■y'r ^° Vr \ ^ t/o 2 t/o 2 ^ 1 /•* 7. Given that = 2 ae-^d + ^da, show that 1 + x' 2 Jo J^=° 1 + coswuc 7r ,„ , , r<°cosmx , 7r — dx = - (1 + e~ m ) and | — dx = - e~ '", m > 0. o 1 + x 2 2 v Jo 1 + x' 2 2 X x x sin n'X — £- dx, by integration by parts and also by substi- tuting x' for ax, in such a form that the uniform convergence for a such that < a ^= a is shown. Hence from Ex. 7 prove /•« X sin ax -k .. ,._ i# i# . I — dx = — e- a , «>0 (by differentiation). Show that this integral does not satisfy the test for uniformity given in the text; also that for a = the convergence is not uniform and that the integral is also discontinuous. 9. If /(x, a, p) is continuous in (x, a, p) for ^ x < cc and for all points (a, p) of a region in the or/3-plane, and if the integral (a, p) = I /(x, a, p)dx con- Jo verges uniformly for said values of (a. p), show that

    0, defines an analytic func- Jo tion of 7 over the whole 7-plane to the right of the vertical + X /• + X the results I cos x 2 sin 2 axdx — / sin x 2 sin 2 axdx = due to the fact that sin x is an odd function, establish the relations I cos x 2 cos 2 axdx — cos ( a' 2 ), | sin x 2 cos 2 axdx = - -- sin ( a 2 ). Jo 2 \4 / Jo 2 \4 / 14. Calculate: (a) | e~ G2a ' 2 cosh frxdx, (/3) j xe~ ax cos 6xdx, Jo Jo and (together) ( 7 ) J^cos (^ ± -~j dx, (5) J^ "sin (^ ± ^ dx. 15. In continuation of Exs. 10-11, p. 368, prove at least formally the relations: .. r {) sinA-x ir 1 f a sin fee lun I /(x) dx = — /(0), Inn- I /(x) dx =/(0), t = x J- a x 2 k=KirJ-a x Jn k' /•» (l p (I n lc /-• (l Sill &X I f(x) cos kxdxdk — I | /(x) cos kxdkdx = / / (x)- dx, J— a J— a Jo J- a X i /^ x /* " 1 /* a sin ftx - | I /(x) cos kxdxdk = lim - | /(x) dx =/(0), 7T Jo J -a A: = oo7rJ-o X - f °° f* f( x ) cos fedxdfc = /(0), - f ' f * /(x) cos ft (x - t) dxdk =f(t) . TT Jo J-x 7T Jo J-x The last form is known as Fourier's Integral; it represents a function /(i) as a double infinite integral containing a parameter. Wherever possible, justify the steps after placing sufficient restrictions on/(x). 1 p » e- ax — e~ bx , , 6 e- ^ J// = - prove / dx = lo ; x Jo x Prove also dx f x n - i t- x dx f a; '"-le- t/o Jo = 2 ( r 2n + 2m - 2 e- r 'dr' 2 ( 2 sin 2 "-i0 cos 2 ™- 1 ^^. Jo Jo 17. Treat the integrals (12) by polar coordinates and show that | /(x, ?/) d^l = ( 2 I f(r cos 0, »* sin 0) rdrd

    and m > 0, and hence define functions of the parameters n or n and m for al l positiv e values, zero not included. Other forms may be obtained by changes of variable. Thus I» = 2f if»^e-y\hj, by x = y% (2) I» = J (log ^J tfy, by «- = y, (3) B(m, ») =f y n ~\l - y)'"- 1 ^ = BO, m), by z = 1 - y, (4; B(/«,7i)= I — ; tt j by .r = , (o) B(m, n) = 2 J sin 2m - 1 ^cos a "- 1 ^rf^, by z = sin 2 <£. (6) i/0 If the original form of T(n) be integrated by parts, then T(n) = | x n - 1 e~ x dx = - £•" / 12 ) Jo T(m + n) Xext let it be required to evaluate the triple integral 1 = fff Xl ~ hjm ~ lzn ~ hlxdydz > X + y + * ~ *» over the volume bounded by the coordinate planes and x + y + >~ = 1, that is, over all positive values of x, y, z such that x + y -f- z S= 1. Then r> 1 p\ — x r>\ — x — y 1= \ \ \ x'- 1 y m - 1 z n - 1 dzdydx Jo Jo Jo = \ff xl ~ v ' i_1(1 ~ * ~ v ^ hjdx - By (12) f Jo tr 1 (i-*-y)"^= r(TO + n + 1) ( 1 - Then J== r(«or(» + i)/- wr(m + w+l)J, r(m)r(w + i) r(/)r(?« +n + l) wr(w< + n + 1) r (£ + »i + ?i + 1) This result may be simplified by (7) and by cancellation. Then There aie simple modifications and generalizations of these results which are sometimes useful. For instance if ir were desired to evaluate I over the range of positive values such that x/u + y/b + z/c = h. the change x = ah%, y — bhr/, z = ch£ gives 1 = C T j x l - 1 y m - 1 z n - 1 dxdydz = a l b'»c n - T(t)T(m)T(n) ,, J . m ,„ x.yz h' + m + n _ + ^+_S/i. (1+ m + n + 1) a 6 c FUNCTIONS DEFINED BY INTEGRALS 881 The value of this integral extended over the lamina between two parallel planes determined by the values h and h + dh for the constant h would be r(l)T(m)T(n) ... . ... dl = a ! h'"c n — — — — — — h l + m + » - hlh. r {I + m + n) Hence if the integrand contained a function /(/<), the reduction would be fffxi-iyn-h"- 1 / ft + ^ + -) dxdydz „ T(l)T{m)T(n) /'", nw;± = a l b m c n — — — ^— !■ — K -J- I / (h) h i + m + n- 1 dh T(l + m + n) Jo w if the integration be extended over all values x/a + y/b + z/c =§ II. Another modification is to the case of the integral extended over a volume '=///••-»--■.■-***. ©'+ g)*+ ©'a K which is the octant of the surface (x/a)i' + {y/h)i + (z/c) r = h. The reduction to J = 1 m ( n a l b m c n h,p « r /•/•/* --1 »-i "-i -fff" 7 ' 1 'J 7 ' * f r * d&W, f + 1? + f ^ 1, i i is made by £A = ( -) , ijft = (-) , £h = (~) , dx = -hi'£i' J= f f f xJ-hj>«-iz»- 1 dxdydz r # — 1 1 a l b m C \p/ \q) \r) - + - + " 1 i r - + _ + _ + i hi' i r . This integral is of importance because the bounding surface here occurring is of a type tolerably familiar and frequently arising ; it includes the ellipsoid, the surface 1111 2 2 2 2 J"-" + y- + z- = "-, the surface set + y* + zs" = a 3. ]}y taking £ = m = n = 1 the volumes of the octants are expressed in terms of the r-function ; by taking first I = 3, m =n = l, and then m = 3. I = n = 1, and adding the results, the moments of inertia about the z-axis are found. Although the case of a triple integral has been treated, the results for a double integral or a quadruple integral or integral of higher multiplicity are made obvious. For example. // x'-^dxdy = „W + - V{l)Tim) , * + v -m h, r (7 + m + 1) a b r I — ii JJ PQ r 11 + m + x \ \aj 1 ! , m \P '1/ 382 INTEGRAL CALCULUS 'k\„ II r ffff xfc-V- 1 *™- 1 *" - l dzdydzdt - pars ^ Ik I m n r -+-+-+-+1 \p q r s 12 n — 1 149. If the product (11) be formed for each of — > — > • • •> j and n n n the results be multiplied and reduced by Ex. 19 below, then *©'©-*( ! Hr 1 )- fl # The logarithms may be taken and the result be divided by w. > logr(-| •- =(--— )log27T--— • ^ \nj n \l 2 nl 2 n Now if n be allowed to become infinite, the sum on the left is that formed in computing an integral if dx — 1/n. Hence ]=* s iog r (**) a,x = r iog r ^ ' / ''' = iog ^*" ^ Then J log r (a + x) dx = a (log a - 1) + log V2^ (15') may be evaluated by differentiating under the sign (Ex. 12 (0), p. 288). By the use of differentiation and integration under the sign, the expressions for the first and second logarithmic derivatives of T(ri) and for log T(ri) itself may be found as definite integrals. By (9) and the expression of Ex. 4 (/t), p. 375, for log x, T'(n)= j x n ~ 1 e~ x logxdx = j x n ~ 1 e~ x J dadx. If the iterated integral be regarded as a double integral, the order of the integrations may be inverted ; for the integrand maintains a posi- tive sign in the region 1 < x < oo, < a < oo, and a negative sign in the region < x < 1, < a < x, and the integral from to oc in x may be considered as the sum of the integrals from to 1 and from 1 to oo, — to each of which the inversion is applicable (§ 146) because the integrand does not change sign and the results (to be obtained) are definite. Then by Ex. 1(a), T '( n )=f f x n ~ 1 e~ x - dxda = T(n)\ 1 \ da (1 + a)"/ a FUNCTIONS DEFINED BY INTEGRALS 383 or r'(n) n) dw is an approximate form for Y(n-\-l), where the quantity e is about 2 ir/^Ja and where the limits ± c of the integral are small relative to V a. But as the integrand falls off so rapidly, there will be little error made in extending the limits to cc after dropping e. Hence approximately Y(n + 1) = 2afe- a I < dw or T(n + 1) = V27r(n + £)"+! e -(» + l)(l + ,), ' (20) where rj is a small quantity approaching as n becomes infinite. EXERCISES 1. Establish the following formulas by changes of variable. la) T(n) = a n f x n ~ l e~ ax dx, a>0, (/3) f' 2 sin" xd.r = B (- + •', - ) , Jo Jo 2 \2 2 2/ (7) B(n, n) = 2 l ~i»B(n, I) by (0), (5) f a:'" -1(1 - x 2 )»-idx = S B( 1 , m, n), Jo /^i x m-i(l — x)»-i _ B(m, n) 1 r(m)r(n) a; ?/ Jo (x + a)"' + " ~ a n {\ + a)'» ~~ a»{\ + «)'» r (m + n) ' x + a ~ 1 + a' „i a-m -i(i _ x )«-i,i x , T(m)T(n) _ % J Jo [ax + 6(1 — x)]'" + » ~ «'"6'T (??i + n) ' ~~ a (1 — ?/) + /;<■/ ' /•ix'»-i(l-x)«-V7x B(m.n) /m /- 1 x"«x a V r (J • ?? + 1) Jo (b + cx)'" + " 6» (6 + c) m Jo Vl- .r- 2 r (2 » + 1) r 1 ,- v ^ i / ., ?»+i\ , , r 1 ilr ~^tt r (»-i) [ x»l-x»*dr = Bb + b- -. / -7= = = - - r7 1 TTT- Jo n \ n J Jo t/] — x" n r ( n + 2) 2. From r (1) = 1 and V (I) = a 7t make a table of the values for every integer and half integer from to 5 and plot the curve // = r(x) from them. 3. By the aid of (10) and Ex. 1 (7) prove the relations VwT(2a) = 2 2a - l T(a)T(a + '), VttT (n) = 2»-VT (\ n) T (I n + I). 4. Given that r(1.75) = 0.0101. add to the table of Ex. 2 the values of r (») for every quarter from to 3 and add the points to the plot. FUNCTIONS DEFINED BY INTEGRALS 385 5. With the aid of the r -function prove these relations (see Ex. 1) : . , A . r\ l-3-5-..(n-l) 7T 2 • 4- 6 • • • (n- 1) (a) I sm" xdx = I cos" xdx = - — or L, Jo Jo 2-4.6-.-n 2 1 • 3 • o - - • n as n is even or odd. io\ T 1 x2 " dx 1-3-5--- (2n — 1) it . , r^x 2 "+hlx 2 ■ 4 • 6 • • • 2 n (p) — = 1 (7) I — = , Jo Vl-x 2 2-4.6--.fln 2 Jo Vl - x 2 1 • 3 • 5 • • • (2n + l) /* ™ t / TTCI^ f n 3 3 TTOp (5) I x 2 Va 2 — x 2 cZx = — , (e) x 2 (a 2 — x 2 )±dx = , Jo 10 v ' Jo 96 — to four decimals, (77) Find / — " -v/l _ r* Jo /' 1 6. Find the areas of the quadrants of these curves : (a) x'2 + 2/ 2 " = «-, (/3) xf + ya = a?, (7) x- + yl = 1, (5) x 2 /a 2 + 2/V'/ 2 = 1, (e) the evolute (ax)f + (by)z = (a 2 — b 2 )s. 7. Find centers of gravity and moments of inertia about the axes in Ex. 6. 8. Find volumes, centers of gravity, and moments of inertia of the octants of (a) j2 + y i + ri = ,72. (£) . r f + y 5 + Z s = as, ( 7 ) j- 2 4- ?/ 2 4 z f = 1. 9. (a) The sum of four proper fractions does not exceed unity ; find the average value of their product. (j3) The same if the sum of the squares does not exceed unity. (7) What are the results in the case of k proper fractions ? 10. Average c~ "■'' ~ ''•"" under the supposition ax' 2 + by' 2 == //. 11. Evaluate the definite integral (15') by differentiation under the sign. 12. From (18) and 1 < — - < 1 + a show that the magnitude of If- log r (n) 1 — c~ a is about \/n for large values of n. 13. From Ex. 12, and Ex. 23, p. 76, show that the error in taking log r in 4- -) for f " log r (x) dx is about log r ( n + ) . \ 2/ Jn 24 n + 12 ° \ 2/ r n + 1 /» 1 14. Show that / logr(x)dx = I log T (n 4- x) dx and hence compare (15'), J 11 Jo (20), and Ex. 13 to show that the small quantity 77 is about (24 n 4- 12) -1 . 15. Use a four-place table to find the logarithms of 5! and 10!. Find the logarithms of the approximate values by (20). and determine the percentage errors. 16. Assume n = 11 in (17) and evaluate the first integral. Take the logarithmic derivative of (20) to find an approximate expression for T'(n)/T (n). and in partic- ular compute the value for n = 11. Combine the results to find 7 = 0.578. By more accurate methods it may be shown that Euler's Constant 7 = 0.577.215.005 17. Integrate (19') from n to n + 1 to find a definite integral for (15'). Subtract -, •. 1 , C ° ea " — e/>.,. • • • , m n be a set of n determinations each made independently of the other and each worthy of the same weight. Then the quantities q l = m 1 — m, q 2 =m 2 -m, ■■-, q n = m n — m, which are the differences between the observed values and the assumed value m, are the errors committed; their sum is q x + q 2 H — + y n = ('"i + m - 2 + ■• • + m n) — >»»■ It will be taken as a fundamental axiom that on the average the errors in excess, the positive errors, and the errors in defect, the negative errors, are evenly balanced so that their sum is zero. In other words it will be assumed that the mean value mn = m + m, + • • • + "'„ or m = — (m, + m, + • • • + m„) (21) i . n \ i i-; the most probable value for in as determined from m v m n , • • • , ?n„. Note that the average value m is that which makes the sum of the squares of the errors a minimum ; hence the term " least squares." Before any observations have been taken, the chance that any par- ticular error q should be made is 0, and the chance that an error lie within infinitesimal limits, say between q and y + dq, is infinitesimal; let the chance be assumed to be a function of the size of the error, and write (q)dq as the chance that an error lie between q and q -f- dq. It may be seen that cf>(q) niay be expected to decrease as q increases ; for, under the reasonable hypothesis that an observer is not so likely to be far wrong as to be somewhere near right, the chance of making an error between 8.0 and 8.1 would be less than that of making an error between 1.0 and 1.1. The function <£('/) is called the error function. It will be said that the chance of making an error y,- is (q { ); to put it more precisely, this means simply that (q,)dq is the chance of making an error which lies between y ; and y ( . -j- dq. FUNCTIONS DEFINED BY INTEGKALS SSI It is a fundamental principle of the theory of chance that the chance that several independent events take place is the product of the chances for each separate event. The probability, then, that the errors q , q , • • • , q n be made is the product * ('A) * (? 2 ) • • • * (in) = <£ 0>h - m) * K - to) ..■ (m, - m). (22) The fundamental axiom (21) is that this probability is a maximum when to is the arithmetic mean of the measurements m 1 , to..,, • • • , to,,: for the errors, measured from the mean value, are on the whole less than if measured from some other value.* If the probability is a maxi- mum, so is its logarithm ; and the derivative of the logarithm of (22) with respect to to is 4>'(m l — to) $' (m. 2 — »/) tj>'(m n — to) _ (m 1 — vi) (m — vi) tf> (m H — vi) when -}- q„ -f- • • • + q n = (jn> x — ni) + (»>., — ni) + • • • + (»*•„ — m ) = 0- It remains to determine <£ from these relations. For brevity let F(rA be the function F = ' '<£ which is the ratio of ' (q) to <£('/). Then the conditions become F(sd + F ( ( / ) = - Cr + K , and ^( 7 ) = # C92 + A '=C^ C ' 2 . This determination of cj> contains two arbitrary constants which may be further determined. In the first place, note that C is negative, for 4>(q) decreases as q increases. Let \ C = — k 2 . In the second place, the * The derivation of the expression for 4> is physical rather than logical in its argu- ment. The real justification or proof of the validity of the expression obtained is a pos- teriori and depends on the experience that in practice errors do follow the law (24). 388 INTEGRAL CALCULUS error q must lie within the interval — cc < q < + x> which comprises all possible values. Hence f + \ (q) dq = 1, gJ + ' e~ ^dq = 1. (23) For the chance that an error lie between q and q + dq is 4>dq, and if an interval a^q ^=b be given, the chance of an error in it is ft b rth 2) ^ ('/) d 'l or > better, lira ^ <£ (q) dq = J (y) fty, and finally the chance that — oo < q < + x represents a certainty and is denoted by 1. The integral (23) may be evaluated (§ 143). Then 0' Vnr/k = 1 and G = /.•/ Vtt. Hence * *(?)« -7= «-**■. ( 24 ) V7T The remaining constant A; is essential ; it measures the accuracy of the observer. If k is large, the function cf>(q) falls very rapidly from the large value /.■/ v7r for q = Q to very small values, and it appears that the observer is far more likely to make a small error than a large one; but if k is small, the function c£ falls very slowly from its value /.•/ V7T for q = and denotes that the observer is almost as likely to make reasonably large errors as small ones. 151. If only the numerical value be considered, the probability that the error lie numerically between q and q -f- dq is 2/.- 2 k r* —,— e~ Wdq, and — ~r~- \ e~ k -'rdq Vtt VttJ,, is the chance that an error be numerically less than $. Xow 2k r$ 2 r k i lK*) = -7= I e~Wdq = - r er*dx (25) '7r J V ir ^ v is a function defined by an integral with a variable upper limit, and the problem of computing the value of the function for any given value of $ reduces to the problem of computing the integral. The integrand may be expanded by Maclaurin's Formula :i A ,.G ,.8 a-lV-fcri e~ x = 1 (26) * The reader may now verify the fact that, with as in (24), the product (22) is a maximum if the sum of the squares of the errors is a minimum as demanded by (21). c * _i l - 2 ! 3 ! 4 ! 5 ! > 0<6< 1, ,r :) x 5 ,/: ; - ,r> " 3 + 10 ~ 42 + 216 ~~ -R, x 11 7 '<1320 FUNCTIONS DEFINED BY INTEGRALS 389 For small values of x this series is satisfactory ; for x == \ it Avill be accurate to five decimals. The probable error is the technical term used to denote that error £ which makes ij/($) = J; that is, the error such that the chance of a smaller error is ^ and the chance of a larger error is also ^. This is found by solving for x the equation ^.1 = 0.44311=/; =a _! + ^_*+*: JC X X^ X° X 7 The first term alone indicates that the root is near x = .45, and a trial with the first three terms in the series indicates the root as between x = AT and x — .48. With such a close approximation it is easy to fix the root to four places as x = k£ = 0.4769 or £ = 0.4769 k~\ (27) That the probable error should depend on k is obvious. For large values of x = A'£ the method of expansion by Maclaurin's Formula is a very poor one for calculating \J/(£) ', too many terms are required. It is therefore important to obtain an expansion according to descending powers of x. Now and J3~ x2 dx = I e-'^dx — I e I) i/0 J x J e~ x 'dx = I — xe~ x dx = tj X %J X - x ~dx = - v 7r — I e~ x2 dx *J X g-a; 2 00 1 r x e~ x \lx 2 x 2 ) x l X *J X The limits may be substituted in the first term and the method of in- tegration by parts may be applied again. Thus 2 dx "'dx r x 27 e~ x ' /„ 1 \ 1.3 T e~ x > J. ,r '" X = ^{ 1 -2*) + -*J. -* ~ 2x\ -2x 2 + 2*x*) 2 3 J x x 6 and so on indefinitely. It should be noticed, however, that the term m l-3-5---(2w-l) e-^ n . T = ■ — r — — diverges as n = oo . 2 n x 2n 2x & In fact although the denominator is multiplied by 2 x 2 at each step, the numerator is multiplied by 2 n — 1, and hence after the integrations by parts have been applied so many times that n > x 2 the terms in the parenthesis begin to increase. It is worse than useless to carry the integrations further. The integral which remains is (Ex. 5, p. 29) 390 INTEGRAL CALCULUS 1 x ' 2n + ' 2 1.3-5--.(2n + l) r x e-**dx 1 ■ 3 • 5 • ■ • (2 n - 1) - — '- I < L e~ x " < T 2«+i Thus the integral is less than the last term of the parenthesis, and it is possible to write the asymptotic scries jf V*V. = 1^-^(1-^ + ^-1^ + ..) (28) J X \ 4 X L'X with the assurance that the value obtained by using the series will differ from the true value by less than the last term which Is used In the series. This kind of series is of frequent occurrence. In addition to the probable error, the average numerical error and the mean square error, that is, the average of the square of the error, are important. In finding the averages the probability ( is — e- k ~ r ~rdrd. is the probability that the shot lies in the ring from /• to /• -+- dr. The most pn>l>uld>' distance v is 392 INTEGRAL CALCULUS that which makes this a maximum, that is, d / ^ n n 1 0.7071 _ (e -,,,. )=0 or ,.=_=-_ ( 30) The mean distance and the mean square distance are respectively V^ _ 0.8802 2 1?e- ***/*& = 2k' ' k - Co j" im., ! /^ 1-0000 r- = I 2 /.-^- k ' r 6 dr = — > Vr = — Tlie probable distance /-| is found by solving the equation (30') l=p 2 ,V-„,, = l-e-?, ,, = ^1 = 0^. (30 ") Hence r p < ?'j < r < V /•-. The chief importance of these considerations lies in the fact that, owing to Maxwell's assumption, analogous considerations may he applied to the velocities of the molecules of a gas. Let u. r, w be the compo- nent velocities of a molecule in three perpendicular directions so that V = (\r + v- 4- iv 2 )- is the actual velocity. The assumption is made that the individual components u, v, w obey the law of errors. The proba- bility that the components lie between the respective limits u and u 4- du, v and v -f- dc, w and w + dw is k s k 3 Wdudrdw, and — ■= e-^T^sin 6dV IT V 7T 7T V 7T is the corresponding expression in polar coordinates. There will then be a most probable, a probable, a mean, and a mean square velocity. Of these, the last corresponds to the mean kinetic energy and is subject to measurement. EXERCISES i. If A- = 0.04475, find to three places the probability of an error £ < 12. 2. Compute f t:~ J "dx to three places for (a) x = 0.2. (,J) s = 0.8. 3. State how many terms of (28) .should be taken to obtain the best value for the integral to x. = 2 and obtain that value. 4. How accurately will (28) determine / e~ x "dx — \ \^tt ? Compute. 5. Obtain these asymptotic expansions and extend them to find the general law. Show that the error introduced by omitting the integral is less, than the last term retained in the series. Show further that the general term diverges when n be- comes infinite. FUNCTIONS DEFINED BY INTEGRALS 393 1 it sinx' 2 cosx 2 1-3 r ''- ., dx , , r '' >, 1 T sinx- cosx- 1-3 r'- (a) cosx-ax = - \ i — | 1 cosx V ' Jo 2 \2 2x 2' 2 x 3 2' 2 Jx , , /** ,, 1 7r cosx 2 sinx' 2 1 • 3 r (B) I sin x'-ax = - -v r— - -1 — I VM/ Jo 2 \2 2x 2 2 x 3 2' 2 Jx sinx' 2 1 • 3 p* . ,dx sinx- — i x* . . r * since /• a: /sinx\ 2 , , (7) I dx, x large, (5) j I c7x, x large. Jo x Jo \ x I 6. (a) Find the value of the average of any odd power 2 n + 1 of the error ; (£) also for the average of any even power; (7) also for any power. 7. The observations 195, 225*, 100. 210, 205, 180*, 170*, 190. 200. 210, 210. 220*, 175*, 192 were obtained for deflections of a galvanometer. Compute k from the mean error and mean square error and compare the results. Suppose the observa- tions marked *, which show great deviations, were discarded ; compute k by the two methods and note whether the agreement is so good. 8. Find the average value of the product qq' of two errors selected at random and the average of the product \q\ ■ \q'\ of numerical values. , , • • 1 t^ 1.0875 9. Show that the various velocities for a gas are V p = -, >£ = , — _ 2 _ 1.1284 /==_ V3 _ 1.2247 k k ~ V^k~ k ' V2fc k 10. For oxygen (at 0°C. and 70 cm. rig.) the square root of the mean square velocity is 402.2 meters per second. Find k and show that only about 13 or 14 molecules to the thousand arc moving as slow as 100 m./sec. What speed is most probable ? 11. Under the general assumption of ellipticity and inclination in the distri- bution of the shots show that the area of the ellipse A:' 2 x 2 + 2 \xy -f k' 2 y- = II is itll (k-k" 2 — \' 2 )~ -, and the probability may be written Ge~ il Tr(k-k" 2 — X-')~ -dll. 12. From Ex. 11 establish the relations (a) G — — V Jc 2 k" 2 — X 2 , 7T 13. Find R p , //$ = 0.093, H, IF- in the above problem. 14. Take 20 measurements of some object. Determine k by the two methods and compare the results. Test other points of the theory. 153. Bessel functions. The use of a definite integral to define func- tions which satisfy a given differential equation may be illustrated by the treatment of xij" + (2n + l)y' + xy = 0, which at the same time will afford a new investigation of some functions which have pre- viously been briefly discussed f§§10~-108). To obtain a solution of this equation, or of any equation, in the form of a definite integral, some special type of integrand is assumed in part and the remainder of the 394 INTEGRAL CALCULUS integrand and the limits for the integral are then determined so that the equation is satisfied. In this case try the form y(x)= I e ixt Tdt, y' = I ite ixt Tdt, y" = J — t 2 e ixt Tdt, where T is a function of t, and the derivatives are found by differen- tiating under the sign. Integrate y and y" by parts and substitute in the equation. Then (1 - f) Te ixt ] - f e ixt [T'(l -?) + (2n- l)tT}dt = 0, where the bracket after the first term means that the difference of the values for the upper and lower limit of the integral are to be taken ; these limits and the form of T remain to be determined so that the expression shall really be zero. The integral may be made to vanish by so choosing T that the bracket vanishes ; this calls for the integration of a simple differential equation. The result then is T = (1 - t 2 ) n ~ i, (1 - ?y + V"'] = 0. The integral vanishes, and the integrated term will vanish provided t = ± 1 or e ixt = 0. If x be assumed to be real and positive, the expo- nential will approach when t = 1 + IK and K becomes infinite. Hence y(x)=l e ixt (l-t 2 ) n -W and z(x)=j e ixt (l - t a ) n ~m (31) are solutions of the differential equation. In the first the integral is an infinite integral when n < -f- J and fails to converge when n = — - 1 . The solution is therefore defined only when n~> — \. The second in- tegral is always an infinite integral because one limit is infinite. The examination of the integrals for uniformity is found below. C +1 • - i Consider J e"*(l — t-)" Kit with u < I so that the integral is infinite. C e ix«(i _ p) n ~ 1 dt= C (1 - t-j" ~ I cosxtdt + i C (1 - l-f ~ I sin xtdt. From considerations of symmetry the second integral vanishes. Then f ' e iM {\-t 2 ) n ~*dt \ = \ f '' (1- t 2 ) n ~i cosxtdt]^ f (1-i 2 )" - ^- This last integral with a positive integrand converges when n > — l, and hence the given integral converges uniformly for all values of x and defines a continuous function. The successive differentiations under the sign give the results FUNCTIONS DEFINED BY INTEGRALS 395 +1 i /.+i -I (1— l 2 ) n ~ it sin xtdt, - (l — t 2 ) n ~it 2 cos x£d£. These integrals also converge uniformly, and hence the differentiations were justi- fiable. The second integral (31) may be written with t = 1 + iu, as du. This integral converges for all values of x > and n > — |. Hence the < iven inte- gral converges uniformly for all values of xs x > 0, and defines a continuous function ; when x = it is readily seen that the integral diverges and could not define a continuous function. It is easy to justify the differentiations as before. The first form of the solution may be expanded in series. e ixt (l - f) n - 1 dt = (1 - t*) n ~ i cos xtdt = 2 f (1 - tf) n - % s cos atfcft (32) «7ll = 2j 1 (l-^)- 1 (l-f + ^-f + «f)*, 0<|«|<1. The expansion may he carried to as many terms as desired. Each of the terms separately may be integrated by B- or T-f unctions. 2 X ^ - f r l w = 2 r(2¥?T) P si " 2 " * cos!l *"* ~ r (2 & + l) r (?i + h + 1) - 2 2 * r (/,• + 1) r (n + A- + 1) ' " w 2-V^r(n + *) fc t/2»+»*r(A + i)r(n + * + i) ^ ; is then taken as the definition of the special function J n (x), where the expansion may be carried as far as desired, with the coefficient 8 for the last term. If n is an integer, the T-functions may be written as factorials. 154. The second solution of the differential equation, namely * (a) = y x (*) + *%(«) = C "-2e i * t (l-t*) n -$dt } (31') where the coefficient — 2 has been inserted for convenience, is for some purposes more useful than the first. It is complex, and, as the equation is real and x is taken as real, it affords two solutions, namely its real part and its pure imaginary part, each of which must satisfy the equation. As y (x) converges for x = and z(x) diverges for x = 0, so that p x (x) or 396 INTEGRAL CALCULUS //.,(.'') diverges, it follows that y(x) and y x {x) or y(x) and //.,(■>•) must be independent ; and as the equation can have but two independent solu- tions, one of the pairs of solutions must constitute a com- plete solution. It will now be shown that y.(x) = y{x) and that Ay(x) + />'//.,(•'') is therefore the complete solu- tion of xy" + (2 n + 1) //' + xy = 0. Consider the line integral around the contour 0, 1 — c. 1 + el, 1 -f cc i, x i, 0, or OPQRS. As the integrand lias a continuous derivative at every point on or within the contour, the integral is zero (§ 124). The integrals along the little quadrant PQ and the unit line RS at infinity may be made as small as desired by taking the quadrant small enough and the line far enough away. The integral along SO is pure imaginary, namely, with t = iu, f -2e ixt (l - t-) n - h -dt = 21 f e-*"(l + m 2 )" - W J so Jo The integral along OP is complex, namely r J 01 2e ixt (l-f-) n -^dt Hence = - 2 = — 2 I (1 — f2 )" * cos xtdt — 2 I I (1 - f 2 )" -' sin xtdt. Jo J it (l — pyl cos xtdt— 21 j (1 - t-f-s sin xtdt 4- ^ J i) J a + where £ and £., are small. Equate real and imaginary parts to zer< separately after taking the limit. (1-t-f ■r 2 T (l-^-isi •tdt = y(x) = i 2 r'"< ■; //,(./•). in xtdt — 2 "(1 + M =)» -s,7, 'X'"- 2 "'"' ! '/> = //.,(■'•)• The signs /^and □? are used to denote respectively real and imaginary parts. The identity of y(x) and j/Ax) is established and the new solu- tion //.,(./') is found as a difference of two integrals. FUNCTIONS DEFINED BY INTEGRALS 397 It is now possible to obtain the important expansion of the solutions y(.r) and //.,(•>') in descending powers of x. For -2e ixt (l-t 2 ) H -*dt= I -2le ix - ,,;r (ir-2ut) n -Klu, t = l + ia. Since x =£ 0, the transformation ux = v is permissible and gives 2 *(-*) V'V si e V SM+— 1 -,(■'■) = P (.r) cos Lx-U+ - Q(ar)sm 1\7T 1\ 7T X Q(.»')cosU- n+- - +P(.>-)si lW >i + are two independent I iessel functions which satisfy the equation (35) of § 107. If n + i is an integer, P and (2 terminate and the solutions are expressed in terms of elementary functions (§108); but if n + I is not an integer, P and Q are merely asymptotic expressions which do not terminate of themselves, but must be cut short with a remainder term because of their tendency to diverge after a certain point; for tolerably large, values of x and small values of n the values of J n (x) and K n {x) may. however, be computed with great accuracy by using the first few terms of P and Q. 398 INTEGRAL CALCULUS The integration to find P and Q offers no particular difficulty. f * e~ V " 2 + k 'dv = Tin + I + k) = (n + * - i)(n + fc — }) •• • (n + J) T(n + J). The factors previous to T (n + |) combine with n— \, n — f , •••, n — k + \, which occur in the fcth term of the binomial expansion and give the numerators of the terms in P and Q. The remainder term must, however, be discussed. The integral form (p. 57) will be used. «/o (ft — 1) ! A -( B _|)...(._, + ^( 1+ - v" *• 2 _ :x/ Let it be supposed that the expansion has been carried so far that n — k — | < 0. Then (1 + vi/2x) n ~ k ~ 2 is numerically greatest when v — and is then equal to 1. Hence IB i< C **"* l< u - ^> • ' • < n - ^ + ^>1 ^ = rt ' 1 (» — i) • • ■ (» - ^ + ^>1 1 kl J (ft- 1)! (2x)* ft! (2 a:)* / n2 _!\ /„. <**-!)" l /.<*> , I \ and Xv*-H ?! — \ the error made in neglecting the remain- der is less than the last term kept, and for the maximum accuracy the series for P + iQ should be broken off between the least term and the term just following. EXERCISES 1. Solve xy" + (2 n + 1) y' — xy = by trying TV as integrand. A C (1 - f 2 ) n - ±e''dt + B f (t 2 - 1)" -i&*dt, x>0, »> — i. 2. Expand the first solution in Ex. 1 into series; compare with y{ix) above. 3. Try T(\ - tx) m on x (1 - x) y" + [y - (a + 8 + 1) i] 2/' - aft/ = 0. One solution is f tf-\l — t)V~P-i(l - tx)- a dt, 8 > 0. 7 > /3, |x| — h 2»V^T(?i + i) Jo ] .r" /* "" (/3) JJx) = I sin-"

    j»6 J*& X^~® 8. Show JJ2 x\ = l- x 2 + + — : - + • • • . oV (2!) 2 (3!) 2 (4!) 2 (5!)« 9. Compute «7 (1) = 0.7652 ; J (2) = 0.2239 ; ^(2. 405) = 0.0000. 10. Prove, from the integrals, Jq(x) = — J y {x) and [x-"J„Y = — x~ n J n + \. 11. Show that four terms in the asymptotic expansion of P + iQ when n = give the best result when x = 2 and that the error may be about 0.002. 12. From the asymptotic expansions compute J (S) as accurately as may be. 13. Show that for large values of x the solutions of J„(x) = are nearly of the form for — -\tt + i nir and the solutions of K n (x) — of the form kir + ± it + £ mr. 14. Sketch the graphs of y = J (x) and y = J x (x) by using the series of ascend- ing powers for small values and the asymptotic expressions for large values of x. 15. From JJx) = - I cos (x cos )d show J e-"'JJbx)dx = — • 7T Jo Jo V' a - + 6' 2 16. Show | e-" x J {x)dx converges uniformly when a s 0. Jo 17. Evaluate the following integrals : (a) I J (bx)dx = &- 1 , Jo /* x £?X 7T O- (/3) I sin axJ (bx) — = - or sin- J - as a > b > or b > 0, */ X a r r- 1 „ (7) I sin ax.J (bx) dx = — = or as a- > b- or b- > a-, Jo V« 2 — 6 2 J-x 1 cos ax J (bx) dx = — or as b 2 > a- or a- > b-. V& 2 — a 2 18. If a = Vx'jJax), show ^ + fa 2 - n '-^)u = 0. If r = VxJ n (bx), dx- \ x-f L<^ _ „ ^T = (j/2 _ a 8 ) f 1 xJ n (ax)J n (bx)dx. L dx dxjo Jo 19. With the aid of Ex. 18 establish the relations: (or) b.T n ( coz xtdt 20. Show J„(x> - - — , 7v,,(x) = - — • V J 2 V^-3- 7rJl W J ~1 CHAPTER XV THE CALCULUS OF VARIATIONS 155. The treatment of the simplest case. The integral / = F(x, y,y') dx $(x, y, dx, dy), /1 x . where <& is homogeneous of the first degree in (t), y = ^( f ), 7 = I $(x, y, dx, dy) = I $(<£, tj/, ', ij/')dt. cJ A Jt„ The ordinal'}* line integral (§ 122) is merely the special case in which 4> = Pdx + Qdy and F = P + OJ//'. In general the value of / will depend on the path C of integration ; the problem of the calculus of variations is to find that path which will make I a. maximum or minimum relative to neighboring jiotJis. If a second path C 1 be y = f(jr) -4- rj(x), where rj(x) is a small quan- tity which vanishes at x Q and x , a whole family of paths is given by y =./'(■'•) + n v (■'')> - 1 = « = !■ i ?^' , o) = ^(■'V = °- and the value of the integral /(a) = / F(x, f+ n v . f + a v ')dx, (V) taken along the different paths of the family, lie- q , r — ' J — ^ comes a function of a ; in particular 1(0) and 7(1) are the values along C and C . Under appropriate assumptions as to the continuity of /'and its partial derivatives F' r . F' v , F' tl ,, the function 1(a) will be continuous and have a continuous derivative which may be found by differentiating under the sign (^ ll'.t) : then •x I \ n ) = I [vK( :r > f+ n v-f' + n v') + v' F ? ■■■•/+ "v-f + "^')]' / '''- 400 Bani* cahu/Oa o*.. CALCULUS OF VARIATIONS 401 If the curve C is to give 7(a) a maximum or minimum value for all the curves of this family, it is necessary that 1 '(<>) = j \vK& v> v') + v'*"X*> y, y')'] & = o ; (2) and if C is to make / a maximum or minimum relative to all neighboring curves, it is necessary that (2) shall hold for any function rj(x) which is small. It is more usual and more suggestive to write 77 (x) = Sy, and to say that 8// is the variation ofy in passing from the curve C or y = f(x) to the neighboring curve C" or y =/(.?•) + 77 (x). From the relations //' =/'(:>■), I/' =/V) + V(*). ¥ = v'(*) = ~?!/, connecting the slope of C with the slope of C , it is seen that the variation of the derivative is the derivative of t he variation. In differential nota- tion this is dSy — 8dy, where it should be noted that the sign 8 applies to changes which occur on passing from one curve C to another curve C,, and the sign d applies to changes taking place along a particular curve. With these notations the condition (2) becomes I (f;s,/ + f;, V) dx = mb- = o, (3) where SFis computed from F, Sy, Sy' by the same rule as the differential dF is computed from F and the differentials of the variables which it contains. The condition (3) is not sufficient to distinguish between a maximum and a minimum or to insure the existence of either; neither is the condition g'(x) = in elementary calculus sufficient to answer these questions relative to a function g(x); in both cases additional con- ditions are required (§ 0). It should be remembered, however, that these additional conditions were seldom actually applied in discussing maxima and minima of g(x) in practical problems, because in such cases the distinction between the two was usually obvious ; so in this cast; the discussion of sufficient conditions will be omitted altogether, as in §§ 58 and 61, and (3) alone will be applied. An integration by parts will convert (3) into a differential equation of the second order. In fact i /^ ' i 7 F'^fdx = F' y , — Sydx = / ' dx KA'/ L„ d , , h Tx F r Jz. Hence J \f$,j + F',V)dx =f ' If;, - £ F^Stydx = 0, (3') 402 INTEGRAL CALCULUS since the assumption that 8y = rj(x) vanishes at x and x 1 causes the integrated term [F,' / ,8//~] to drop out. Then d , cF c-F c 2 F , c~F „ F F , = v ?/" = (±) v dx v cy cxdij' oyOy' J if' 1 J For it must be remembered that the function hj = r}(x) is any function that is small, and if F' y — F' y , in (3') did not vanish at every point of the interval x ^ x ^ x v the arbitrary function hy could be chosen to agree with it in sign, so that the integral of the product would neces- sarily be positive instead of zero as the condition demands. 156. The method of rendering an integral (1) a minimum or maximum is therefore to set up the differential equation (4) of the second order and solve it. The solution will contain two arbitrary constants of inte- gration which may be so determined that one particular solution shall pass through the points A and B, which are the initial and final points of the path C of integration. In this way a path C which connects A and B and which satisfies (4) is found ; under ordinary conditions the in- tegral will then be either a maximum or minimum. An example follows. Let it be required to render I = I - Vl + y'-dx a maximum or minimum. x o v a 1 /r~, — To cF 1 ,- cF y' 1 F(x, y, y') = - VI + ?/ 2 , — .= - — VI + y" V cy y 2 cy' y Vl + Hence Vl + y" 1 + — y' y" = or yy" + y' 2 + 1 = V' V- Vl + y' 2 2/(1 + ? /'-)i is the desired equation (4). It is exact and the integration is immediate. {yy'Y +1 = 0, yy' + x = c v y 2 + (x - r x y 2 = c 2 . Tlie curves are circles with their centers on the .r-axis. From this fact it is easy by a geometrical construction to determine the curve which passes through two given points A (z , y ) and B(x v yA; the analytical determination is not difficult. The two points A and li must lie on the same side of the x-axis or the integral I will not converge and the problem will have no meaning. The question of whether a maximum or a minimum has been determined may be settled by taking a curve (\ which lies under the circular arc from A to B and yet has the same length. The integrand is of the form ds/y and the integral along ('■ is greater than along the circle C if y is positive, but less if y is negative. It therefore appears that the integral is rendered a minimum if A and B are above the axis, but a maximum if they are below. For many prohlems it is more convenient not to mul;c the choice of x or y as independent variable in the frst jdufe, but to operate symmetri- cally with both variables upon the second firm of (1). Suppose that the integral of the variation of $ be set equal to zero, as in (3). CALCULUS OF VARIATIONS 403 f 8$ = f [*;&/■ + &,fij + & dx 8dx + $;,„£%] = 0. Let the rules 8dx = d&x and Srfy = d&y be applied and let the terms which contain d8x and dhy be integrated by parts as before. J 8$= J [(<*>;. - d*' dx )8x + (*; - rfcQSy] + U^S* + *^Syl* = 0. As .4 and /J are fixed points, the integrated term disappears. As the variations 8.e and 8// may be arbitrary, reasoning as above gives <*>; - ^ = o, <*>; - r/4>; 7y = o. (4') If. these two equations can be shown to be essentially identical and to reduce to the condition (4) previously obtained, the justification of the second method will be complete and either of (4') may be used to deter- mine the solution of the problem. Now the identity (/. y. dx. dy) = F(x. y. dy/dx)dx gives, on differentiation, ' — F' dx $' — F' dx 4>', — F' ', — — F' — 4- F by the ordinary rules for partial derivatives. Substitution in each of (4') gives <*>' _ cft>' = F'dx - dF' = ( F' - — F'\ dx = 0. K ~ <»*, = F' x dx -d(F- F'y,,/) = Fflx - dF + F' y/ dy' + y'dF' y , = F' x dx - F' x dx - Fydy - Fy,di/ + F' y ,dy' + y'dF'y, = - F'ydy + y'dF'y, = - (f' v - £ F^ dy = 0. Hence each of (4') reduces to the original condition (4). as was to be proved. Suppose this method be applied to I — = I — • Then J y J y f 5 ,h l _ f 5 ^ ,lc ' 2 + dl J~ _ C \ llr5 ' lr + dySdy _ ds 1 J y J y J [ yds y- J = -/[<£* + (*&<>]• where the transformation has been integration by parts, including the discarding of the integrated term which vanishes at the limits. The two equations are , dx „ , dy ds . , dx 1 d = 0. d— -\ = 0; and = — yds yds y- yds c 1 is the obvious first integral of the first. The integration may then be completed to find the circles as before. The integration of the second equation would not be so simple. In some instances the advantage of the choice of one of the two equations offered by this method of direct operation is marked. 404 INTEGRAL CALCULUS EXERCISES 1. The shortest distance. Treat J (1 + y" 2 )- dx for a minimum. 2. Treat I VcZr 2 + r 2 d0 2 for a minimum in polar coordinates. 3. The brachistochrone. If a particle falls along any curve from A to iJ, the velocity acquired at a distance h below A is v = ~V'2gh regardless of the path fol- lowed. Hence the time spent in passing from A to B is T = I ds/v. The path of quickest descent from A to B is called the brachistochrone. Show that the curve is a cycloid. Take the origin at A. 4. The minimum surface of revolution is found by revolving a catenary. 5. The curve of constant density which joins two points of the plane and has a minimum moment of inertia with respect to the origin is (\r' A = sec (3

    and 8// are interpreted as differentials along the curves F and I\. CALCULUS OF VARIATIONS 405 /(Is p v ux~ -{■ (it) — = I treated above, the integrated V J V terms, which were discarded, and the resulting conditions are V dxdx dydy l B dxSx + dy8y l B _ Q dxSx + di/Si/ l _ L yds yds J a ' yds J yds J a Here Jx and cZy are differentials along the circle C and Sx and §?/ are to he inter- preted as differentials along the curves r o and I\ which respectively pass through A and B. The conditions therefore show that the tangents to G and r o at A are perpendicular, and similarly for C and T 1 at B. In other words the curve which renders the integral a minimum and has its extremities on two fixed curves is the circle which has its center on the x-axis and cuts both the curves orthogonally. To prove the rule for finding the conditions at the end points it will be suffi- cient to prove it for one variable point. Let the equations C:x = 4>(t), y = f(t), C 1 -.x = (t) + l (s) f(«), y = f (t) + m (s) t? (t). x' = . and where the argument of* is as in (0). Now if y (s) has a maximum or minimum when s = 0, then 5r'(0) = J '' [i'(0) f 4>;,.(.r. y, x', ?/') + m'(0) ^; + Z'(0) f ' *;., + m'(0) ,%] * ^'(0) f*^ + »i'(0) „*;. I' + j^ '' ^'(o) f (*; - 1 *;,) + m'(0) *, (*;. - |*;,) j The change is made as usual by integration by parts. Now as $>(;r. y, x\ y')dt = $(x, y, dx, dy), so & x dt = '.. *', = $,',,,, etc. = 0: ctt = 0. 406 INTEGRAL CALCULUS Hence the parentheses under the integral sign, when multiplied by dt, reduce tc (4') and vanish ; they could be seen to vanish also for the reason that f and ?? are arbitrary functions of t except at t = t and t = t v and the integrated term is a constant. There remains the integrated term which must vanish, *'(0) f ( + y s *' y >J = [*' dx te + *;„tyj l = 0- The condition therefore reduces to its appropriate half of (5), provided that, in interpreting it, the quantities 8x and by be regarded not as a = f (i x ) and & = ??(( 1 ) but as the differentials along T 1 at B. 158. In many cases one integral is to be made a maximum or minimum subject to the condition that another integral shall have a fixed value, 1=1 F ( x >y>y') dx ™^> J= \ G(x,i/,y')dx = const (7) For instance a curve of given length might run from A to B, and the form of the curve which would make the area under the curve a maxi- mum or minimum might be desired ; to make the area a maximum or minimum without the restriction of constant length of arc would b? useless, because by taking a curve which dropped sharply from A, in- closed a large area below the x-axis, and rose sharply to B the area could be made as small as desired. Again the curve in which a chain would hang might be required. The length of the chain being given, the form of the curve is that which will make the potential energy a minimum, that is, will bring the center of gravity lowest. The prob- lems in constrained maxima and minima are called isoperimetrie prob- lems because it is so frequently the perimeter or length of the curve which is given as constant. If the method of determining constrained maxima and minima by means of undetermined multipliers be recalled (§§58, 61), it will appear that the solution of the isoperimetrie problem might reasonably be sought by. rendering the integral I + XJ=f \F{x, y, y') + XG (x, y, y'^dx (8) a maximum or minimum. The solution of this problem would contain three constants, namely, X and two constants e , c 2 of integration. The constants c , c could be determined so that the curve should pass through A and B and the value, of A would still remain to be determined in such a manner that the integral J should have the desired value. This is the method of solution. CALCULUS OF VARIATIONS 407 To justify the method in the case of fixed end-points, which is the only case that will be considered, the procedure is like that of § 155. Let C be given bj y =f(x) ; consider V = /(») + a v (x) + /3f (05), v = V X = f = h = 0, a two-parametered family of curves near to C. Then g (a, p) = J *F(x, y + a v + /3f, y' + a v '+ j8f ') dx, g (0, 0) = I h(a, p) = f s G(x, y + a v + jSf, ?/ + ari?' + jSf) dx = J= const. would be two functions of the two variables a and p. The conditions for the mini- mum or maximum of g(a, p) at (0, 0) subject to the condition that h(a, p) = const, are required. Hence flr^O, 0) + X/< r (0, 0) = 0, ^(0, 0) + X/^0, 0) = 0, or f % (F,; + \G' y ) + v '(F' y> + \G' y ) dx = 0, f'' i uK + xg ;) + i-w + xG v) da: = °- "'o By integration by parts either of these equations gives (F+XC?);-l(F+XG); / = 0; (9) the rule is justified, and will be applied to an example. Required the curve which, when revolved about an axis, will generate a given volume of revolution bounded by the least surface. The integrals are I = 2 tv I yds, min., J = v I y 2 dx, const. X a 't r x i (yds + \y'-dx) min. or I 5 (yds + Xy-dx) = 0. •o J * £ *' S (yds + \y*dx) = fj' Isyds + y lUSdx + d{,S ' l! ' + 2 \ySydx + \y*8dx\ = = f" 1 hx I- \d (y-) - d ild ^-\ + 8y (ds - d ^ + 2 XydxVi. Hence \d (y 2 ) + d — = or ds - d — + 2 \ydx = 0. ds (is The second method of computation has been used and the vanishing integrated terms have been discarded. The first equation is simplest to integrate. x o 1 X(r. — i/-)dy X?/ 2 + y - — == = CjX, ± — v 1 J ' = dx. VI + if* Vy* - X 2 (q - y*y* The variables are separated, but the integration cannot be executed in terms of elementary functions. If, however, one of the end-points is on the a;-axis, the 408 INTEGRAL CALCULUS values x , 0, y' or x 1 , 0, y\ must satisfy the equation and, as no term of the equa- tion can become infinite, c x must vanish. The integration may then be performed. ydy =dx, 1 - A'-V 2 = X 2 (x - c,) 2 or (x - e 2 ) 2 + y 2 = 1 Vl - X 2 // 2 x ' 2 In tliis special case the curve is a circle. The constants c 1 and X may be deter- mined from the other point (x L , y^) through which the curve passes and from the value of J = v ; the equations will also determine the abscissa x of the point on the axis. It is simpler to suppose x = and leave x x to be determined. With this procedure the equations are 9 * / \ o . 9 1 ^ ^ 1 ^ / 3 n 9 . o 9 \ c| = ■;, • K ~ Cg)" + 2/f = - . - = ci - S ( X l ~ 3 W + 3 ^l); A" A" W A~ o or Xi + 3 2/fXj = 0, c„ = , 7T 2x, and x : = 7T- i [(3 u + Vl) d 2 + 7r 2 yf) 3 + (3 v — Vo u 2 + ir-ij^) a] . EXERCISES 1. Show that (a-) the minimum line from one curve to another in the plane is their common normal ; (fi) if the ends of the catenary which generates the mini- mum surface of revolution are constrained to lie on two curves, the catenary shall be perpendicular to the curves ; (7) the brachistochrone from a fixed point to a curve is the cycloid which cuts the curve orthogonally. 2. Generalize to show that if the end-points of the curve which makes any inte- gral of the form J I<"(x, y)ds a maximum or a minimum are variable upon two curves, the solution shall cut the curves orthogonally. 3. Show that if the integrand (x, ?/, dx, dy, x t ) depends on the limit x x , the condition for the limit B becomes 3>'/ x 5x -f $' d J8y + Sx I x $>' x = 0. L '0 J 4. Show that the cycloid which is the brachistochrone from a point A, con- strained to lie on one curve T , to another curve r\ must leave r o at the point A where the tangent to r o is parallel to the tangent to r x at the point of arrival. 5. Prove that the curve of given length which generates the minimum surface of revolution is still the catenary. 6. If the area under a curve of given length is to be a maximum or minimum, the curve must be a circular arc connecting the two points. 7. In polar coordinates the sectorial area bounded by a curve of given length is a maximum or minimum when the curve is a circle. 8. A curve of given length generates a maximum or minimum volume of revolution. The elastic curve Bsz ii + v*)t = _± or ,fr = _(£r_£i)jr_ . V" 2 y V\2 - (y 2 - c x ) 2 CALCULUS OF VARIATIONS 409 9. A chain lies in a central field of force of which the potential per unit mass is V(r). If the constant density of the chain is p, show that the form of the curve is dr J r[ c *(pV + \)V-lJ 10. Discuss the reciprocity of / and J, that is, the questions of making I a maxi- mum or minimum when J is fixed, and of making / a minimum or maximum when / is fixed. 11. A solid of revolution of given mass and uniform density exerts a maximum attraction on a point at its axis. Ans. 2\(x 2 -f y 2 )± + z — 0, if the point is at the origin. 159. Some generalizations. Suppose that an integral / = C F(x, y, y', z, «', • • •) dx =f <*» (,; dx, y, dy, z, dz, • • •) (10) (of which the integrand contains two or more dependent variables y, z, ■ ■ • a/id their derivatives y', z', ■ ■ ■ with respect to the independent variable x, or in the symmetrical form contains three or more variables and their differentials) were to be made a maximum or minimum. In case there is only one additional variable, the problem still has a geo- metric interpretation, namely, to find y=f(x), z = (/ (x), or x = 4>(t), y = +{t), * = x(0> a curve in space, which will make the value of the integral greater or less than all neighboring curves. A slight modification of the previous reasoning will show that necessary conditions are F' - — F', = and F' — — F' = " dx " ~ dx z (11) or 3/ - d& dx = 0, ; — d& (lu = 0, ; — d<$>' dz , = 0, where of the last three conditions only two are independent. Each of (11) is a differential equation of the second order, and the solution of the two simultaneous equations will be a family of curves in space dependent on four arbitrary constants of integration which may be so determined that one curve of the family shall pass through the end- points ,1 and B. Instead of following the previous method to establish these facts, an older and perhaps less accurate method will be used. Let the varied values of //, z. //', z'. be denoted by y + hy, z + Sz, y' + ty, z> + &z>, hj' = (8y)', &' = (&)'. 410 INTEGRAL CALCULUS The difference between the integral along the two curves is A/ = f \f(x, y + By, y' + 8y', * + Bz, z' + 8,.') - F(x, y, y', z, z')-\dx = f l AFdx = f '(F,% + *V'%' + F & + ^S*') ete + • • • , where F has been expanded by Taylor's Formula* for the four variables y, y', z, z' which are varied, and " + ..." refers to the remainder or the subsequent terms in the development which contain the higher powers of By, By', Bz, Bz'. For sufficiently small values of the variations the terms of higher order may be neglected. Then if AT is to be either positive or nega- tive for all small variations, the terms of the first order which change in sign when the signs of the variations are reversed must vanish and the condition becomes f \f;&j + F' y ,By' + F:Bz + F' z ,Bz')dx = f ^BFdx = 0. (12 Integrate by parts and discard the integrated terms. Then f[( -S r 'V + fa"!*"' 0. (13) * In the simpler ease of § 155 this formal development would run as A / = f " ' (Ffoj + F' y ,8y') dx + ~ f \F yy 8tj 2 + 2 F yy ,Sy8y' + F^Si/*) dx + higher terms, and with the expansion A/ = 5/ -| 8-1 + — 5 3 / + • • • it would appear that 81= C\F' y 8y + F y ,8y')dx, 8*1 = f "\f^W + 2 F yy ,8y8y' + F y , y ,8y'*)dx, J 'o J o 53/= f X \F y 7Sy s + 3 FZ,8y*8y' + ?> F yyn 8ij8y'* + F y %8y'z)dx, ■■■. The terms 5/, 5-/. 5 : >/, • • • are called the Jirst, second, third, ■ ■ ■ variations of the integral /in the case of fixed limits. The condition for a maximum or minimum then becomes 5/= 0, just as (If/ = is the condition in the case of g (./•). In the case of variable limits there are some modifications appropriate to the limits. This method of procedure sug- gests the reason that dx, 8y are frequently to be treated exactly as differentials. It also suggests that 5-/ > and S-I < would be criteria for distinguishing between maxima and minima. The same results can be had by differentiating (T) repeatedly under the sign and expanding 1(a) into series: in fact, 5/= /'(0), 5 2 /= /"(0), ■ • • . No emphasis has been laid in the text on the suggestive relations 81 = ( SFdx for fixed limits or 5/= I 5 for variable limits (variable in x, y, but not in t) because only the most ele- mentary results were desired, and the treatment given has some advantages as to modernity. CALCULUS OF VARIATIONS 411 As 8y and 8s are arbitrary, either may in particular be taken equal to while the other is assigned the same sign as its coefficient in the parenthesis ; and hence the integral would not vanish unless that coeffi- cient vanished. Hence the conditions (11) are derived, and it is seen that there would be precisely similar conditions, one for each variable y, z, ■ • ■, no matter how many variables might occur in the integrand. Without going at all into the matter of proof it will be stated as a fact that the condition for the maximum or minimum of I $ (x, dx, y, dy, z, dz, . . .) is / S<3> = 0, which may be transformed into the set of differential equations &x - d& dx = o, $; - d& dy = 0, $; - d& (lz = o, • • • , of which any one may be discarded as dependent on the rest ; and & dx 8x + <5>' dlJ 8y + & dz 8z H = 0, at A and at B, where the variations are to be interpreted as differentials along the loci upon which A and B are constrained to lie. It frequently happens that the variables in the integrand of an inte- gral which is to be made a maximum or minimum are connected by an equation. For instance / *(.r, dx, y, dy, z, dz) min., S(x, y, z) = 0. (14) It is possible to eliminate one of the variables and its differential by means of £ = and proceed as before ; but it is usually better to introduce an undetermined multiplier (§§ 58, 61). From S(x,y,z) = follows S x 8x -f- S y 8y + S' z 8z = if the variations be treated as differentials. Hence if / [(<&; - d& dx ) 8x + (<*>; - d# dy ) 8y + (<&; - d& dg ) &] = o, [(*; - d& dx + \S' X ) 8x + (% - d& dy + XS' y ) 8y no matter what the value of A. Let the value of A. be so chosen as to annul the coefficient of S,~. Then as the two remaining variations are independent, the same reasoning as above will cause the coefficients of 8x and 8y to vanish and ' x -d<3?, h: + \$ x = o, $;_^; /?/ + x.s; = o, <*>; - ,/$;„ + \s' z = (15) 412 INTEGRAL CALCULUS will hold. These equations, taken with S = 0, will determine y and z as functions of x and also incidentally will fix A. Consider the problem of determining the shortest lines upon a surface S(x, y, z) = 0. These lines are called the geodesies. Then /' 8ds = dxSx -}- dySy + dzBz ds S d^&r + d'^hf + d'^z ds ds /(*i+*sM'!+**M'sH ds 8* = ,(16) 0, dx d — + an;. ds + an: = % = d—+\S' z = 0, as and as dz In the last set of equations A lias been eliminated and the equations, taken with S = 0, may be regarded as the differential equations of the geodesies. The denominators are proportional to the direction cosines of the normal to the surface, and the numerators are the components of the differential of the unit tangent to the curve and are therefore pro- portional to the direction cosines of the normal to the curve in its oscu- lating plane. Hence it appears that the osculating plane of a, geodesic carer contains the normal to the surface. The integrated terms dxSx + dydy + r /> r r r r \p I ds = const., A ds = 0, 8 ds = 0, | 5ds = = dxdx + dySy + dzdzl . Jo Jo Jo Jo The integral in (16) drops out because taken along a geodesic. This final equality establishes the perpendicularity of the lines. The fact also follows from the state- ment that the geodesic circle and its center can be regarded as two curves between which the shortest distance is the distance measured along any of the geodesic radii, and that the radii must therefore be perpendicular to the curve. 160. The most fundamental and important single theorem of mathe- matical physics is Hamilton's Principle, which is expressed by means of the calculus of variations and affords a necessary and sufficient con- dition for studying the elements of this subject. Let T be the kinetic energy of any dynamical system. Let X-, Y h Z ; be the forces which act at any point x t , //,-, z t of the system, and let o> ( -, hy { , 8z { represent displacements of that point. Then the work is sir 2} (A-,o>,. + us.//,. + ^,s.-./). CALCULUS OF VARIATIONS 413 Hamilton's Principle states that the time integral J \ST + SW)dt = f \8T + ^ (X8x + 3% + Zhz)~]dt = (17) vanishes for the actual motion of the system. If in particular there is a potential function V, then 8W — — SV and f 8(r- V)dt = 8 f \t- V)dt = 0, (17') and the time integral of the difference between the kinetic and potential energies is a maximum or minimum for the actual motion of tit e system as compared with any neighboring motion. Suppose that the position of a system can be expressed by means of n independ- ent variables or coordinates q t , q 2 , ■ ■ ■. q n . Let the kinetic energy be expressed as T = V 1 "»•■"? = f l v ' 2dm = r ( f /i, 9 2 « ' " "1 '!>" 4n 4 8 i • • •' W. a function of the coordinates and their derivatives with respect to the time. Let the work done by displacing the single coordinate q r be 5 W = Q,-8q r , so that the total work, in view of the independence of the coordinates, is Q 1 8q l + Q./lq.,+ ■ ■ ■ + Q n dq n - Then = f'\sT+8W)dt = f l \T r ' l 8q l + TgSq s + ■■■+ T^Sq n + T' h tq x + T^Sq 2 + ■■■ + T ')J f ln + QMx + Q % Sq s + ■■■ + Q n Sq„)dt. Perforin the usual integration by parts and discard the integrated terms which vanish at the limits t = t and t = l v Then = C [( r « + "• - £ 7;; .) Sl " + ( t: - + "■- " I ''-) s,,s In view of the independence of the variations 8q l . 07.,. • • •. 8q n , d cT cT r d IT cT , d cT IT r = Q. . = Q„, • • •. — - — = Q„. (18) dt cyj fry j r?< fj) of the center of gravity and the angle

    where R is the com- ponent of the force perpendicular to the radius ?• ; but Rr = 3> is the moment of the force about the center of gravity. Hence T=\ 3[{i 2 + y 2 ) + \ I$' 2 : S W = Xdx + YSy + *50 and M — = X , 31 — = 1 , I = . dt 2 dt 2 dt 2 by substitution in (18), are the desired equations, where X and Y are the total components along the axis and is the total moment about the center of gravity. A particle glides without friction on the interior of an inverted cone of revo- lution; determine the motion. Choose the distance r of the particle from the ver- tex and the meridional angle as the two coordinates. If I be the sine of the angle between the axis of the cone and the elements, then d* 2 = dr 2 + r-V 2 d- and r' 2 = ?■'- + r' 2 l-'•- + r 2 l 2 ip 2 ). dW = — ?»(/x 1 - l 2 dr or V = mgVl — Pr. d 2 r _ M fd(p\ 2 ^ /- d f .-,,„d(p\ A _ „d4> Then rP — = -a VI - P, - r 2 P — = dt 2 \dtj J dt\ dt] dt The remaining integrations cannot all be effected in terms of elementary functions. 161. Suppose the double integral 1= \\ F(.r, y, z, p, q) dx(hj, p = ~ , ,j = j- f > (19) extended over a certain area of the cry-plane wore To be made a maxi- mum or minimum by a surface z = z(x, if), which shall pass through a given curve upon the cylinder which stands upon the bounding curve of the area. This problem is analogous to the problem of § loo with CALCULUS OF VARIATIONS 415 fixed limits ; the procedure for finding the partial differential equation which z shall satisfy is also analogous. Set jj SFdxdy = ff(F£z + F;,8p + F' q §q)dxdy = 0. (:?>•' d?\z Write 8p — — - > 8/ = -~ and integrate by parts. 1 ex Cij J l jJF> g dxdy =JV> \hj -jj^Udxdy. The limits A and B for which the first term is taken are points upon the bounding contour of the area, and 8z = for A and B by virtue of the assumption that the surface is to pass through a fixed curve above that contour. The integration of the term in 8'j is similar. Hence the condition becomes Jf m i xdy =ff( K -i%-±%) ***» = (20) ^--r^-ffr-Oi W cz dx cjj dy cq x J by the familiar reasoning. The total differentiations give K-K,-K- Kp -Kn- K> - 2 K> - Kf = o. The stock illustration introduced at this point is the minimum surface, that is, the surface which spans a given contour with the least area and which is physically represented by a soap him. The real use, however, of the theory is in connection with Hamilton's Principle. To study the motion of a chain hung up and allowed to vibrate, or of a piano wire stretched between two points, compute the kinetic and potential energies and apply Hamilton's Principle. Is the motion of a vibrating elastic body to be investigated ? Apply Hamilton's Principle. And so in electrodynamics. In fact, with the very foundations of mechanics some- times in doubt owing to modern ideas on electricity, the one refuge of many theorists is Hamilton's Principle. Two problems will be worked in detail to exhibit the method. Let a uniform chain of density p and length I be suspended by one extremity and caused to execute small oscillations in a vertical plane. At any time the shape of the curve is y = y{x). and y = y(x, t) will be taken to represent the shape of the curve at all times. Let y' = cy/dx and y = cy/ct. As the oscillations are small, the chain will rise only slightly and the main part of the kinetic energy will be in the whipping motion from side to side ; the assumption dx = ds may be made and the kinetic energy may be taken as 416 INTEGRAL CALCULUS The potential energy is a little harder to compute, for it is necessary to obtain the slight rise in the center of gravity due to the bending of the chain. Let X be the shortened length. The position of the center of gravity is f K z { l+ly*)dz >V + fjxy> 2 dx 1 1 x = — — = - X j -X x ) y -ax. r K r k 2 X Jo \4 2 / f (1+ \y' 2 )dx X+ / \v' 2 dx X ' Jo Jo Here ds = Vl + y' 2 dx has been expanded and terms higher than if" have been omitted. r*l 1 1 /* M /1\ l = X+ -y ,2 dx, -l-x= ~(\-x)y' 2 dx, V = Ipg(-l-z)- Jo 2 2 \t/i) 2 \2 / Then f\T- V) (21 > provided X be now replaced in V by I which differs but slightly from it. Hamilton's Principle states that (21) must be a maximum or minimum and the integrand is of precisely the form (19) except for a change of notation. Hence d[ ,, .dyl d( ?y\ 1 c 2 y .c"y dy dx\ cx_\ dt\ ctf act- ex 1 ex The change of variable I — x = u 2 , which brings the origin to the end of the chain and reverses the direction of the axis, gives the differential equation c 2 y , ley ic 2 y d 2 P ldP -in 2 — ^ H = or 1 1 P = if y = P (u) cos nt. cu 2 u du g ct' 1 du 2 u du g As the equation is a partial differential equation the usual device of writing the dependent variable as the product of two functions and trying for a special type of solution has been used (§ 194). The equation in P is a Eessel equation (§ 107) of which one solution P(u) = A,T (2ng~ 2u) is finite at the origin u = 0, while the other is infinite and must be discarded as not representing possible motions. Thus y (x, t) = AJ (2 ng~ \ u) cos nt, with y (I. t) = A J (2 ng~ M) = as the condition that the chain shall be tied at the original origin, is a possible mode of motion for the chain and consists of whipping back and forth in the peri- odic time 2 7r/;i. The condition J () (2ng~-l-) = limits n to one of an infinite set of values obtained from the roots of J (j . Let there be found the equations for the motion of a medium in which V = I llffftf* + 0, b > 0, is the Maclaurin expansion of a surface tangent to the plane z = at (0, 0), find and solve the equations for the motion of a particle gliding about on the surface and remaining near the origin. 9. Show that r (1 + q 2 ) + t(l + p' 2 ) — 2pqs = is the partial differential equa- tion of a minimum surface ; test the helicoid. 10. If /o and S are the density and tension in a uniform piano wire, show that the approximate expressions for the kinetic and potential energies are HX'dh HI'*©- Obtain the differential equation of the motion and try for solutions y — P(x) cos nt. 11. If |, 7j, fare the displacements in a uniform elastic medium, and a = d A, & = **, c = d l, f=( d i + ^, g = (. 14. Trove Stokes's Formula 7 = f F-dr = ffvxF.dS of p. 345 by the calculus of variations along the following lines : First compute the variation of I on pass- ing from one closed curve to a neighboring (larger) one. 5/ = 5 f F-dr = f (SF.rfr - dF-dx) + f d(F.5i) = f (VxF).(Srxdr), Jo Jo Jo Jo where the integral of d (F.5r) vanishes. Second interpret the last expression as the integral of VxF«tZS over the ring formed by one position of the closed curve and a neighboring position. Finally sum up the variations 5/ which thus arise on passing through a succession of closed curves expanding from a point to final coin- cidence with the given closed curve. 15. In case the integrand contains y" show by successive integrations by parts that 5 7 (x, y, y'. y")dx = \ Y'w + 3 "«'- — - « + I [Y — + — ,- Mx, Jj- dx J ii J.r n \ dx dx- I w = dy. dF cF cF v'here Y = dy ' Y' : w Y" ~ dy" PART IV. THEORY OF FUNCTIONS CHAPTER XVI INFINITE SERIES 162. Convergence or divergence of series.* Let a series ]T U = l, Q + lt x + U 2 + • • • + »„_! + U„ +■■■, (1) the terms of which are constant but infinite in number, be given. Let the sum of the first n terms of the series be written S« = "o + "l +»,+ ••• + "n-1 = X U - ( 2 ) Then S v S 2 , 5„ ••■, S n , S n+l , ••■ form a definite suite of numbers which may approach a definite limit lira S n = S when n becomes infinite. In this case the series is said to converge to the value S, and S, which is the limit of the sum of the first n terms, is called the sum of the series. Or S n may not approach a limit when n becomes infinite, either because the values of S n become infinite or because, though remaining finite, they oscillate about and fail to settle down and remain in the vicinity of a definite value. In these cases the series is said to diverge. The necessary and sufficient condition that a series converge is that a value of n may be found so large that the numerical value of S n+p — S n shall be less than any assigned value for every value of p. (See §21, Theorem 3, and compare p. 356.) A sufficient condition that a series diverge is that the terms u n do not approach the limit when n becomes infinite. For if there are always terms numerically as great as some number r no matter how far one goes out in the series, there must always be successive values of S n which differ by as much as /• no matter how large n, and hence the values of S n cannot possibly settle down and remain in the vicinity of some definite limiting value S. *It will be useful to read over Chap. II. §§ 18-22, and Exercises. It is also advisable to compare many of the results for infinite series with the corresponding results for in finite integrals (Chap. XIII). 419 420 THEORY OF FUNCTIONS A series in which the terms are alternately positive and negative is called an alternating series. An alternating scries in which the terms approach as a limit when n becomes Infinite, each teem being less titan Its predecessor, will converge and the difference between the sum S of the series and the sum S n of the first n terms Is less than the next term v n . This follows (p. 39, Ex. 3) from the fact that| S n+p - S n \ < u n and u n = 0. For example, consider the alternating series 1 - x- + 2 x 4 — 3 x* + ■ ■ ■ + (- l)"nx 2n + ■ ■■ . If \x\ = 1, the individual terms in the series do not approach as n becomes infinite and the series diverges. If \x\ < 1, the individual terms do approach ; for i- > i- n i- 1 n Inn nx 2n = Inn = Inn = 0. n = x n = ooX~' 2n n=x — 2x -2B logSC And for sufficiently large* values of n the successive terms decrease in magnitude since -, -, nx 2n < (h — l)x 2n_2 gives > x- or n > x- Hence the series is seen to converge for any value of x numerically less than unity and to diverge for all other values. The Comparison Test. If the terms of a series are all positive (or all negative) and each term Is numerically less than the em-responding term of a series of positive terms which Is known to converge, the series con- verges and the difference S — S n Is less than the corresponding difference for the series known to converge. (Cf. p. 355.) Let and v[ t -f v[ ■+- v' 2 + • • • + u' n _ x + u' n H be respectively the given scries and the series known to converge. Since the terms of the first arc less than those of the second, S« +P ->% = *'» + ■•• + «„+„_! < < + ••• + < +P -i=S' n+p -S' n . Now as the second quantity S' n + p — S' H can be made as small as desired, so can the first quantity S n+p — S H , which is less: and the series must converge. The remainders i: n = s - s n = u n + »/„+, H =2 u, s' — s: "'n + K+l H =2 "' * It should be remarked that the behavior of a scries near its beginning is of no con- sequence in regard to its convergent ■ divergence: the first X terms may be added and considered as a finite sum N v and the series may be written as S v + ".y + ".v-m + • • • : it is the properties of u y + «.v+i H which are important, that is, the ultimate behavior of the series. INFINITE SERIES 421 clearly satisfy the stated relation R n < R' n . The series which is most frequently used for comparison with a given series is the geometric, a -f- ar -f- or -f- ar z -f- R. = < r < 1, 1-r which is known to converge for all values of r less than 1 For example, consider the series (3) , , l l l 1 + 1 + - + - - + ~^ + and 1 + - + + 2-3-4 1 1 + — + 2 + ••• + + Here, after the first two terms of the first and the first term of the second, each term of the second is greater than the corresponding term of the first. Hence the first series converges and the remainder after the term 1/n ! is less than 1 1 B n < — + 2" 2" + + ••• = 1 1 2»-i' 2» 1 — _ A better estimate of the remainder after the term 1/n ! may be had by comparing 11 ..,1.1 1 11, = (n + 1) ! (n + 2) ! with + (n + l)!(n + l) n\n (n + 1) : 163. As the convergence and divergence of a series are of vital im- portance, it is advisable to have a number of tests for the convergence or divergence of a given series. The test by comparison with a series known to con- verge requires that at least a few types of convergent series be known. For the estab- lishment of such types and for the test of many series, the terms of which are positive, Cauclnjs integral test is useful. Suppose that the terms of the series are decreasing and that a function f(n) which decreases can be found such that u n =f(n\ Xow if the terms u n be plotted at unit intervals along the ?z-axis, the value of the terms may be interpreted as the area of certain rectangles. The curve y=f(n) lies above the rectangles and the area under the curve is f f(n)dn > u 2 + u 3 -\ \- u n . (4) Hence if the integral converges (which in practice means that if I f(n)da = F(n), then / f(n) = F(zc) — -F(l) is finite], 422 THEORY OF FUNCTIONS it follows that the series must converge. For instance, if be given, then u n = f(n) = l/n p , and from the integral test 1 1 / f* x dn ^- + ^ + - • < 1 2" 3" ./ , n p (p-iy -i provided p > 1. Hence the series converges if p > 1. This series is also very useful for comparison with others ; it diverges if p =§ 1 (see Ex. 8). The E.\tio Test. If the ratio of tiro successive terms in a series ofj/osi- tioe terms approaches a limit which is Jess than 1, the series converges ; if the ratio approaches a limit which is greater than one or if the ratio becomes infinite, the series diverges. That is if lim — — = y < 1, the series converges, n=x. " n if lim ~ JL±1 = y' > 1, the series diverges. n = x " n For in the first case, as the ratio approaches a limit less than 1, it must be pos- sible to go so far in the series that the ratio shall be as near to y < 1 as desired, and hence shall be less than r if r is an assigned number between y and 1. Then m„+i < ru n , u„ + 2 < ru» +i < r-u„. ■ ■ ■ and u„ + u n +i + "« + 2 + • • ■ < "»(1 + r + r- + •••) = u n 1 — r The proof of the divergence when u n +i/u„ becomes infinite or approaches a limit greater than 1 consists in noting that the individual terms cannot approach 0. Note that if the limit of the ratio is 1. no information relative to the convergence or divergence is furnished by this test. If the series of numerical or absolute values KI + K1 + KI + --- + KI + --- of the terms of a series which contains positive and negative terms converges, the series converges and is said to converge absolutely. For consider the two sums S n+P - S n = v n H h«, + ,-i and \u n \-\ h|w» +P -i|. The first is surely not numerically greater than the second; as the second can be made as small as desired, so can the first. It follows therefore that the given series must converge. The converse proposition INFINITE SERIES 423 that if a series of positive and. negative terms converges, then the series of absolute values converges, is not true. As an example on convergence consider the binomial series m(m— 1) „ m(m— l)(?n— 2) „ ??i(to — 1) ■•• (to — n + 1) 1 + mx + — -x 2 + — - '- '-X s + h — '- i Z—l x n + .. 1-2 1-2-3 l-2-.-n where ' — — = x, lim [U n \ 71+1 n=x |w„| It is therefore seen that the limit of the quotient of two successive terms in the series of absolute values is |x|. This is less than 1 for values of x numerically less than 1, and hence for such values the series converges and converges absolutely. (That the series converges for positive values of x less than 1 follows from the fact that for values of n greater than ??i -f 1 the series alternates and the terms approach 0; the proof above holds equally for negative values.) For values of x numerically greater than 1 the series does not converge absolutely. As a matter of fact when |x| > 1, the series does not converge at all ; for as the ratio of successive terms ap- proaches a limit greater than unity, the individual terms cannot approach 0. For the values x = ± 1 the test fails to give information. The conclusions are there- fore that for values of |x|l it diverges, and for |x] = 1 the question remains doubtful. A word about series with complex terms. Let U + "l + V 2 H ^ V n - 1 + U n H = K + u'i + "-i-\ — + K - 1 + K -\ — + l( M o' + "l + V 2 H 1" "n- 1 + Vn -\ ) be a series of complex terms. The sum to n terms is S n = S' H -+■ iS„. The series is said to converge if S n approaches a limit when n becomes infinite. If the complex number S n is to approach a limit, both its real part S' n and the coefficient S„ of its imaginary part must approach limits, and hence the series of real parts and the series of imaginary parts must converge. It will then be possible to take n so large that for any value of p the simultaneous inequalities I K + P - s' n I < i- e and ( s'; +p - s;;\ < * e , where e is any assigned number, hold. Therefore \s a+p -s a \=z \s;^ p - s;,\ + \is;;. p - ts;;\ < c . Hence if the series converges, the same condition holds as for a series of real terms. Now conversely the condition \S n + p - S n \ < e implies \S' n+p - S' H \ < e, \S',; +p - S£\ < e. Hence if the condition holds, the two real series converge and the com- plex series will then converge. 424 THEORY OF FUNCTIONS 164. As Cauchy's integral test is not easy to apply except in simple cases and the ratio test fails when the limit of the ratio is 1, other sharper tests for conver- gence or divergence are sometimes needed, as in the case of the binomial series when x = ± I. Let there be given two series of positive terms Mo + Mi + • ■ • + u„ + ■■■ and v + Vi + • • • + v n + ■ • ■ of which the first is to be tested and the second is known to converge (or diverge). If the ratio of two successive terms u n + i/u n ultimately becomes and remains less (or greater) than the ratio v n + i/v H , the first series is also convergent (or divergent). For if M n + 1 V,i + \ U)i + 2 Vn + 2 . . U n Mn-f-1 Un + % - <^ , <^ . . . inen s> ^> ^> • * * a M„ Vn M„ + i V n + i V n V n + X V H + 2 Hence if u n = pv n , then u n + i< pv n + i, w„ + 2 < pv n + 2 , •••, and Un + Un + i + Un + 2 + •■• < p (v n + v n + i + v n + 2 + •••)• As the ^-series is known to converge, the pu-series serves as a comparison series for the it-series which must then converge. If u n + \/u„ > v n + \/v„ and the u-series diverges, similar reasoning would show that the u-series diverges. This theorem serves to establish the useful test due to Haabe, which is if lim n(- — 1 )> 1 , S n converges ; if limnl — 1 )< 1, S„ diverges. n = x \M«+1 / n = » \U n +\ j Again, if the limit is 1, no information is given. This test need never be tried except when the ratio test gives a limit 1 and fails. The proof is simple. For /— = is finite »(logn) 1 +" a (logn) a J / x du ~1 " - - = log logn is infinite, n log n J 1 1 1 1 hence h • • • H — — h • • • and h • • • H h • • ■ 2 (log 2)!+ « n(logn) 1 + a 2(log2) n(logn) are respectively convergent and divergent by Cauchy's integral test. Let these be taken as the u-series with which to compare the it-series. Then v„ _ n + 1 /log(n + 1)\ 1 + (/ \ log?i and _^^/ 1+ l\lo. g (l + n) »»+i \ nj logn in the two respective cases. Next consider Raabe's expression. If first !im n (—2- - l\ > 1, then ultimately n (-" - l) > y > 1 and -""- > 1 + 7 « = » \Un + i I \M n + i / u»+i n ,■ /l«u(l + n)V + « , , .. , /log(l + n)V + a , Now bin — '-) =1 and ultimately — ^A_-^--^) < 1 + e, ,i = *\ logn / \ logn / INFINITE SERIES 425 where e is arbitrarily .small. Hence ultimately if 7 > 1, / 1\ /log(l + n)V + « "« l + « « , 7 ( 1 + - ) ( , - < 1 + — — + — < 1 + - . \ nj \ log n / n n- n or »„/«„ j_ 1 < w„/w„ + 1 or u n + i/w„ < i>„ + i/b„, and the M-series converges. In like manner, secondly, if lira n(— 1 )< 1, then ultimately — - < 1 + -, 7 < 1 ; and 1 + - < ( 1 + - ) " v - ; or — — < — — or -^ > -iti , logn i<„ + i r„4i Hence as the ?>series now diverges, the w-series must diverge. Suppose this test applied to the binomial series for x = — 1. Then u n n+1 .. In + 1 \ ,. m + 1 = — — , Inn n{ — 1 ) = urn = m + 1. ii n + i n — m » = x \n — ?n / « = » m It follows that the series will converge if m > 0, but diverge if m < 0. If x = + 1, the binomial series becomes alternating for n > m + 1. If the series of absolute values be considered, the ratio of successive terms | u n /u n + t| is still (n + l)/(n — m) and the binomial series converges absolutely if m > ; but when m < the series of absolute values diverges and it remains an open question whether the alternat- ing series diverges or converges. Consider therefore the alternating series 1 + m + m ( m - 1 ) , m ( m -l)(m-2) ro(m-l)-.- ( m-n + 1) 1-2 1-2-3 1 • 2 • • • ji This will converge if the limit of w„ is 0, but otherwise it will diverge. Now if m =1 — 1, the successive terms are multiplied by a factor |?/i — 11 + 1 \/n s 1 and they cannot approach 0. When — 1 < m < 0, let 1 + m = 8. a fraction. Then the nth term in the series is m=(i-*>H)-K) - logK I = - log (1-0)- log (l - ^ log (1 - ?\ . and Each successive factor diminishes the term but diminishes it by so little that it may not approach 0. The logarithm of the term is a series. Now apply Cauchy's test. f X - log (l - ?) dn = l-n log (l - °-\ + 6 log (n - (9)1 * = oo. The series of logarithms therefore diverges and lim|w„]=e- x = 0. Hence the terms approach as a limit. The final results are therefore that when x — — 1 the binomial series converges if m > but diverges if m < ; and when x = + 1 it con- verges (absolutely) if in > 0, diverges if m < — 1, and converges (not absolutely) if — 1 < m < 0. 2" ./•' 5 2 7 x 7 - -=- + t 2- 2 3 426 THEORY OF FUNCTIONS EXERCISES 1. State the number of terms which must be taken in these alternating series to obtain the sum accurate to three decimals. If the number is not greater than 8, compute the value of the series to three decimals, carrying four figures in the work : I 1 1 1 1 1 1 1 (a) 3 ~~ 2^ + 3-3 3 ~ IT3* + " ' ' ( ' 2 ~ 2^ + 3T2 3 ~ r^ 1 + " ' ' 2 3 4 log 2 log 3 log 4 ( e ) 1-I + I-I + ..., (f) e -i_2e-2 + 3e-3-4e-4 + ..-. 3^ o- 1 '- 2. Find the values of x for which these alternating series converge or diverge: 1,1, , . , x 2 x 4 x 6 (a) 1 - x* + -x* - -jfi + ..-, (B) 1 - — + J-- — + ■ ■ • , x 3 x° x" , „ x 3 x G x 7 (7 J -- + --_,+••• , (S)X--+--- + "-, 3 ! 5 ! 1 1 3 -j 1 ( .,i_£. + |--| + ..., >p 3" 4" VP ' 2(log2)* 8(loe3)" 4(log4)p INFINITE SERIES 427 CO 1 T. -J (7)1+V , (3)1+ V , *H n log n log log n *H n log n (log log n)i> 2 3 4 (e) cot- 1 1 + cot" 1 2-| , (0 1 + 1 1 + • • • • K ' ' V ; 2 2 + 1 3- + 2 4 2 + 3 5. Apply the ratio test to determine convergence or divergence : 12 3 4 2 2 2 3 2 4 < a )l + ^ + P + 2i + ""' (/?) 2To + 3T« + iro + '--' , . 2! 3! 4! 5! ... 2 2 3 3 4 4 ( *>* + F + IS + tf + -' (5) 2l + 3- ! + 4l + ---' 910 510 410 (e) Ex. 3(a), (/J), ( 7 ). (5) ; Ex. 4(a), (f), (f) — + — + — + ..., a -2 a- 3 x 4 . , 1 6x' b 2 x 3 2 3 4 a a 2 a 3 6. Where the ratio test fails, discuss the above exercises by any method. 7. Prove that if a series of decreasing positive terms converges, lim nun = 0. 8. Formulate the Cauchy integral test for divergence and check the statement on page 422. The test has been used in the text and in Ex. 4. Prove the test. 9. Show that if the ratio test indicates the divergence of the series of absolute values, the series diverges no matter what the distribution of signs may be. 10. Show that if V«„ approaches a limit less than 1, the series (of positive terms) converges; but if a u„ approaches a limit greater than 1, it diverges. 11. If the terms of a convergent series u + u { + u. 2 + • • • of positive terms be multiplied respectively by a set of positive numbers a , a,, o„. ■ ■ • all of which are less than some number G, the resulting series a o u + a^u,\ + n.,u„ + ■ ■ ■ converges. State the corresponding theorem for divergent series. What if the given series has terms of opposite signs, but converges absolutely ? • ir , „. ,_, ,_ j., ■ sin x sin2x sin3x sin 4a; . la. Show that the series 1 — h ■ ■ ■ converges abso- l 2 2 2 3 2 4 2 lutely for any value of x. and that the series 1 + x sin 6 + x' 2 sin 2 + x 3 sin 3 4- • • ■ converges absolutely for any x numerically less than 1, no matter what 6 may be. 13. If o , a v a 2 , ■ • • are any suite of numbers such that "\/|#»| approaches a limit less than or equal to 1, show that the series a + a x x + a„x 2 4- ■ • • converges absolutely for any value of x numerically less than 1. Apply this to show that the following series converge absolutely when |xj < 1 ; 1 1-3 1-3.5 (a) 1 + -X 2 + x 4 + x 6 + • ■ • , (8) 1 - 2 x + 3 x 2 - 4 x 3 + • • • , y ' 2 2-4 2-4-6 y ' (y) 1 + x + 2px 2 + 3px s + 4;'x 4 + • • • , (5) 1 - x log 1 + x 2 log 4 - x 3 log 9 4- • ■ • . 428 THEORY OF FUNCTIONS 14. Show that in Ex. 10 it will be sufficient for convergence if -\/u n becomes and remains less than y < 1 without approaching a limit, and sufficient for diver- gence if there are an infinity of values for n such that -\/w„ > 1. Note a similar generalization in Ex. 13 and state it. 15. If a power series a + a } x + a 2 x 2 + a 3 x 3 + ■ ■ ■ converges for x = A~>0, it converges absolutely for any x such that \x\ < A', and the series a x + I rtjX 2 + I a 2 x s + ■ ■ ■ and a 1 + 2 a 2 x + 3 1 or a = 1. /3 > 1, but diverges if a < 1 or a = 1, /3 ^ 1. This test covers most of the series of positive terms which arise in practice. Apply it to various instances in the text and previous exercises. "Why are there series to which this test is inapplicable ? 17. If p (l . pj. p .- ■ • is a decreasing suite of positive numbers approaching a limit X and S , S x , »S.,,- • • is any limited suite of numbers, that is, numbers such that |S„| s G. show that the series (p - p t ) S + (p 1 - p 2 ) S 1 + (p., - p..) S. 2 + • • • converges absolutely, and ]£(p„-p H + 1 )S„ =s G(p -\). 18. Apply Ex. 17 to show that, p,,. p x . p.,. • • • being a decreasing suite, if "n + w i + u 2 + ' ' ' converges, p » + p x >t l + p.,u„ + ■ ■ ■ will converge also. X.B. p u + p 1 u l + • • • + p„M„ = p S 1 + pj (S., - Sj) H + p„ (S n+ i - S„) = Sj (p - pj) + • • • + S„ (p„_i - p„) + p»S„ + i. 19. Apply Ex.18 to prove Ex. 15 after showing that p M " ( , -f p,*/, + ••• must converge absolutely if p„ + p t + • • • converges. 20. If Gfj, '(.,. "... • • • . rc„ are )i positive numbers less than 1. show that (1 + ",) (1 + ",) •■■(1 + ft,,) > 1 + ra, + "._, + • • • + ",« and (1 — ttj) (1 — «.,)■ . . (1 — «„) > 1 — r/j — r/ — ■ . • — r/„ by induction or any other method. Then since 1 -f n l < 1/(1 — ",) show that 1 1 — {a x + a 2 + ■ ■ ■ + a») 1 1 + (n. + ft, + • •• + (',,) > + «,) (1 + ft,) ■■■{1 + n n ) > 1 + (a, + a 2 + • • • + a»), > (1 - oj) (1 - «„) ■ • • (1 - «„) > 1 - (a t + «, + ••• + a,), INFINITE SERIES 429 if «j + a 2 + ■ ■ ■ + a n < 1. Or if JJ be the symbol for a product, (i-^«) >TT(i + «)>i + 2}«, ( 1+ X") >TT( 1 - r ')> 1 -^"- 21. Let TT(1 + »i) (1 + »o)- •■ (1 +w„) (1 +w»i+i)- •• be an infinite product and let P„ be the product of the first n factors. Show that j P n +p — P« i < « is the neces- sary and sufficient condition that P„ approach a limit when n becomes infinite. Show that u„ must approach as a limit if P n approaches a limit. 22. In case P„ approaches a limit different from 0. show that if e be assigned, a value of ?( can be found so large that for any value of p |^t£ _ l ' = "ff" (i + „.) _ 1 1 < 6 or "fr' (1 + ut) =l + ii, hi < e. I Pn I l» + l I »+l Conversely show that if this relation holds, P„ must approach a limit other than 0. The infinite product is said to converge when P„ approaches a limit other than ; in all other cases it is said to diverge, including the case where lim P„ = 0. 23. P>y combining Exs. 20 and 22 show that the necessary and sufficient con- dition that P„ = (1 + 0l ) (1 + a 2 ) ■ • • (1 + o„) and Q n = (1 - a x ) (1 - a s ) • • • (1 - c, n ) converge as n becomes infinite is that the series n l + H)H)K) •• «K)K)K)- (7) fiTl - ( ? 7^)"1 • ( 5 ) 0- + J ")( 1 + • r2 )( 1 + - r4 H.l + * 8 ) • • • • \ log2/\ (log 4)2/ \ (log8) 8 / VIA c + nf I 1 u- 1 1 x u- 26. Given J - or u" < » — log(l -f u) < u 2 or — according as u is a posi- 1 + u 2 2 1 + « tive or negative fraction (see Ex. 20. p. ]1). Prove that if 2m; ( converges, then U n + i + M„ + 2 + • • ■ + Mn+p - 10g(l + W n + l) (1 + "» + 2) • • • (1 +"n+p) = (S n+i ,-S n )-(logP n+p -logP H ) can be made as small as desired by taking n large enough regardless of p. Hence prove that if Zu* converges. TT(1 + u n ) converges if 2»„ does, but diverges to x if 2w„ diverges to -f x . and diverges to if S«„ diverges to — x ; whereas if 2?/j; diverges while 2u„ converges, the product diverges to 0. 430 THEORY OF FUNCTIONS 27. Apply Ex. 26 to : (a) (l + ^ (l - ^ (l + ^)(l - ^j . . . , "b-M+W-ti-- «H)K)K)K)- 28. Suppose the integrand /(x) of an infinite integral oscillates as x becomes in- finite. What test might be applicable from the construction of an alternating series ? 165. Series of functions. If the terms of a series S (x) = u (x) + u x (x) +■■■ + u n (x) + • • • (6) are functions of x, the series defines a function S(x) of x for every value of x for which it converges. If the individual terms of the series are continuous functions of x over some interval a =2 x =g b, the sum S n (.?•) of n terms will of course be a continuous function over that interval. Suppose that the series converges for all points of the interval. Will it then be true that S(x), the limit of S n (x), is also a continuous function over the interval ? Will it be true that the integral term by term, u (x)dx-\- J u x (x) dx + ■ • • , converges to I S(x)dx? Will it be true that the derivative term by term, Uq(x) + ti[(x) + • • •, converges to S'(x) ? There is no a priori reason why any of these things should be true ; for the proofs which were 1 , given in the case of finite sums will not apply to the case of a limit of a sum of an infinite number of terms (cf. § 144). These questions may readily be thrown into the form of questions concerning the possibility of inverting the order of two limits (see $ 44). For integration : Is I lim S n (x)dx — lira I S n (x)dx? For differentiation : Is — lim S n (x) = lim —S n (x)? dx » = x n = x dx For continuity : Is lim lim S n (x) = lim limS„(x)? x = x n n = y- n = x x = ar fl As derivatives and definite integrals are themselves defined as limits, the existence of a double limit is clear. That all three of the questions must be answered in the negative unless some restriction is placed on the way in which S n (x) converges to S(x) is clear from some examples. Let =e= x =E 1 and S n (x) = xn' 2 e-" r , then lim S„ (x) = 0, or S(x) = 0. n= x No matter what the value of s. the limit of S n (x) is 0. The limiting function is therefore continuous in this case ; but from the manner in which S n (x) converges INFINITE SEEIES 431 to 8 (x) it is apparent that under suitable conditions the limit would not be con- tinuous. The area under the limit S (x) = from to 1 is of course ; but the limit of the area under S n (x) is lim J xrfie-nxdx = lim e~ nx (— nx — 1) =1. The derivative of the limit at the point x = is of course ; but the limit, lim — {xn i e~ nx ) » = oo \_dX Ja:=0 = lim n 2 e- ** (1 — nx)\ = 1 T Mob) / ' \ / i \ / ' \ / ' \ / \Aa(*) / y\ i \ / / \ ! 2 3. / / ' ^-^ — " — ' \ i O Vi x 1 .Y = lim w = oo, of the derivative is infinite. Hence in this case two of the questions have negative answers and one of them a positive answer. If a suite of functions such as S^x), S 2 (x), • • • , S n (x), ■ ■ ■ converge to a limit S(x) over an interval a S x ^ b, the conception of a limit requires that when e is assigned and x is assumed it must be possible to take n so large that \R n (x )\ = \S(x ) — S n (x )\ < e for this and any larger n. The suite is said to converge uniform/// toward its limit, if this condition can be satisfied simultaneously for all values of x in the interval, that is, if when e is assigned it is possible to take n so large that \R n (x)\ < e for every value of x in the interval and for this and any larger n. In the above example the convergence was not uniform ; the figure shows that no matter how great n, there are always values of x between and 1 for which S n (x) departs by a large amount from its limit 0. The uniform convergence of a continuous function S n (x) to its limit is sufficient to Insure the continuity of the limit S(x). To show that S(x) is continuous it is merely necessary to show that when € is assigned it is possible to find a Ace so small that \S(x + A.?') — S(x)\ < e. But \S(x + Ax) - S (,r) | = | S H (x + Ax) - S n (x) + R n (x + Ax) - R n (./•) | ; and as by hypothesis R n converges uniformly to 0, it is possible to take n so large that | R n (x + Ax)\ and \R n (x)\ are less than i c irrespective of x. Moreover, as S n (x) is continuous it is possible to take Ax so small that [ S n (x + Ax) — S n (x) | < I e irrespective of x. Hence | S (x + Ax) — S (x) | < e, and the theorem is proved. Although the uniform convergence of S n to S is a sufficient condition for the continuity of S, it is not a necessary con- dition, as the above example shows. The uniform convergence of S n (x) to its limit insures that lim Xb pb S n (x) dx —IS (x) dx. U a 432 THEORY OF FUNCTIONS For in the first place S(x) must be continuous and therefore integrable. And in the second place when e is assigned, n may be taken so large that | R n (.r) |< e/(b - «). Hence Xb r*h J r* b r*b S(x)dx- S H (x)dx\ = \f R n (x)dx < / — *—- dx = e, and the result is proved. Similarly if S' n (x) Is continuous and, converges uniformly to a limit T(x), thru T(x) = S' (x). For by the above result on integrals, f U a T(x)dx = lim I S' n (x)dx = lim S n (x)-S n (a) = S(x)-S(a). Hence T(x) = S'(x). It should be noted that this proves incidentally that if S' n (x) is continuous and converges uniformly to a limit, then S(x) actually has a derivative, namely T (,/■). In order to apply these results to a series, it is necessary to have a test for the uniformity of the convergence of the series ; that is, for the uniform convergence of S n (x) to 5 (./■). One such test is Weierstra ss' 's M-test : The series u (x) + i' l (x) H 1- u n (x) H (7) will converge uniformly provided a convergent series M + M x + • • • + M n + • • • (8) of positive terms may be found such that ultimately hi i (x)\ =!= M { . The proof is immediate. For I k „(■'■)! = KO) + "•-+i( 3 -) + •• -I = M * + 3/ »+i + • • • and as the Af-series converges, its remainder can be made as small as desired by taking n sufficiently large. Hence any series of continuous functions defines a continuous function and may be integrated term by term to find the integral of that function provided an J /-test series may be found ; and the derivative of that function is the derivative of the series term by term if this derivative series admits an il/-test. To apply the work to an example consider whether the series „. . rns.r eos2x cos3x cos not defines a continuous function and may be integrated and differentiated term by term as sin 2.r sin -]f sin nx V J" x , , sin.r sin 2 a: sin 3/ sin nx S (x) = - - + - — + - — H -\ 1 (7'"> o 1 :1 2 3 3 3 » 3 . d ,, . . sinx sin2x sin3x sin nx ,..,, and (x) = a Q + a x x + a. 2 x 2 + • • • + a n x" + • • • (9') is a series which surely converges for x = 0. It may or may not con- verge for other values of x, but from Ex. 15 or 19 above it is seen that if the series converges for A', it converges absolutely for any x of smaller absolute value ; that is, if a circle of radius X be drawn around the origin in the complex plane for x or about the point a in the complex plane for z, the series (9) and (9') respectively will converge absolutely for all complex numbers which lie within these circles. Three cases should be distinguished. First the series may converge for any value x no matter how great its absolute value. The circle may then have an indefinitely large radius ; the series converge for all values of x or z and the function defined by them is finite (whether real or complex) for all values of the argument. Such a function is called an integral function of the complex variable; z or x. Secondly, the series may (ton- verge for no other value than x = or z = a and therefore cannot define any function. Thirdly, there may be a definite largest value for the radius, say R, such that for any point within the respective circles of radius R the series converge and define a function, whereas for any point outside the circles the series diverge. The circle of radius Jl is called the circle of convergence of the series. As the matter of the radius and circle of convergence is important, it will be well to go over the whole matter in detail. Consider the suite of numbers [fljj, v|a 2 |, vftfsl' '■'' v|a„|, •••. Let them be imagined to be located as points with coordinates between and + co on a line. Three possibilities as to the distribution of the points arise. First they 434 THEORY OF FUNCTIONS may be unlimited above, that is, it may be possible to pick out from the suite a set of numbers which increase without limit. Secondly, the numbers may converge to the limit 0. Thirdly, neither of these suppositions is true and the numbers from to + co may be divided into two classes such that every number in the first class is less than an infinity of numbers of the suite, whereas any number of the second class is surpassed by only a finite number of the numbers in the suite. The two classes will then have a frontier number which will be represented by 1/R (see§§19ff.). In the first case no matter what x may be it is possible to pick out members from the suite such that the set V | q t - 1 , V | «/1, v |a^|, • • • , with i R. For if |x| < R, take e < R — jxtso that|x| < R — e. Now proceed in the suite so far that all the subsequent numbers shall be less than \/{R — e), which is greater than 1/R. Then \a n + p x n +p\< l X| " + P < 1, and V - |j| " +;i 1 +P ' (R- e )»+P ' Zf(R-e) n + P will do as a comparison series. If |x| > R, it is easy to show the terms of (9') do not approach the limit 0. Let a circle of radius r less than R be drawn concentric with the circle of convergence. Then within the circle of radius r < A' the power .series (9') converges uniformly and defines a continuous function : the integral of the function may be had by integrating the scries term by term, r x 11 i 3> (.r) = I <£ (x) dx = a x + ^ ajt? + - « j? + •■• + - "„_ r ''" H ; Jo '*> n and the scries of derivatives converges uniformly and represents the, derivative of the function, <£'(./•) = a 1 + 2 a 2 x -h 3 a s x 2 -\ + na n x n ~ l -\ . To prove these theorems it is merely necessary to set up an ^/-series for the series itself and for the scries of derivatives. Let A' be any number between r and R. Then |« | + jttj X + |",| X 1 -f- ■ • ■ + |// w | A'» + • ■ • (10) INFINITE SERIES 435 converges because A r < it ; and furthermore \a n x n \ < \a n \ X" holds for any x such that \x\< X, that is, for all points within and on the circle of radius r. Moreover as \x\< X, \na n x^\ = \a n \^(^J~ } > <\a n \X» holds for sufficiently large values of n and for any x such that \x\ =§ r. Hence (10) serves as an AA-series for the given series and the series of derivatives ; and the theorems are proved. It should be noticed that it is incorrect to say that the convergence is uniform over the circle of radius R, although the statement is true of any circle within that circle no matter how small 11 — r. For an apparently slight but none the less important extension to include, in some cases, some points upon the circle of convergence see Ex. 5. An immediate corollary of the above theorems is that any p>ower series (9) in the complex variable which converges for other values than z = a, and hence has a finite circle of convergence or converges all over the complex plane, defines an analytic function f{z) of z in the sense of §§ 73, 126; for the series is differentiable within any circle within the circle of convergence and thus the function has a definite finite and continuous derivative. 167. It is now possible to extend Taylor's and Maclaurin's Formulas, which developed a function of a real variable x into a polynomial plus a remainder, to infinite series known as Taylor's and Maclaurin's Series, which express the function as a power series, provided the remainder after n terms converges uniformly toward as n becomes infinite. It will be sufficient to treat one case. Let /(•'') =/(0) +/'(<>)* + j,f"(0).r + • • • + ^—L-^-^o),.--! + r m v yd — 6)"- 1 i r x '■■■ = Si^'W = (»-i): >*"<*> = oTZif! J[ '""'^^ - '>'"> lim R n (x) = uniformly in some interval — h =i x == h, where the first line is Maclaurin's Formula, the second gives differnet forms of the remainder, and the third expresses the condition that the remainder converges to 0. Then the series /(0)+.f(0)^ + ^/"(0)^ 2 + • • • + (^ri)!^"" 1 ^ ) 35 "" 1 + h f(n)(0)xn + • ' ' (11) 486 THEORY OF FUNCTIONS converges to the value /(a - ) for any x in the interval. The proof con- sists merely in noting that/(rr) — R n (x) = S„(z) is the sum of the first n terms of the series and that \ll n (x)\ < e. In the case of the exponential function e x the nth derivative is e x , and the re- mainder, taken in the first form, becomes E n (x) — — e^x", | R n (x)\< — e h h". | x I s; ft. As n becomes infinite, /.'„ clearly approaches zero no matter what the value of h ; aild , X-2 X 3 X» e* = l + x + — + — + ... + _ + ... 2 ! 6 ! n ! is the infinite scries for the exponential function. The series converges for all values of x real or complex and may be taken as the definition of e^ for complex values. This definition may be shown to coincide with that obtained otherwise (§ 74). For the expansion of (1 + a:)" 1 the remainder may be taken in the second form. w 1.2-..(n-l) \l + 8xJ , -^ , m \m(m — 1) • • • (m — n + 1)1 , ,_ \Kn{x)\<\ v 1 7 ; — ^— — — ' /<» ( i + »)- - 1 , ft < i . I 1 • 2 • - • (?i — 1) I Hence when h < 1 the limit of Z?„ (x) is zero and the infinite expansion m(m — 1) „ ?n(m — 1) (m — 2) „ (1 + x)'" = 1 + mx + — — '- x* + — y - -^ '- x 3 + • • • is valid for (1 -f x)" 1 for all values of x numerically less than unity. If in the binomial expansion x be replaced by — x 2 and m by — |, 1 , 1 , 1 • 3 . 1 • 3 • 5 , 1 • 3 • 5 • 7 Q 1 + x 2 H x 4 + - - — x 6 + — — x 8 + ■ • • . Vl _ x 2 2 2 • 4 2-4.6 2 • 4 • (5 • 8 This series converges for all values of x numerically less than 1. and hence con- verges uniformly whenever ,X| =§ ft < 1. It may therefore be integrated term by term ' . 1 x 3 1 • 3 x 5 1 • 3 • 5 x? 1 • 3 • 5 • 7 x 9 sm-ix = x -1 1 1 1 !-•••• 23 2-45 2- 4- 67 2 • 4 • 6 • 8 9 This series is valid for all values of x numerically less than unity. The series also converges for x — ± 1 , and hence by Ex. 5 is uniformly convergent when — 1 = x =§ l . But Taylor's and Maclaurin's series may also be extended directly to functions f (z) of a complex variable. If f(z) is single valued and has a definite continuous derivative f'(f-) at every point of a region and on the boundary, the expansion /(*) =/(«) +f(a) (z-a) + ... +f n -'\«) { \~"X)\ + 7 '» lias been established (§ 120) with the remainder in the form f(t) dt 1 ?•" ML i*.«i=R?i i) H (t — z)\ 2ir p n p — r INFINITE SERIES 487 for all points z within the circle of radius r (Ex. 7, p. 306). As n becomes infinite, R n approaches zero uniformly, and hence the infinite series /(*) = /(«) + /'(*) (*-«) + ■••+ f w («) &^f } - + • • • (12) is valid at all points within the circle of radius r and upon its circum- ference. The expansion is therefore convergent and valid for any z actually within the circle of radius p. Even for real expansions (11) the significance of this result is great because, except in the simplest cases, it is impossible to compute f 00 (x) and establish the convergence of Taylor's series for real variables. The result just found shows that if the values of the function be considered for complex values z in addition to real values x, the circle of conver- gence will extend out to the nearest point where the conditions imposed on f(z) break down, that is, to the nearest point at which f(z) becomes infinite or otherwise ceases to have a definite continuous derivative/' (z). For example, there is nothing in the behavior of the function (1 + a: 2 )" 1 = 1 - x- + x* - x 6 + x s , as far as real values are concerned, which should indicate why the expan- sion holds only when \x\ < 1 ; but in the complex domain the function (l + '-r) -1 becomes infinite at z= ± i, and hence the greatest circle about z = in which the series could be expected to converge has a unit radius. Hence by considering (1 + ?r) _1 for complex values, it can be predicted without the examination of the nth derivative that the Mac- laurin development of (1 -4- a: 2 ) -1 will converge when and only when x is a proper fraction. EXERCISES 1. (a) Does x + x (1 — x) + x (1 — x) 2 + ■ ■ ■ converge uniformly when == x ==j 1 ? 1 j £ n k) (1 2 k) (/3) Does the series (1 + k)k = 1 + 1 -f _ \- ± L± ' + . . . converge uni- formly for small values of k ? Can the derivation of the limit e of § 4 thus be made rigorous and the value be found by setting k = in the series ? 2. Test these series for uniform convergence ; also the series of derivatives-. (a) 1 + x sin 9 + x" sin 2 9 + x 3 sin 3 9 + ■ ■ ■ , |x| S A" < 1, , sinx sm-x sura; sin 4 a; (P) 1 + ^ir + -5«- + -Z*r- + ~^r- + •■-, M = A < co , 0"2 l-l 2 2 2 3 2 2 4ri . , x - 1 1 [x - 1\ 2 1 (x - 1\ 3 1 _ _^ _ y ' x + l 3\x + l/ 5\x + l/ ' '~ ~ (e) Consider complex as well as real values of the variable. 438 THEORY OF FUNCTIONS 3. Determine the radius of convergence and draw the circle. Xote that in prac- tice the test ratio is more convenient than the theoretical method of the text: (a) x - I x- + i x s - \ x 4 + ■ • ■ , (p) x - \ x 3 + i x 5 - 1 x~ + ■ ■ ■ , , v IT bx b-x* /Ar 3 1 ,.. , „ x 4 x 6 x» a L a a- a 3 J 2 ! 3 ! 4 ! (O l^-(! + i)^ 2 + (} + ^ + i)^- 3 -(i + 1 + i + i)^ 4 + •••, , v , 3 2 + 3 , 3 4 + 3 , 3 6 + 3 fi , ( t ) 1 — x- + — ! — X 4 X G + • • •, U ; 4 ■ 2 ! 4 • 4 ! 4-6! ' (ij) 1 - x + x 4 - x 3 + x 8 - ./ 9 + x 12 - x 13 + • • • , {0 ) (x - I)' - J (X - 1)5 + I (X - 1)3 - 1 (X - I) 4 + ■ ■ ■ , . . (m - 1) (m + 2) ., (»i - 1) (m - 3) (m + 2) (m + 4) ( i ) x — — .r j H — x 5 — • • • , 3 '. 5 ! x 2 x 4 x 6 (k) 1 h + • • ■ , 2 2 (?ti + 1) 2 4 • 2 : (m + 1) (m + 2) , 2 6 • 3 ! (>/i + 1) (m + 2) (m + 3) X 2 x 4 a 1\ x 6 /l 1 1\ x 8 (I 1 1 1\ ( X ) 2 , " 24^ \\ + 2 j + 2^ ll + 2 + 3J _ 2^(4^ (j + 2 + 3 + 4 ) + ' ' ' ' a/3 «(a + l)ff(/3 + l) a (a + 1) (a + 2)/3Q8 + 1) (/3 + 2) I U I 1 -f - X "+* — ' " " ~ *C "T" *C "T" * ' * ♦ 1-7 1-2-7(7 + 1) 1.2-3. 7 (7 + l)(7 + - > ) 4. Establish the Maclaurin expansions for the elementary functions: (a) log (1 — x), (/3) sinx, (7) cosx, (5) coshx, (e) a 1 , (f) tan-^x, (77) sinb-ix, {$) tanh-ix. 5. Abel's Theorem. If the infinite series a + a x x + a 2 x 2 + (i.,x 5 + • • ■ converges for the value X, it converges uniformly in the interval =j= x ^= X. Prove this hy showing that (see Exs. 17-19, p. 428) |R„(x)| = |a B x" + a„+ix»+i + • • • | < (~ X | a»X» + •■• + a n + p X*+p\, when p is rightly chosen. Apply this to extending the interval over which the aeries is uniformly convergent to extreme values of the interval of convergence wherever possible in Exs. 4 (cr), (f), (6). 6. Examine sundry of the series of Ex.3 in regard to their convergence at ex- treme points of the interval of convergence or at various other points of the circum- ference of their circle of convergence. Xote the significance in view of Ex. 5. 1 7. Show that/(x) = e * 2 , /(0) = 0. cannot be expanded into an infinite Mac- _ \_ laurin series by showing that /.*„ = e * 2 , and hence that R„ docs not converge uniformly toward (see Ex.9, p. 60). Show this also from the consideration of complex values of x. 8. From the consideration of complex values determine the interval of con- vergence of the Maclaurin series for (a) taiur:^, (/3) —5—i (7) tanhx, (5) log(l + e*). INFINITE SERIES 439 9. Show that if two similar infinite power series represent the same function in any interval the coefficients in the series must be equal (cf. § 32). 10. From 1 + 2 r cosx + r 2 = (1 + re ix ) (1 + re~ ix ) = r 2 (l + — \ (l + -—\ I r- r s \ prove . log (1 + 2 r cos x + r 2 ) = 2 I r cos x cos 2 x + — cos 3 x — • • ■ I , Xx / r -2 r 3 \ log (1 + 2 r cos x + r 2 ) dx = 2 1 r sin x — — sin 2 x 4 — r sin 3 x — • • ■ 1 11/1,0 , ox -,1 /cosx cos2x cos3x \ and loo; (1 + 2 r cos x + r 2 ) = 2 loir r + 2 1 • • • . \ r 2r 2 3r 2 /' T ' 1 /1 , -. , o N , n , , „ /win j^- sin2x sin3x \ I log (1 + 2 r cos x + ?-) dx — 2 x log r + 2 ( 1 ... J i) \ r 2'-V 2 3 2 r 2 / T<\ r >1 /* J (X ( (X CC Sill "^ J" I log (1 + sili or cos x) dx = 2 x log cos — h 2 ( tan — sin x — tan 2 (- Jo 2 \ 2 2 2 2 2 2 2 1 dx r 1 dx , 1 1-3 1 ■ 3 • 5 r * tZx Ju V 1 + x^ 2-0 2 ■ 4 • !) 2 • 4 ■ *\ 3 1 /r\ 3 r (or) / etc = ---(-) +-(-) =tan-i-, Jo x q 3 V// 5 V// (/ ;(1 + k cosx) + ,„, f r 10g(l + A COSX) /"r xSinX , 7T 2 (/3) I — '-dx = irsm-ik, (7) / dx = — , Jo cosx Jo 1 + cos 2 x 4 e- a fV^ Jo (1 — i a cos x + a 2 2 sin 2 xdx (1— 2 a cosx+ a 2 )(l — 2/3 cosx + /3 2 ) 15. In Ex. 14 (7) let a = 1 — ///»; and x = z/m. Obtain by a limiting process, and by a similar method exercised upon Ex. 14 (a) : r x z sin zdz _ 7r _ A r x cos zdz _ Jo /r + z 2 ~ 2 6 " Jo h 2 + z 2 _ Can the use of these limiting processes be readily justified ? 440 THEORY OF FUNCTIONS 16. Let h and x be less than 1. Assume the expansion f(x, h) = * = 1 + hP^x) + 7i 2 P (x) + • • • + h"P n (x) + • • . . Vl - 2 x/i + h 2 Obtain therefrom the following expansions by differentiation : h {l-2xh + /i 2 )2 / A ' = ^—^ =P X + 2//P., + 3FP 3 + .. . + ,i/i«-ip n + . . . . (l-2xA + A 2 )a Hence establish the given identities and consequent relations: ^/* = a*i + h (xP-2 - p[) + ■ ■ ■ + *■* ~K*K - K -i) + • • • = f h = P 1 + h(2P 2 ) + . . . + h*-i(nP„) +-.., (l±p.f> x - f = - 1 + p; + h(p: 2 - p x ) + . . . + *(f; +1 + p,^ _ p w ) + . . . = 2 xhf = A (2 x) + • ■ • + h\2 xP n _i) . Or nP n = xP' n -P' n _ 1 and p; +1 + P^ - P„ = 2xP;. Hence xP' n - P' n +1 - (n + 1) P„ and (x 2 - 1) P^ = n (xP„ - P n _ a ) . Compare the results with Exs. 13 and 17, p. 252, to identify the functions with the Legendre polynomials. Write 1 1 1 (1 _ 2 xh + h 2 )^ (1 - 2 7i cos + Ifl)^ (1 - he^)\ (1 - he~ •'»)* 1 + I h e id + — ifie-i is + . . .\ /i + I /, e - tt + — / t 2 e - 2 '0 + ■ • ■ V 2 2-4 A 2 2-4 / and show P„(cos 0) = 2 ll 3 ' ' ' ( 2n ~ ] ) j cos n # + — cos (n - 2) + • • • 1 • 2 • 4 • • ■ 2 n [_ 1 • (27i — 1) J 168. Manipulation of series. If an Infinite series S = " + " l + i(. 2 -\ h i'n-i + u n H (13) converges, the series obtained by grouping the terms in parentheses with- out altering their order will also converge. Let S' = U + tf, + • • • + ?/„,_, + tf„, + ■ ■ • (13') and S[, S' a) ■ ■ ■ , S' K ,, ■ ■ ■ be the new series and the sums of its first n' terms. These sums are merely particular ones of the set S, $.„■■■, S n , ■ ■ •, and as n' < n it follows that n becomes infinite when n' does if n be so chosen that S n = S' n ,. As S n approaches a limit, S' n , must approach the same limit. As a corollary it appears that if the series obtained by removing paren- theses in a given series converges, the value of the series is not affected by removing the parentheses. INFINITE SERIES 441 If two convergent infinite series he given as S = u + u t -\ , and T = v Q + i\ -\ , then (\u + fiv ) + (A^ + fx<\) -\ will converge to the limit XS -+- /xT, and will converge absolutely provided both the given series converge absolutely. The proof is left to the reader. If a given series converges absolutely, the series formed by rearranging the terms in any order without omitting any terms will converge to the same value. Let the two arrangements be S = u Q + «! + i'.,-\ + >'n - 1 + ii n H and S = u a , + u v + u. t , -+-•••+ 'V-i + u n> "+" ' " • • As S converges absolutely, n may be taken so large that |««| + K + i|H — < e ; and as the terms in S' are identical with those in S except for their order, n' may be taken so large that S' n , shall contain all the terms in S n . The other terms in S' n , will be found among the terms v n , u H ± l} Hence , „, „ , As \S — S n \ < e, it follows that [ S — S^| < 2 e. Hence S' n , approaches S as a limit when n' becomes infinite. It may easily be shown that S' also converges absolutely. The theorem is still true if the rearrangement of S is into a series some of whose terms are themselves infinite series of terms selected from S. Thus let s , = ^ + ^ + ^ + + ^ _ i + ^ + _ ^ where U i may be any aggregate of terms selected from S. If U i be an infinite series of terms selected from S, as u i = v m + «.-i + «« H h "{» H , the absolute convergence of L* ( - follows from that of ,V (ef. Ex. 22 below). It is possible to take n' so large that every term in S n shall occur in one of the terms U Q , U , ■ ■ •, U n ,_ 1 . Then if from S- U - U x U„_ x (14) there be canceled all the terms of S n , the terms which remain will be found among u n , i( n+1 , • ••, and (14) will be less than e. Hence as n' becomes infinite, the difference (14) approaches zero as a limit and the theorem is proved that S = U Q + U x + • • • + U n ,_ t + U n , + ■ • • = S'. 442 THEOKY OF FUNCTIONS If a series of real terms is convergent, but not absolutely, the number of posi- tive and the number of negative terms is infinite, the series of positive terms and the series of negative terms diverge, and the given series may be so rearranged as to comport itself in any desired manner. That the number of terms of each sign cannot be finite follows from the fact that if it were, it would be possible to go so far in the series that all subsequent terms would have the same sign and the series would therefore converge absolutely if at all. Consider next the sum S„ — Pi — X m . I + m = n, of n terms of the series, where Pi is the sum of the positive terms and N m that of the negative terms. If both Pi and y m converged, then P t + X m would also converge and the series would converge absolutely ; if only one of the sums Pi or N m diverged, then S would diverge. Hence both sums must diverge. The series may now be rearranged to approach any desired limit, to become positively or negatively infinite, or to oscillate as desired. For suppose an arrangement to approach L as a limit were desired. First take enough positive terms to make the sum exceed L. then enough negative terms to make it less than L. then enough positive terms to bring it again in excess of L. and so on. But as the given scries converges, its terms approach as a limit ; and as the new arrangement gives a sum which never differs from L by more than the last term in it. the difference between the sum and L is approaching and L is the limit of the sum. In a similar way it could be shown that an arrangement which would comport itself in any of the other ways mentioned would be possible. If two absolute/// convergent series be multiplied, as S == u + »j -f- k 2 -\ + u n -\ , T=r {) + v 1 + c, + --- + v n + .-., and W = u v + u^ + u. 2 v H + u H v H + 'Vi + 'Vi + u 2 v i + ■ • • + 'V'i H — + + 'V'„ + "l' - » + »-2 r n H 1" V n V n ~\ + and if the terms in W be arranged in a simple series as Vo + ('Vo + 'Vi + Vi) + ( >V'o + U 2 V i + "/, + "i'"a + ? V'o) + ■ • • or in "u// other manner ichatsoerer, tin- series is absolutely convergent and converges to the value of the product ST. In the particular arrangement above, ST, N, 7'.,. .S', ( 7' n is The sum of the hrst, the iirst two, the first n terms of the series of parentheses. As lim S n T n = ST, the series of parentheses converges to ST. As S and T are absolutely convergent the same reasoning could be applied to the series of absolute values and I "ol I r ol + I "i 1 1 v o\ + I ? 'il I l 'il + i w ol I v il + I "-2 ! i '"o I + • ' " would be seen to converge. Hence the convergence of the series Vo + 'Vo + 'Vi + 'Vi + 'V'o + 'Vi + 'V; + "i r i + "/2 + ■■• INFINITE SERIES 443 is absolute and to the value ST when the parentheses are omitted. Moreover, any other arrangement, such in particular as Vo + OV'o + V ; i) + OVo + Vi + v 2 ) + • • • > would give a series converging absolutely to ST. The equivalence of a function and its Taylor or Maclaurin infinite series (wherever the series converges) lends importance to the operations of multiplication, division, and so on, which may be performed on the series. Thus if /(•*') = "0 + a i x + "^ + %x s + ■■■, \x\ < R v 9 (•'') = \ + h x + ¥ >2 + h x% + • ' '> \ x \ < R v the multiplication may be performed and the series arranged as f(x)g(x) = a Q b + { % \ + a^)x + (a b 2 + afc + a 2 b Q )x 2 + ■■■ according to ascending powers of x whenever x is numerically less than the smaller of the two radii of convergence R , /.'.,, because both series will then converge absolutely. Moreover, Ex. above shows that this form of the product may still be applied at the extremities of its inter- val of convergence for real values of x provided the series converges for those values. As an example in the multiplication of series let the product sin x cosx be found. 1,1- ,1,1 1 c sin x = x x 3 -\ x" — • • ■ , cos x = 1 — ■ x 2 -| x 4 x 6 + • • • . 3 ! 5 ! 2:4:6! The product will contain only odd powers of x. The first few terms are / 1 1\ „ /l 1 1 \ . /I 1 1 1 1 x- \- — )x 3 + h 1 )x a — [— H h h — \3 ! 2 !/ \o ! 3 ! 2 ! 4 !/ \7!5:2!8!4!6 The law of formation of the coefficients gives as the coefficient of x 2k + 1 w r 1 i i ■ 1 11 (- 1)M -H + + ••• + h — - v ' l{2k + l)\ (2i-l)!2! (2A-3)!4! 3! (2 A -2)! (2A')U ( - V)l ' r 2 , (-'/■ +1)2 A- | ( 2A-+l) ( 2/.-)(2fr-l)(2fc-2) ; , (2 k 4-1) (2t + i):L 2: 4: 1 But 2** + i = (1 + 1) 2A ' + 1 =l + (2k + l)+ (2A -^-^- 2 i' + • • . + (2 k + 1) + 1 . Hence it is seen that the coefficient of x 2i+1 takes every other term in this symmet- rical sum of an even number of terms and must therefore be equal to half the sum. The product may then be written as the series ,'9 r\3 sinx cosx If (2x) 3 (2x) 5 "I 1 . 1 = - 2x — + ■ • • = - sin 2x. 2L 3! 5: J 2 444 THEORY OF FUNCTIONS 169. If a function f(x) be expanded into a power series f(x) = a Q + a x x + a.p? + a g x s -\ , \x\ < R, (15) and if x = a is any point within the circle of convergence, it may be desired to transform the series into one which proceeds according to powers of (x — a) and converges in a circle about the point x = a. Let t — x — a. Then x — a + t and hence x 1 = a 2 + 2 at + /", x*= a 3 + 3 aH + 3 art 2 + f ! , • • • , f(x) = a Q + «., (a + + " 2 (« 2 + 2 a* + / 2 ) + ■ • • . (15') Since |«r] < R, the relation [a| + \t\ < 7? will hold for small values of t, and the scries (15') Avill converge for x = \a\ + \t\. Since % + a x (\a\ + \t\) + a 2 (\a\* + 2\a\\t\ + \tf) + ... is absolutely convergent for small values of t, the parentheses in (15') may be removed and the terms collected as fix) = <£ (t) = (« + a x « + «./r + « 3 « 8 + •••) + K + 2 «./* + 3 »y + •••)< + (a J + 3a, ff + ...)< , + (a i + ...K + ..., or /(*) = <£(* - or) = ,1 + ^(x - a) + ^(3 - rr) 2 + A s (x-ay + ..., (16) where vl , -4 l5 ,1.,, ■ ■ • are infinite series ; in fact \ = /(«), ^ = /'(«)> ^ = |j /», ^ = 1 /'», .... The series (16) in x - a will surely converge within a circle of radius R —\oc\ about x = a ; but it may converge in a larger circle. As a matter of fact it will converge within the largest circle whose center is at rr and within which the function has a definite continuous derivative. Thus Maclaurin's expansion for (1 + x 2 )~ l has a imit radius of convergence; but the expansion about x = \ into powers of x — \ will have a radius of convergence equal to |V5, which is the distance, from x = \ to either of the points x = ± i. If the function had originally been defined by its development about x = 0, the definition would have been valid only over the unit circle. The new development about x =■ \ will therefore extend the definition to a considerable region outside the original domain, and by repeating the process the region of definition may be extended further. As the function is at each step defined by a power series, it remains analytic. This process of extending the definition of a function is called analytic continuation. INFINITE SERIES 445 Consider the expansion of a function of a function. Let f(x) = a + a x x + a, 2 x 2 + a^x 3 -\ \x\ < R v x = (y) = b + b lV + by + b^f +■■■, \y\< R 2 , and let \b \ < II x so that, for sufficiently small values of y, the point x will still lie within the circle i? r By the theorem on multiplication, the series for x may be squared, cubed, • • • , and the series for x 2 , x 3 , ■ ■ ■ may be arranged according to powers of y. These results may then be sub- stituted in the series for fix) and the result may be ordered according to powers of y. Hence the expansion for /[<£(?/)] is obtained. That the expansion is valid at least for small values of y may be seen by considering h! + h!£ + hl£ 2 + h|f + ---, € which are series of positive terms. The radius of convergence of the series for /[(//)] may be found by discussing that function. For example consider the problem of expanding e C08X to five terms. c'J = 1 + y + I ?y 2 + l y 3 + & ?/ 4 + • • • , y = cos x = 1 - \ x 2 + & x* + ■ ■ ■ , 2/2 = 1 _ X 2 + 1 x i ? y 3 - l _ 3 X 2 + 7 x i ) yi - I _ 2 X 2 + If X 4 , C'J = 1 + (1 - I X 2 + ^ X 4 ) + 1(1 _ x 2 + ^x 4 ) + i (1 _ i x 2 + |x 4 ) + 2 V(l-2a- 2 + lfx 4 ) + ... = (i + 1 + \ + I + & + • • •) - (-v + i + \ + A + • • -)* 2 + (i + 1 + A+ i + ---)* 4 + ---, c?/ _ c eoea: _ 21| _ 11- X 2 + \ ;f X* . It should be noted that the coefficients in this series for e cosx are really infinite series and the final values here given are only the approximate values found by taking the first few terms of each series. This will always be the case when y = b + b x x + • • • begins with b ^ ; it is also true in the expansion about a new origin, as in a previous paragraph. In the latter case the difficulty cannot be avoided, but in the case of the expansion of a function of a function it is some- times possible to make a preliminary change which materially simplifies the final result in that the coefficients become finite series. Tims here c cobx _ e l+ z =ee z ? z — cos x — 1 = — \ X 2 + nV xi — tIo Z 6 + • • • •> 2 2 = ^X 4 -J ? X6 + ..., Z 3 = _l x 6 + ... ) Z 4 , Z 5, Z 6 = + ..., e* = 1 + (- A* 2 + ^x 4 - T i^x<5 + •••) + i (|x 4 - fox* +•••) + H- i* 6 + •••) + ••• , e cosx _ ee z _ e (1 _ 1 x 2 + T x 4 _ JM_ X 6 _! ). The coefficients are now exact and the computation to x 6 turns out to be easier than to x 2 by the previous method ; the advantage introduced by the change would be even greater if the expansion were to be carried several terms farther. 446 THEORY OF FUNCTIONS The quotient, of two power series f(x) by (/(■>'), if y(0) =#= 0, may be obtained l>y the ordinary algorism of division as For in the first place as g(0) =#= 0, the quotient is analytic in the neigh- borhood of x = and may be developed into a power series. It there- fore merely remains to show that the coefficients c Q , c , c 2 , ■■■ are those that would be obtained by division. Multiply K + a x x + a^ + •••) = (c + ^ + c.^ + ■ ■ •) (b Q + /y + /y 2 + • • •) = & o c o + (V'o + W * + OVo + * A + W x * + • • • » and then equate coefficients of equal powers of a\ Then is a set of equations to be solved for c , c l3 c 9 , • • •. The terms in /(a;) and ■.■■■ c„ - 1 " in are therefore precisely those obtained in dividing the series. If y is developed into a power series in x as y = /(■'■) = "0 + "x x + "•/' + • • • > "1 ^ °> ( 17 ) then x may be developed into a power series in // — a as x =f- 1 (y - a ) = \{y - a.) + b,(y- « Q f + • ••. (18) For since er =£ 0, the function f(x) lias a nonvanishing derivative for x = and hence the inverse function/ -1 (// — a ) is analytic near x = or ?/ = « and can be developed (p. 477). The method of undetermined coefficients may be used to find b , b ,->-. This process of finding (18) from (17) is called the reversion of (17). For the actual work it is simpler to replace (y — " V"i b y t so that t = x + a'./- + a' y r* + r^.r 4 -| , a' { = fff/rt^, and ./• = / + A:/- + /,./' + /^ 4 H , &;. = l, i u[ . Let the assumed value of x be substituted in the series for t ; rearrange the terms according to powers of t and equate the corresponding coef- ficients. Thus ,,..-., t = t + (h', + «: z )t- + (Ju + 2 A>; + a' 3 , r + {l>\ + 2 b' 9 a' 3 + Khu + 3 a>; + »; ) f 4 + • • • or b' 2 = — ■ a!,, b' ?> = 2 ".," — a~, b\ = — 5 ".,'' + o "..".", — "4, • • •. INFINITE SERIES 447 170. For some few purposes, which are tolerably important, a formal ojjemtional method of treating series is so useful as to be almost indis- pensable. If the series be taken in the form 1 + a x x + ^x 2 + ^x 3 + • • • + -^ x» + ■-, 1 2 . 6 . n . with the factorials which occur in Maclauriir's development and with unity as the initial term, the series may be written as ir , a 3 „ a n e°* = 1 + o\r + — x 2 + — x s + • • • 4- — x n + • • • , 2 . 6 . n . provided that a' be interpreted as the formal equivalent of " ; . The product of two series would then formally suggest e ax e bx _ e (a + b).c _ I _j_ („ _j_ h y x _J_ _ („ _|_ / y y2,.o _| ^ ^C)^ and if the coefficients be transformed by setting a'b J = a/jj, then ( i + «x* + fr ^ + • • •) (i + M + fr •'" + • • •) f/ a- 2 "A + //., n = 1 + (^ + / V ,' 4 2+ 2 ; 1+ 2 x* +■■-. This as a matter of fact is the formula for the 'product of two series and hence justifies the suggestion contained in (19). For example suppose that the development of x , B, , B % , = 1 4- B x 4 x 2 4 — x 3 4 e x — 1 x 2 ! 3 ! were desired. As the development begins with 1, the formal method may be applied and the result is found to be X — p Bx j. _ p (B+l)x _ p Bx (20) e — 1 x = x + [(5 4 l r - £' 2 ] ^ + [(£ 4- If - z,"] ^ + • • • , (21) (B + l)' 2 - B 2 = 0, (5 + 1 f - /? 3 = 0, • • • , (B + 1/ - S* = 0, ■ • ■ , or 2^ + 1 = 0, 3 /J, + 3 ^ + 1 = 0, I £ 3 + 6 /J, + 4 7i 1 + 1 = 0, • • • , or B x =~h B. 2 = l, B s = 0, B t = -&,-•-. The formal method leads to a set of equations from which the suc- cessive .B's may quickly be determined. Note that x x x e r + 1 x , . x x ,. I x\ /OOA ^Ti + 2 = 2 ?Ti = 2 C ° th 2 = ~ 2 C ° th I" 2J (22) 448 THEORY OF FUNCTIONS is an even function of x, and that consequently all the B's with odd indices except B l are zero. This will facilitate the calculation. The first eight even B : s are respectively 1 1 1 1 5 __ _6 9 J 7 36 17 C>'\\ £> ~515> 4 25 :i0 5 5? J 2 730) <5"> .3 1 • \~ ' ) The numbers B, or their absolute values, are called tlie Bernoullian numbers. An independent justification for the method of formal cal- culation may readily be given. For observe that e r i J ' x = r (/3xl)a ' of (20) is true when B is regarded as an independent variable. Hence if this identity be arranged according to powers of B, the coefficient of each power must vanish. It will therefore not disturb the identity if any numbers whatsoever are substituted for B l , B' 2 , B 9 , • ■ • : the particular set B , B 2 , B % , ■■■ may therefore be substituted ; the scries may be rear- ranged according to powers of x, and the coefficients of like powers of x may be equated to 0, — as in (21) to get the desired equations. If an infinite series be written without the factorials as 1 + a t x + ",.<•- -ky' 3 "i + ""■''" H > a possible symbolic expression for the series is — = 1 + (^x + a % x~ + "V + • • • , c' : = a { . 1 — ax If the substitution y = #/(l + ' 7 ') or x = >//(! — y) be made, 1 1 - // l-(l + a)y' (24) Now if the left-hand and right-hand expressions be expanded and J + (1 + ")Y +•■•]> provided that both series converge absolutely for a t = a*. Then 1 + a x x + ".,./■- + a/' + ... = l4-«y + a (l+ a) if + " (1 + «ff + i if + i y 6 + • • • • To compute log 2 to three decimals from the first series would require several hundred terms ; eight terms are enough with the second series. An additional advantage of the new series is that it may continue to converge after the original series has ceased to converge. In this case the two series can hardly be said to be equal ; but the second series of course remains equal to the (continuation of the) function defined by the first. Thus log 3 may be computed to three decimals with about a dozen terms of the second series, but cannot be computed from the first. EXERCISES 1. By the multiplication of series prove the following relations: (a) (1 + x + x 2 + x s + ...)2 = (l + 2x + 3x 2 + 4x 3 + • • •) = (1 - x)- 2 , (£) cos 2 x + sin 2 x = 1, (7) e*ev = e x +y, (5) 2 sin 2 x = 1 — cos 2 x. 2. Find the Maclaurin development to terms in x 6 for the functions: (a) e^cosx, (/3) e*sinx, (7) (1 + x)log(l + x), (5) cosxsin^x. 3. Group the terms of the expansion of cosx in two different ways to show that ens 1 > and cos 2 < 0. Why does it then follow that cos £ = where 1 < £ < 2 ? 4. Establish the developments (Peirce's Xos. 785-789) of the functions: (a) e 8in - r , (j3) e tanx , (7) e 6 " 1-1 ^, (5) e tan ~ lx . 5. Show that if g(x) = b m x m + 6 m + 1 x m + 1 + • • • and/(0) ^ 0, then fix) a n + «,x + a„x 2 + • ■ • C- m c_ m+ i c_i J -±-t = -r 1 -r 2 -r _ m + m + 1 + ... + 1 _ + c + c r + ■ ■ ■ g{x) h m x>» + 6 TO+ ix™ + i + ••• x m x"'" 1 x and the development of the quotient has negative powers of x. 6. Develop to terms in x 6 the following functions: (a) sin(tsinx), (j3) logcosx, (7) Vcosx, (5) (1 — k 2 sin 2 x)~ 2. 7. Carry the reversion of these series to terms in the fifth power: (a) ij = sinx = x — \ x 3 + • • • , (/3) y = tan- 1 x = x — \ x z + ■ ■ ■ , (7) y = e* = 1 + x -f I x 2 + • • • , (5) y = 2 x + 3 x 2 + 4 x 3 + 5 x 4 H . 450 THEORY OF FUNCTIONS 8. Find the smallest root of these series by the method of reversion: («) 5 = f X er*dx = x - -x 3 4- -x 5 - — xT + • • •, 2 Jo 3 .5.0 o.i „ arc the coefficients in the expansion of sechx. Establish the defining equations and compute the first four as — 1. 5. — 61, 1385. 12. Write the expansions for sec x and log tan (\tt + ?, x). 1 2 1 13. From the identity = derive the expansions: ■ t r _ i <.--'•' - 1 e* + 1 (a) — - = I + 7?,(2 2 -1)^-4- B 4 (2* - 1) ~ + ■ ■ - + B,J2^ _ \)*^± + . t-' + 1 2 2 ! 4 ! 2 n ! + 11 x r 3 r 2/i-i = - - 7J.,(2 2 - 1) — - 7M2 4 - 1) 7io„(2 2 « - ] ) : + 12 " ' 2 ! 4 4 '. 2 ji : (7) tanh x = (2 2 - 1 ) 2 2 7i„ — + (2 4 - 1 ) 2 4 /?— + ... + (2 2 » - 1 ) 2 2 »7? 2 „ — + " 2 ! 4 ! 2 ?i ! (5) tanx =.r + -r - + "-4- + -^ + ■•■ + (- 1 )" - 1 1-' 2 '' - 1) 2 2 »7Jo„ — — + ■•• 3 lo 31 o 2 n '■ x' 2 X 4 X 6 X 2 " ( e ) log cos x = - 1 )" -i(2 2 » - 1) 2 2 "7*.,„ 2 12 45 - 2n-2n! X 2 7 ,/' 4 r 2 " (f) log tanx = logx + (- (-... + (_ i)«-i(oi,,-i _ i)2 2 "7J..„ _ + 3 60 n ■ 2 n ! ( 77 ) esc x = - (cot ' r 4- tan X ) = - + — 4- • ■ . + (- 1 )» -1 2 (2 2 " - 1 — 1 ) Bo „ ' r ~ , 2 \ 2 2 ' x 3 1 2 n ! ((9) log cosh x, (t) log tanh x, (*) cschx, (X) sec 2 x. INFINITE SERIES 451 Observe that the Bernoullian numbers afford a general development for all the trigonometric and hyperbolic functions and their logarithms with the exception of the sine and cosine (which have known developments) and the secant (which re- quires the Eulerian numbers). The importance of these numbers is therefore apparent. 14. The coefficients P^y), P 2 (y), ■ ■ ■ • Pn(y) in the development e'j* — 1 e r - 1 y + P x {y)x + P 2 (y)x 2 + ■■■ + P„(y)x« + are called Bernoulli's polynomials. Show that (n -f 1) ! P„(y) = (B + y) n + 1 — B n + l and thus compute the first six polynomials in y. 15. If y = X is a positive integer, the quotient in Ex. 14 is simple. Hence n : P n (X) = 1 + 2» + 8* + • • • + (X- 1)« is easily shown. With the aid of the polynomials found above compute: (a) 1 + 2* + 3 4 + • • • + 10 4 . 0) 1 + 2< + 3 5 + ■ • • + 6 , ( 7 ) 1 + 22 + 8 a + • • + (X - l) 2 , (5) 1 + 23 + 3 8 4- • + (X - 1 )«. (a - 6) | ,_ T 11 1 1 1 ^a» + i 16. Interpret — = — = > 1 — ax 1 — bx x (a — b) [_l — ux a — 6xJ *-/ « /* x 1 17. From | e-( 1 - ax ) t dt = — - establish formally Jo 1 — ax 1 + «,x + a„x 2 + a.,x 3 + ■ ■ ■ = t-'F(xt)dt = - I e •' ' F(u)du, Jo x Jo where P( u ) = 1 + «i w H «..w 2 + — « 3 m 3 + • • ■ • Show that the integral will converge when < x < 1 provided |a,-| ^ 1. 18. If in a series the coefficients a,- = I l'f(l)dt, show 1 + «,x + «.>x 2 + « 3 x 3 + •••=/ Jo 1 f{ " dt. xt 19. Note that Exs. 17 and 18 convert a series into an integral. Show («)i + ^+^ + ^ + ...=-iY— -°' : -^ ** - 0,) =fv^-^. v ' 2? 3? iP T(p)Jo 1-xt ' ui' Jo , , 1 .r ./•- /» i sin log £ , , 1 r x , . (j8) + + — +■■• = -/ — ^-dt by- — = / e-»*sn£df, ; 1 + 1- 1 + 2 2 1 + 8* Jo l-z« * 1 + n- Jo .r ./•- p r ! sin log2 ,, , + 2 2 1 + 3- Ju 1 a (a + 1) _ 2 , a(« + 1) (a + 2) _ 3 (7) 1 + - x + —5 — : — £ x- + — — — x 3 + v; b 6(6 + 1) 6(6 + 1) (6 + 2) = m r r(a) I(b -a) Jo («)I(6-a)Jo 1-xt 452 THEORY OF FUNCTIONS 20. In case the coefficients in a series are alternately positive and negative show that Euler's transformed series may be written a x x — o 2 x 2 + a 3 x 3 — a 4 a- 4 + • ■ ■ = a x y + Aaiy' 2 + A 2 aiy 3 + A 3 ai// 4 H where Aai = a.\ — a- 2 , A-t/i = Aai — Aa 2 = (ii — 2 c/ 2 + « 3 , • • ■ are the successive first, second, • • • differences of the numerical coefficients. 21. Compute the values of these series by the method of Ex. 20 with x = 1, y = \. Add the first few terms and apply the method of differences to the next few as indicated : ( a ) i 1_ __ h • • • = 0.09315, add 8 terms and take 7 more, V ' 2 3 4 (/3) 1 1 + • • • = 0.0049, add 5 terms and take 7 mure, V2 V'3 V4 ( 7 ) ? = i_ 1 + l_ 1 + ... = 0.78539813. add 10 and take 11 more, 4 3 5 7 / 111 \ 2/'- 1 / 111 (5) Prove (l + _+_ + _ + ...) = 1 + + . K ' \ 2p 3/' 4/' / 2p - 1 — 1 \ 2p 3/' -i'' and compute fovp = 1.01 with the aid uf five-place tables. 22. If an infinite series converges absolutely, show that any infinite series the terms of which are selected from the terms of the given series must also converge. What if the given series converged, but not absolutely ? 23. Note that the proof concerning term-by-term integration (p. 432) would not hold if the interval were infinite. Discuss this case with especial references to justifying if possible the formal evaluations of Exs. 12 (a), (5), p. 439. 24. Check the formula of Ex. 17 by termwise integration. Evaluate - f e * JJbu) du =1—4 b-x 2 + I • 3 = (1 + 7Ac-)~ * x Jo 2 ! by the inverse transformation. See Exs. 8 and 15. p. 399. CHAPTER XVII SPECIAL INFINITE DEVELOPMENTS 171. The trigonometric functions. If m is an odd integer, say in = 2 n + 1. De Moivre's Theorem (§ 72) gives sin ?//<& (m — 1) (m — 2) . „ — ^=™»» + - -|i i««->.m-* + ... 1 (1) where by virtue of the relation cos 2 = 1 — sin 2 <£ the right-hand mem- ber is a polynomial of degree n in sin' 2 cj>. From the left-hand side it is seen that the value of the polynomial is 1 when sin = and that the n roots of the polynomials are sin 2 7r/??i, sin 2 2 7r/w?, •■•, svcPntr/m. Hence the polynomial may be factored in the form sin m _ L _ sin' 2 \ A _ sin 2 y sin tt/?uJ y ain-'J 7r//n J y sin^mr/m If the substitutions <£ = x/m and <£ = i#/ra be made, sin.r _ / sin' 2 ./■'/// \ / sin 2 a"/m \ / sin 2 x/m \ in a\n x/m. y sin 2 ir/inj \ sin' 2 2 it/ m] y sin 2 nnr/m/ _j*mh* = / sjnh^A / + sinh^/m X A + simh^A msinhay/M, y sm 7r/?>t / y smzTr/mJ y sin mr/mj 7 Now if ?>j be allowed to become infinite, passing through successive odd integers, these equations remain true and it would appear that the limiting relations would hold: sm x -v 1 -? x sinh x X -(»■=: lim sin 2 — . ,/.-7T snr — m TTd-^l- W .r 1 .r 3 7H <) //^ /.•7T 1 //C7r\ 3 Y 2 ^' 2 T 2 r — + ' 453 454 THEORY OF FUNCTIONS In this way the expansions into infinite products sin x = x TT ( 1 — —, — , |> sink a; = a- TT ( 1 + 71— ; ) (5) 1 \ IriT-J 1 \ /i-7T7 would be found. As the theorem that the limit of a product is the prod- uct of the limits holds in general only for finite products, the process here followed must be justified in detail. For the justification the consideration of sinhx, which involves only positive quantities, is simpler. Take the logarithm and split the sum into two parts P I sinh 2 - \ „ j si nh 2 — \ sinh x sr\ , I , , m \ v^ , I ., m msinh — 1 \ sin- — / p + 1 \ sin 2 — / m m m J As log(l + a) < a. the second sum may be further transformed to n I sinh 2 — \ „ sinh 2 — n li = > luK 1 + <> = smh 2 -> 4< • J^ ^ ■ M mAi . Jen P+i \ sin 2 — / i> + 1 sin 2 — J'- 1 sm 2 — Now as n < \ m, the angle kir/m is less than \ it, and sin| > 2 £/7r for £ < -J it, by Ex. 28, p. 11. Hence „ . , ., x v\ »( 2 m' 2 . . a x -\-\ 1 m- . ,.x r E = — sinh 2 — > — < — sinh 2 — | in Jm 4, 4 k 2 -1 m ^~~t, k' 2 4 m Jp P T i p + 1 ?n 2 . . _ J r* dk V 2 ' i s i nn £ "ST 11 I -i , '"1 "< 2 ■ i o x Hence log > 1 -| < sinh 2 —. • , •'' ^ • , kir \ 4 n m in sinh — i \ sin- — / Now let m become infinite. As the sum on the left is a finite, the limit is simply sinh x -sr\ /, x' 2 \ x 2 . . sinh j- v-\ A j2 \ log— > 1 + < — ; and log— - = > 1 + ) then follows easily by letting/* become infinite. Hence the justification of (4'). By the differentiation of the series of logarithms of (5), sin./- ^ , / ./•- \ , sinh./' ^ , /. x 2 \ log — = v i^ i-^V io g . - = yi..g i + ^, (0) the expressions of cot x and coth x in series of fractions l v> -•'' i 1 ^ -•'■ eot x = > ,., ., — ;,) coth ,/• = - + > — — , ( i ) ./■ y ,r7r ~ - J •*' T 7< ^ + •' SPECIAL INFINITE DEVELOPMENTS 455 are found. And the differentiation is legitimate if these series converge uniformly. For the hyperbolic function the uniformity of the conver- gence follows from the J/-test 7 -» o . ., < tt-v and 2t^ c Irir 4- j- kit ^"1 irir on verges. The accuracy of the series for cot x may then be inferred by the substi- tution of ix for x instead of by direct examination. As 2 x Irir 2 — j-- 1 1 - utt .'• 4- KIT cot ^ X - h (8) In this expansion, however, it is necessary still to associate the terms for k = 4- n and k = — n : for each of the series for k > and for /.■ < diverges. 172. In the series for cotha- replace x by \x. Then, by (22), p. 447, - - *y 4 /.' it 4- x *r* 1 III (9) If the first series can be arranged according to powers of x, an expres- sion for B. lu will be found. Consider the identity which is derived by division and in which 6 is a proper fraction if t is positive. Substitute t = x' 2 /-i Irir' 1 ; then 4 /.'-7T- + ./•- 4 Z-V- it=l |_^=1 \ J- ' -*s ^2£-™^;s£- Let S / t; = 1 + ^ + ^+-=-v _ coth __ 1 = _ 2 2.s 2p — " V 4 7T' * The # is still a proper fraction since each 6 k is. The interchange of the order of summation is legitimate because the series would still converge if all signs were positive, since "S.k~' p is convergent. 456 THEORY OF FUNCTIONS As S. 2n approaches 1 •when n becomes infinite, the last term approaches if x < 2 7r, and the identical expansions are 2 1 SW- 1)" " ' ^37 = | *, gj = 1 «*>> 1 " !• (1°) 2 (2 />■) ! Hence ^^ty-i-LJJLs,, (11) and l COth i = 1 + S JS ^2y! + ^ jB2 »2T!' (12) The desired expression for B. ln is thus found, and it is further seen that the expansion for \ x coth \ x can be broken off at any term with an error less than the first term omitted. This did not appear from the formal work of § 170. Further it may be noted that for large values of n the numbers B. 2n are very large. It was seen in treating the T-function that (Ex. 17, p. 385) log r (n) = (n — J) log n — n + log v 2 ir + w (?i), where w (n) = J I- coth - — 1 J e"* —^ • r° . r m o , r(2» + i) 2»! As I x 2p e ax dx=l x 2p e~ nx dx = - ,*p+i the substitution of (12), and the integration gives the result w( ^ == T^ + T^r + --- + (2 y ,-3)(2 7 ,-2) + (2 7 >-l)2 7 / < 13 > For large values of w this development starts to converge very rapidly, and by taking a few terms a very good value of «(») can be obtained ; but too many terms must not be taken. Compare §§ 151, 154. EXERCISES 1. Prove cos x = ™\ 2 ' = Tl(\ *** 2sinj; \ (2k + 1)%- 2. On the assumption that the product for sinhic may be multiplied out and collected according to powers of- x, show that w 2 ^ = ^ ■ ( 5 > 2 2 ^ = ^ • if k ma y ^ ual * SPECIAL INFINITE DEVELOPMENTS 457 111 2 3. By aid of Ex. 21 (d), p. 452, show : (a) 1 + — + — 4- — + ... = — , , , 111 7T 2 , . , 1 1 1 7T 2 <« 1 + p + S + P + -=8" W l -? + ?-S + - = 5- 4. Prove, (a) /' ^ = - £, « f ' >S£*, = _^. Jo 1 — x t/o 1 + x 12 . , r 1 logx . 7r 2 . „, r 1 , 1 + x dx ir 2 (7) / — IL - 9 dx = -—,. (5 log-± = — .. Jo 1 — x- 8 Jo 1 — xx 4 5. From tan x = — cot ( x it) = — 7 — \ 2 / ^ X -(fc + |-)7T 1/ x x\ tW-1)* 1 £W-l)*2x show esc x = - ( cot - + tan - ) = > - — — = - + > 2 \ 2 2/ ±4 x - kir x ^ x 2 - fc% 2 n-l n-l 6. From = V (- »)* + (- 1)»-?— - = V (- at)* + (- l)»fcc" ~1 x a-l JL (_ \)k I show I — dx = 7 — , and compute for a = - by Ex. 21, p. 452. Jol + x ^ a + k 4 J 7. If a is a proper fraction so that 1 — a is a proper fraction, show Jo 1 + x *-ia — k J 1 1 + x Jo 1 + x sin aw 1 8. When n is large B-2 n = (— l)"- 1 4 V7r?i( — ) approximately (Ex. 13). \iref Q* T ^ — \ 2 X 9. Expand the terms of - coth - = 1 + 7 by division when x < 2 it 1 2 2^4 & 2 tt 2 + x 2 and rearrange according to powers of x. Is it easy to justify this derivation of (11) ? 10. Find «'(n) by differentiating under the sign and substituting. Hence get T'(n) = loo . n _ J_ _ A _ A Bo -p-°- _ 91i *P . r (n) 2 >i 2 n 2 4n 4 (2_p — 2) n 2 ''~ 2 2pn 2 P T'(n) />i i_ (v»-i 11. From — — + 7=1 da of § 149 show that, if n is integral, T (n) Jo 1 — a r'(«) ,11 1 , r'(l) n rnncmnnA* — LZ+ 7 = i + _-| 1 _| , and 7 = ^ = 0.5772156649 •• • r (n) 2 3 n-l r (1) by taking n = 10 and using the necessary number of terms of Ex. 10. 12. Prove log r (n + i) = n (log n— 1) + log V2 7r + w t (n), where / I \ w i (n) = J » \x ~ ~rzi/ ena; "7 ' Wi (n) = w (») - ^ ( 2 ») , o>, (n) = — ? 1 + — * (1 ) + — 6 ( 1 H . lV ' 1.2 \ 2/ 3-4 V W 5.6 26/ T 458 THEORY OF FUNCTIONS n + l\ n + 13. Show n! = V2 7T)i (- ) e n or V2 tr ( — —^) ~e '*" + ". Note that the results of § 149 are now obtained rigorously. n-l 1 ^-\ e~" r ^-\ e~ ( n— V x 14. From => e~ kx -\ — — = > e- i ' x +^ — — , and the formulas of § 140, prove the expansions W s '"»" r <»> = I 3 sin 3 x + • • • converges over an interval of length 2 7r in ,r, say =s x < 2 7r or — 7r < ./■ =§ 7r, the series will converge for all values of x and will de- fine a periodic function f(x + 2 7r) = f( x ) of period 2 7r. As J' "" /° " cos kx COS /.£ cos kx sin £/■ (x) be a 460 THEORY OF FUNCTIONS function which coincides with f(x) during the interval a < x < ft, over which the expansion of f(x) is desired, and which has any value whatsoever over the remainder of the interval (3 < x < a + 2 tt, the expansion of <£ (a;) from a to a + 2 tt will converge to /(.'') over the interval a < x < (3. In practice it is frequently desirable to restrict the interval over which f(x) is expanded to a length ir, say from to tt, and to seek an expansion in terms of sines or cosines alone. Thus suppose that in the interval < x < tt the function <£ (x) be identical with f(x), and that in the interval — tt < x < it be equal to /(— x) ; that is, the func- tion (x) is an even function, $(x) = (—x), which is equal to f(x) in the interval from to tt. Then v / + TT /~>TT r*TT cf> (x) cos kxdx = 2 J (x) cos kxdx = 2 j f (x) cos kxdx, TT Jl) J() /+ TT r>TT r>TT (x) sin kxdx — I 4> (x) sin kxdx — / <£ (,/•) sin kxdx = 0. Hence for the expansion of (x) from — tt to + tt the coefficients b k all vanish and the expansion is in terms of cosines alone. As f(x) coin- cides with (.'■) from to tt, the expansion « r> ^tt /(•'')= 2 "* f, os /.••>•, a k = ~ / f(x) cos kxdx (1,) Jo oif(x) in terms of cosines alone, and valid over the interval from to tt, has been found. In like manner the expansion f(x)=^0 k sinkx, b k = - f"f(x)sinkxdx (18) 1 J in term of sines alone may be found by taking <£(•'') equal \>o f(x) from to it and equal to — /(— .'') from to — tt. Let \x be developed into a scries of sines and into a .series of cosines valid over the interval from to w. For the series of sines 2 r*l . . , (— 1)*' x A sin fee ui- = - x sm fccax = — , = > - ttJo 2 A; 2 ±f k or J x = sin x — \ sin 2x + \ sin 3x — J sin 4x + • • • . (A) „ „ f 0. fc even 2 /" r 1 tt 2/"M Also a = — I -rdx. = -, ctjc = ■ I - x cos todx = X > — 7T. 174. For discussing the convergence of the trigonometric series as formally calculated, the sum of the first 2n + 1 terms may be written as S n -- f~ \-+ cos(t-x) + cos2(t-x) + ■■■ + cosn(t-x)]f(t)dt 7T Jo |_2 J t — X sin (2 n + 1) = - I ; ; f(t)dt = - / x ~f(x + 2u) TT Jo ~ . t — X TT J - - 2 sm - du, 462 THEORY OF FUNCTIONS where the first step was to combine a* cos kx and b k sin kx after replacing x in the definite integrals (f6) by t to avoid confusion, then summing by the formula of Ex.9, p. 30, and finally changing the variable to u = \{t — x). The sum S„ is therefore represented as a definite integral whose limit must be evaluated as n becomes infinite. Let the restriction be imposed upon/(x) that it shall be of limited variation in the interval < x < 2 tt. As the function f(x) is of limited variation, it may be regarded as the difference P(x) — N(x) of two positive limited functions which are constantly increasing and which will be continuous wherever /(x) is continu- ous (§ 1*27). If f(x) is discontinuous at x = x , it is still true that/(x) approaches a limit, which will Vie denoted by/(x — 0) when x approaches x from below ; for each of the functions P(x) and -V(x) is increasing and limited and hence each must approach a limit, and f(x) will therefore approach the difference of the limits. In like manner /(x) will approach a limit /(x + 0) as x approaches x from above. Furthermore as/(x) is of limited variation the integrals required for »S'„, a^, b k will all exist and there will be no difficulty from that source. It will now be shown that 1 r w ~~ 7t sin (2 n 4- \\ u HmS I ,(x ) = lim - f - '/(x +2u) ' . du = - [f(x + 0)-/(x - 0)]. H = ao n = » 7T >J — - Sin U 2 This will show that the series converges to the function wherever the function is con- tinuous and to the mid-point of the break wherever the function is discontinuous. _ . .. , _ , sin (2 n + 1) u u sin(2n + l)u smku Let /(x + 2u) ; '— =f(x + 2u)- — = F(u) sin u sm u u u the 1 r n ~^ _. . sin ku . 1 r' 1 _ . . sin ku , n S„(x ) = - I , " F(u) du = - I F(u) du. - tt< a < < b < ■ 7T J— -- U 7T Jn U As/(x) is of limited variation provided — tt < a == u === l> < it. so must/(x + 2 u) be of limited variation and also F(u) = »//sin u. Then F(u) may be regarded as the difference of two constantly increasing positive functions, or, if preferable, of two constantly decreasing positive functions ; and it will be sufficient to investigate the integral of F(u) u -1 sin ku under the hypothesis that F(u) is constantly de- creasing. Let n be the number of times 1ir/k is contained in b. 2t in 2nir r h „ sin/iu , rjr C k C~k C -^, . sin&u , Jo u Jo J — J -- - - J _ u k I: k = | +1 +•••+/ l-(-) -du+ I F(u) du. Jo J -2- J-2(n-l)v \k/ U J — U As F(u) is a decreasing function, so is u- l F(u/k), and hence each of the integrals which extends over a complete period 2 tt will be positive because the negative ele- ments are smaller than the corresponding positive elements. The integral from 2 mr/k to b approaches zero as k becomes infinite. Hence for large values of k, r h „. sinfcw , r-' ,7T ,,(u\ sin u , , . /• (") -du > t I '/". p iixed and less than n. Jo u J o v fr ' u SPECIAL INFINITE DEVELOPMENTS 463 sin ku Again, { F{u) du = + + Jo u Jo Jit JZtt ~(2»-l)ir /u\siiiu /**> + - + JL,>. f l)^r"" + /<---'« sin few. , au. Here all the terms except the first and last are negative because the negative ele- ments of the integrals are larger than the positive elements. Hence for k large, r b r ,, . sin ku , r i2p ~ 1):T „/i<\ sin u , _ I i< (m) du < I i* I - 1 aw, p nxea and less than n. Jo u Jo \k/ u In the inequalities thus established let k become infinite. Then u/k = from above and F(u/k) = F(+ 0). It therefore follows that r.-. r < '- /,_1)7! 'sin m , ,. r b „, , sinfru , _, „ r-P n sinu , F(+0)| — du < Inn | 1(h)— — du>F(+0)l —du. Jo a i-= x Jo u Jo u Although p is iixed, there is no limit to the size of the number at which it is fixed. Hence the inequality may be transformed into an equality lim f V(h)^«h = F{+ 0) f* *^du = ?F(+ 0). k=r, Jo u Jo u 2 Likewise lim f F(u) ^^ du = Fl- 0) C ^-^du = -F(- 0). a- = x J« u J o u 2 Hence lim f F(u) ^^ du = - [F(+ 0) + F(- 0)] a- = « J« u 2 1 /* "■ — -- sin ( 2 j? 4- I 1 ) i/. or lim - f - " /(*„ + 2 u) l . ; du = - [/(j + 0) +f(x - 0)]. n = x 7T J— — Sill H 2 Hence for every point x in the interval < x < 2 7r the series converges to the function where continuous, and to the mid-point of the break where discontinuous. As the function f(x) has the period 2 tt. it is natural to suppose that the con- vergence at x = and x = 2 ir will not differ materially from that at any other value, namely, that it will be to the value \ [/(+ 0) +/(2tt— 0)]. This may be shown by a transformation. If A- is an odd integer, 2 n + 1. sin (2 n + 1) u — sin (2 n + 1) (ir — w) = sin (2 n + 1) u', r f'rv x sin(2n + l)u .. r*~ h sin(2» + 1) u' *-,,,, , ™ lim I F(u) — — du = \im F(u ) — — d« = — -F(u = + 0). n = x> Jb U H = oo Jo It' 2 z"' 1 ' sin (2 ?i -4- ] ) h /• * /" "^ 7r Hence lim f F(w) ' ^ '-du = Ym\ / + / = - [F(+ 0) + F(tt - 0)]. n = a> Jo 7i n = =o Jo J?) 2 sin (2 71 + 1) H 1 r* Now for j = or j = 2 7r the sum S n = — I /(2 u) 7r Jo sin u will therefore be | [/(+ 0) +/(2 7r— 0)] as predicted above. The convergence may be examined more closely. In fact du, and the limit 1 C »-p t , •-> \ u sin fat 1 r bCx ^ v/ .smi S„{x) = - _ 7(j; + 2u)- du = - F(z,u) 7TJ-- S1I1H U 7rJa(x) u sin fa/. , au. 464 THEOKY OF FUNCTIONS Suppose 0 sinu , , r h(r) -^, , sinfcu , _, „. r x sin?/ , F(x, + 0) I dit + e' < / F(x, u) du < F(x, + 0) / — dw + V ^o w Jo w Jo it where, if 5 > is given. K may be taken so large that |e'| < 5 and |?/| < 5 for A - > K ; with a similar relation for the integration from a(x) to 0. Hence in an}- interval 0(— n + l-Aa)Aa + \- (n- Aa)Aa + • • •] = lim ^? $ (k ■ Aa) Aa = lim V| — 3 r i 5 r 3 P — 3 1 "> r 2 4- 8 "> r-4 P — 15 r 3 5 P 3 _i_ 6 3 r 5 L 3 — 2 ~ 2 ' 4 — 8 I ' 8 t ~> — 8 ¥" ~ ~ 8 * r l 2 / 6 \ 2 Compute I x' shi7rxcZx = 0, — ( 1 . 0, - .0 when i = 4, 3. 2, 1. 0. Hence show J -l 7T \ it' 1 I' TV that the polynomial of the fourth degree which best represents. sin ttx from — 1 to + 1 reduces to degree three, and is sin 7TX = - x - - ( — = l\ ('- x s - - x | = 2.69x - 2.89x 3 . TV \TT- Show that the mean square error is 0.004 and compare with that due to Maclaurin's expansion if the term in x 4 is retained or if the term in x 3 is retained. n T , . . 1 12 „ 168 /]0 \ _ „ Fro r ^. , „ 11. Expand sm - ttx = P. — - - — 1 P 3 = 1.553,/- — 0.562x 3 . 12. Expand from — 1. to 4-1, as far as indicated, these functions : (a) cos7rx toP 4 , (fi) i- r toP 5 , (7) log(l-fx) to P 4 , ( 5 ) VI - x- toP 4 , (e) cos-ix to P 4 , (f) tan-'x to P 5 , (,) —1- to P 3 , (0) -L^ to P 3 , (,) 1 to P 3 . \ 1 + x VI- x 2 Vl + x 2 "What simplifications occur if /(x) is odd or if it is even ? SPECIAL INFINITE DEVELOPMENTS 467 175. The Theta functions. It has been seen that a function with the period 2 ir may be expanded into a trigonometric series ; that if the function is odd, the series contains only sines ; and if, furthermore, the function is symmetric with respect to x = \ ir, the odd multiples of the angle will alone occur. In this case let f(x) = 2 [a sin x — a x sin 3 x -\ + (— 1)" a n sin (2 n + 1) x -\ ]. As 2 sin nx = — i (e* 3 * — e~ nxi ), the series may be written f(x) = 2^ (- 1)X sin (2n + l)x = -ij£(- lfa^ 7i+l)3 a This exponential form is very convenient for many purposes. Let lp be added to x. The general term of the series is then a g (2«-l)(.T+ ip)i __ (( „- (2« _l)p _ 2xi e (2n + l)xi Hence if the coefficients of the series satisfy a n _ 1 e~ 2np = a n , the new general term is identical with the succeeding term in the given series multiplied by — e p e~ 2xi . Hence f(x + lp) = - e p e~ 2xi f(x) if a n _ x = a n e* n ». The recurrent relation between the coefficients will determine them in terms of a Q . For let q = e~ p . Then il n = a n-\ ( f' 1 = a n-2 ( f n< f U ~ 2 = ' ' * == « ^ 2 " -/(H). *— " ? - < 19 > ff 00 = 2 qi sin ^ - 2 2 « sin ^ + 2 ,/V sin ^ - -. (20) The function //(«), called eta of ?/, has therefore the properties //(« + 2 A") = - #(m), H(u + 2 iK') = - q~ l ^ KU H{ti), (21) ■ #(w + 2 wiA~ + 2 inK') - (— 1)'" + "7- "e - *" "//((/), t», n integers. The quantities 2 K and 2 i/v' are called the periods of the function. They are not true periods in the sense that 2 it is a period (A fix) ; for when 2 A" is added to it, the function does not return to its original value, but is changed in sign ; and when 2 iK' is added to a, the function takes the multiplier written above. Three new functions will be formed by adding to u the quantity A' or iK' or K -+- iK', that is, the half periods, and making slight changes suggested by the results. First let H 1 (u) = H(u + K). By substitution in the series (20), ~ 1 7T// , a 3 7T» _ 25 5 7T// # x (m) = 2 y* cos — + 2 7 * cos — + 2 '/ 4 ™s — - + • • ■ . (22) By using the properties of II, corresponding properties of H , // a (a + 2 K) = - II X («), //, (u + 2 /A") = + y - V" A ' "/^ («), (23) are found. Second let iK' be added to u in II (u). Then |(2/«+l)'- (2h+1)7j.(m + i'A'0 »- + »+' -w(n+j)-p' (2;i + 1)--'-m 7 « - A =7 e " e - is the general term in the exponential development of II (it + iK') apart from the coefficient ± i. Hence //(> + iK') = ;^(-i)" 7 ""%~- h '",> 2 "- K " SPECIAL INFINITE DEVELOPMENTS 469 . t*jr co „ iri Let ®(m) = - i( 1 e TRU H{u + iA") = ^ (- l)"//"" • The development of ©(«) and further properties are evidently 0(.) = 1- 2 2 cos |^ + 2,/ cos |^-2,/ cos |^ + ---, (24) ®(u + 2K) = ®(u), ®(u + 2iK') = -q- 1 e~^ w ®(u). (25) Finally instead of adding K + iA' to u in H (u), add A' in (?<)• / n - « 2 7TM n , 4 7T?< „ Q G 7T» ,„_ N ©^h) = 1 + 2 gr cos — + 2 ,/ cos — + 2 ? » cos — + • • • , (26) ©^ + 2 A) = ©jO), ©> + 2 iA') = + 2 - *e-£" ®». (27) For a tabulation of properties of the four functions see Ex. 1 below. 176. As -ff(w) vanishes for n = and is reproduced except for a finite multiplier when 2 mK + 2 nlK' is added to w, the table H (u) = for u = 2 mK + 2 mA', ^(tt) = for u = (2 to + 1) A + 2 m'A', ® (u) = for m = 2 wiA + (2 n + 1) iA', ©^m) = for u = (2 m + 1) A + (2 n + 1) iK', contains the known vanishing points of the four functions. Now it is possible to form infinite products which vanish for these values. From such products it may be seen that the functions have no other vanish- ing points. Moreover the products themselves are useful. It will be most convenient to use the function ®.(w). Now -== (2 mK + K + 2 » iK' + iK') in „ _l i \ ^ ^ «A = — qV»+V } — x < 71 < X . Hence e% u -f q-^+v and e"^" + y- (2 " +1) , n S 0, are two expressions of which the second vanishes for all the roots of ® x (") for which n i= 0, and the first for all roots with n < 0. Hence TT = C TT (l + q 2n+1 e^ u ) (l + q 2 ' l + l e~ lJ ^) is an infinite product which vanishes for all the roots of ®j(m)- ^ ne product is readily seen to converge absolutely and uniformly. In par- ticular it does not diverge to and consequently has no other roots than those of © a (») above given. It remains to show that the product is identical with ® x (^) with a proper determination of C. 470 THEORY OF FUNCTIONS in* Let G 1 (u) be written in exponential form as follows, with z = e K : 0(2) = e^u) = 1 + q (z + ^j + q* (z* + i\ + • • • + Q" 2 U + £\ + • • • , f (z) = C-iTT(M) = (l + ?z)(l + g 3 z)(l + ? 5 z)- • -(l + 9 2 »-iz). • • *H)K)K)"K-^)-- A direct substitution will show that

    ■<"). (28) SPECIAL INFINITE DEVELOPMENTS 471 H x (u) = C2 q i cos ~ TT (l + 2 enn = -v- • dn u . = Va; — ! (3o) V/.- ©0') N'/.' ©00 ®(u) The functions sn u, en ?/, dn ;/ are called elliptic functions* of u. As 7/ is the only odd theta function, sn u is odd but en u and dn u are even. J// three functions hare t/co actual ijerlods in the same sense that sin./' and cus x have the period 2 ir. Thus dn // has the periods 2 7v" and 4 t*A'' by (25), (27); and sn u has the periods 4 7v and 2 IK' by (25), (21). That en /' has 4 7v* and 2 K -\- 2 IK' as periods is also easily verified. The values of a which make the functions vanish are known: they are those which make the numerators vanish. In like manner the values of u for which the three functions become infinite are the known roots of ©(«). If q is known, the values of VA and ~\k' may be found from their definitions. Conversely the expression for \k', V/ ' - 0^0) - 1 + 2, + V + LV+... ' ( } * The study of the elliptic functions is continued in Chapter XIX. 472 THEORY OF FUNCTIONS is readily solved for q by reversion. If powers of q higher than the first are neglected, the approximate value of q is found by solution, as 1 1 _ VP q + g 9 + ■ • 2 1+ Vk> 1-2,/ + q - 2 q 5 + 5 rf + 1 1 - a /•' 2 /l - V^V 15 (1 - Va-'V Hence o- = = + — - — — I + — ( — I + ■ • ■ (6 < ) is the series for q. For values of A-' near 1 this series converges with great rapidity; in fact if k 1 ' 2 ^ ^, k' > 0.7, VP > 0.82, the second term of the expansion amounts to less than 1/10 6 and may be disregarded in work involving four or five figures. The first two terms here given are sufficient for eleven figures. Let # denote any one of the four theta series //, H , 0, © r Then *(«) = *(*) = 5^, « = «-»" (38) may be taken as the form of development of >'/' 2 : this is merely the Fourier series for a function with period 2 K. But all the theta func- tions take the same multiplier, except for sign, when 2 iK' is added to u; hence the squares of the functions take the same multiplier, and in par- ticular (q 2 z) — q~ 2 z~ 2 (z). Apply this relation. It then is seen that a recurrent relation between the coefficients is found which will determine all the even coefficients in terms of b Q and all the odd in terms of b . Hence ,(z). Moreover <£ and ^ are iden- tical for all such functions, and the only difference is in the values of the constants b Q , b r As any three theta functions satisfy (38') with different values of the constants, the functions <5 and ^ may be eliminated and at 2 (it) + p&%(it) + y&i(u) = 0, where a, /3, y are constants. In words, the squares of any three theta functions satisfy a linear homogeneous equation with constant coeffi- cients. The constants may be determined by assigning particular values to the argument it. For example, take //. H , 0. Then* *For brevity the parenthesis ahout the arguments of a function will frequently be omitted. SPECIAL INFINITE DEVELOPMENTS 473 aH-(u) + /3H? (u) = y®' 2 (ft), f3H{0 = y©*0, aH 2 K = y® 2 K, e-K H-(ii) 0-0 HHu) . FT('fz) = q~ 2 z~ 2 C(z) if log 6' = iira/K. Reasoning like that used for treating # 2 then shows that between any three such expressions there is a linear relation. Hence aH(u)H(u - a) -f /Sfl, (?<) ^ (« - ft) = y© («)©(« - a), u = 0, pll x (0) #, (a) = y© (0) © (a), v = K, «H X (0) II x (a) = y © 1 (0) 0, (ft), ®0® 1 0® 1 aH(u)H(u — a) Q 2 H^itjII^u - a) _ ©0 //^/ // 2 O®ft0(ft)0(> - ft) 77fO ©(ft) ©(ft - ft) ~ HjO ©ft ' or dn a sn ft sn (ft — a) + en ft en (ft — ft) = en ft. ' (41) In this relation replace ft by — r. Then there results en u en (ft + r) + sn ft dn v sn (ft -+- /■) = en v, or en v en (« + v) + sn r dn u sn (ft + /•) = en u, en 2 ft — en 2 v = sn 2 v — sn 2 ft and sn (ft + v) = — » (42) sn v en ft dn ft — sn a en r dn r by symmetry and by solution. The fraction may be reduced by multiply- ing numerator and denominator by the denominator with the middle sign changed, and by noting that sir v en 2 ft dn 2 it — sn 2 a en 2 v dn 2 v — (sir v — sir ft) ( 1 — A' 2 sir ft sn 2 v). sn // en r dn v 4- sn r en ft dn ft ..„, Then sn(ft + v) = - — — ^ - a , (43) 1 — Ir sir ft sir v , . s sn ft en r dn c — sn v en a dn ft and sn(ft — r) = — : : > 1 — A- sir ft sir r ., . 2sn /• en ft dn u .... and sn(ft + v) — sn(u — v) = - -, — s 5—- (44) 1 — A-sn- ft. sir v The last result may be used to differentiate sn ft. For sn(ft + Aft) — sn a _ sn i A^ en (ft + \ Aft)dn(ft + \ Aft) Aft 1 Aft 1 — Ir sir \ Aft sir(ft + \ Aft) ' d . sn ft — sn ft = r/ en ft dn ft, a = lun • (±0) du J J u ± » 474 THEORY OF FUNCTIONS Here g is called the multiplier. By definition of sn u and by (33) ^(0)^(0) _J[_ 9 ~ H x {0) 0(0) ~2if 0l(O) - (4o) The periods 2 K, 2iK' have been independent up to this point. It will, however, be a convenience to have g = 1 and thus simplify the formula for differentiating sn a. Hence let g = l, A j^ = 1 (O) = l + 2y + 2 7 4 + ---. (46) Now of the five quantities K, K', k, k', q only one is independent. If q is known, then k' and K may be computed by (36), (46) ; k is de- termined by k* + k n = 1, and K' by ttK'/K = - log q of (19). If, on the other hand, k' is given, q may be computed by (37) and then the other quantities may be determined as before. EXERCISES ?V iir _ 1 « u 1. With the notations X = q i e, 2K ,fi = q~ 1 e K establish: H (- u) = - H(u), II (u + 2 A) = - II (u), II (u + 2 IK') = - (nH(u), II \ (- u) = + II x (it), //j (u + 2 A) = - /^ (it), If, (u + 2 IK') = + (uffj (u), e(-«) = + e(«), e(w + 2A') = + e(w), e(« + 2<7r) = - At e(«), e 1 (-M) = + e 1 (u), e 1 (« + 2A') = +e 1 ( M ), e 1 (u + 2iK'') = + M e 1 (u), //(u + A) = + Hj («), BT(m + IK') = l\e (u), II (u + K + IK') = + X9 1 (it), II x (u + K) = — H(u), II \ (u + IK') = + X9 1 (u), II \ (u + K + IK') = — i\0 (u), e (m + A) = + e x (u), e (u + ik') = ;\// («), e (a + k + ik') - + \n 1 («), Gj (it + K) = + G (u), Bj (u + t A"') = + X/7j (it), Gj (u + K + iJT) = + iX# (m). 2. Show that if w is real and (/ =s £, the first two trigonometric; terms in the series for //, II V G, G p give four-place accuracy. Show that with q s= 0.1 these terms give about six-place accuracy. 3. Use — = q sin a + q 2 sin 2 rr + ty 3 sin 3 a + • • • to prove (. TTH , . 2 7T« „ . 3 iru \ (i sin — (/-sin- i/' 1 sin — \ K K K 1 - g 2 1 - ( / 4 1 - r/« / 4. Prove the double periodicity of en u and show that : en u , . „,, 1 . r „,, dn u sn (» + A ) = , sn (u + %K ) — — — , sn (;/ + 7\ + / A ) = -■■ — , dn a k sn u A' en u — k' sn u , .,,„ — idnu , ,, ..... — ik' cn(it+A)=: - - , en (u + iK ) = — — , cn(u + A + zA ) = , dn u k sn u A' en « n / ,- &' ! , .^^,v . cnu , . ,, . ,,„ ...sum dn (« + A ) = , dn (» + *7v ') = — z , dn (u + A + tA ) = ik dn u sn u en u SPECIAL INFINITE DEVELOPMENTS 475 5. Tabulate the values of sn u, en u, dn u at 0, A", iK', K + IK'. 6. Compute A - ' and A; 2 for ^ = J- and g = 0.1. Hence show that two trigonometric terms in the theta series give four-place accuracy if k' == \. „ T , , en w en v — sn w sn v dn u dn v 7. Trove en (m + v) = — and dn(w + r) = 1 — k' 2 sn 2 u sn 2 v dn it dn v — A: 2 sn w sn v en w en v 1 — k' 2 sn 2 w sn 2 v 8. Trove — cnu=— snwdnw, — dn u =— k 2 snu en u, = 1. d(i du 9. Trove sn~ a M = / — — from (45) with g = 1. •/o V(l - m 2 ) (1 - & 2 u 2 ) 10. If r/ = 1, compute k, k', K, K', for q = 0.1 and q = 0.01. 11. If g = 1, compute A:', g, A~, A'', for A; 2 = i, |, ^. 12. In Exs. 10, 11 write the trigonometric expressions which give sn w, en u, dnu with four-place accuracy. 13. Find sn 2 w, en 2 «, dn 2 ?/, and hence sn \ u, en * u, dn -'■ w, and show sn |- A" = (1 + k')~ 2, en £ A' = Vk' (1 + k')~ i, dn | A = Vk'. 14. Trove — k ( sn u dn = log (dn u + A: en u) ; also e 2 (0)//(« + a)H(u- a) = e 2 (it)ii 2 (u)- 7/ 2 («)e 2 (w), e 2 (0) e (u + a) e (« - «) = e 2 («) e 2 (a) - J/ 2 («) m (a) . CHAPTER XVIII FUNCTIONS OF A COMPLEX VARIABLE 178. General theorems. The complex function u (x, y) + lr (x, ?/), where u (x, //) and v (x, //) are single valued real functions continuous and differentiable partially with respect to x and y, has been defined as a function of the complex variable z = x + iy when and only when the relations v x = v' y and u' y = — v' x are satisfied (§73). In this case the function has a derivative with respect to z which is independent of the way in which Az approaches the limit zero. Let to = f(z) be a function of a complex variable. Owing to the existence of the deriva- tive the function is necessarily continuous, that is, if c is an arbitrarily small positive number, a number 8 may be found so small that |/(*)-/(*.)l<« when |*-*ol<«» W and moreover this relation holds uniformly for all points z of the region over which the function is defined, provided the region includes its bounding curve (see Ex. 3, p. 92). It is further assumed that the derivatives u x , u' u , v' x , v' y are continuous and that therefore the derivative /'('-) is continuous.* The function is then said to be an analytic function (§ 12G). All the functions of a complex variable here to be dealt with are analytic in general, although they may be allowed to fail of being analytic at certain specified points called singular points. The adjective "analytic" may therefore usually be omitted. The equations w—f{z) or i( = n(a;y), r = r(.r,y) define a transformation of the .Ty-plane into the ///--plane, or, briefer, of the .v-plane into the //--plane; to each point of the former corresponds one and only one point of the latter (§ 63). If the Jacobian = ocf + (o a =iA*)r (2) * It may be proved that, in the ease of functions of a complex variable, the continuity of the derivative follows from its existence, but the proof will not be given here. 470 COMPLEX VARIABLE 477 of the transformation does not vanish at a point z , the equations may be solved in the neighborhood of that poinb, and hence to each point of the second plane corresponds only one of the first : x = x(u, v), y = y(u, v) or z = (iv). Therefore it is seen that if w = f(z) is analytic in the neighborhood ofz = z , and if the derivative f (z^ does not vanish, the function may be solved as z = cf>(ic), where is the inverse function of /, and is like- wise analytic in the neighborhood of the point w = w Q . It may readily be shown that, as in the case of real functions, the derivatives f\z) and 4>'( w ) are reciprocals. Moreover, it may be seen that the transforma- tion is conformal, that is, that the angle between any two curves is unchanged by the transformation (§ 63). For consider the increments *» = [/' (*o) + C] *« = f (*o) [1 + C/f (^o)] A*- /' (*o) * 0. As Az and Aiv are the chords of the curves before and after transforma- tion, the geometrical interpretation of the equation, apart from the infin- itesimal £, is that the chords Az are magnified in the ratio |/'(.v )| to 1 and turned through the angle of f'(z ) to obtain the chords Air (§ 72). In the limit it follows that the tangents to the ^--curves are inclined at an angle equal to the angle of the corresponding .^-curves plus the angle of f'(z ). The angle between two curves is therefore unchanged. The existence of an inverse function and of the geometric interpre- tation of the transformation as conformal both become illusory at points for which the derivative f(z) vanishes. Points where f'(z) = are called critical points of the function (§ 183). It has further been seen that the integral of a function which is ana- lytic over any simply connected region is independent of the path and is zero around any closed path (§ 124) ; if the region be not simply con- nected but the function is analytic, the integral about any closed path which may be shrunk to nothing is zero and the integrals about any two closed paths which may be shrunk into each other are equal (§ 125). Furthermore Cauchy's result that the value /«-»W^* (3> of a function, which is analytic upon and within a closed path, may be found by integration around the path has been derived (§ 126). By a transformation the Taylor development of the function has been found whether in the finite form with a remainder (§ 126) or as an infinite series (§ 167). It has also been seen that any infinite power series 478 THEORY OF FUNCTIONS which converges is differentiable and hence defines an analytic function within its circle of convergence (§ 166). It has also been shown that the sum, difference, product, and quotient of any two functions will be analytic for all points at which both func- tions are analytic, except at the points at which the denominator, in the case of a quotient, may vanish (Ex. 9, p. 163). ■ The result is evidently extensible to the case of any rational function of any number of analytic functions. From the possibility of development in series follows that if two functions are analytic in the neighborhood of a point and have identical rabies upon any curve drawn through that point, or even upon any set of points which approach that point as a limit, then the functions are identically equal within their common circle of convergence and over all regions which can be readied by (§ 169) continuing the functions analyti- cally. The reason is that a set of points converging to a limiting point is all that is needed to prove that two power series are identical pro- vided they have identical values over the set of points (Ex. 9, p. 439). This theorem is of great importance because it shows that if a function is defined for a dense set of real values, any one extension of the defi- nition, which yields a function that is analytic for those values and for complex values in their vicinity, must be equivalent to any other such extension. It is also useful in discussing the principle of permanence of form : for if the two sides of an equation are identical for a set of values which possess a point of condensation, say. for all real rational values in a given interval, and if each side is an analytic function, then the equation must be true for all values which may be reached by ana- lytic continuation. For example, the equation sin z = cos(| n — s) is known to hold for the values = x ==j \ ir. Moreover the functions sin z and cos z arc analytic for all values of z whether the definition be given as in >; 74 or whether the functions be considered as defined by their power series. Hence the equation must hold for all real or complex values of /. In like manner from the equation c r O' = c x + " which holds for real rational exponents, the equation e-t"' = e- r + "' holding for all real and im- aginary exponents may be deduced. For if y be given any rational value, the functions of x on each side of the sign are analytic for all values of ./' real or com- plex, as may be seen most easily by considering the exponential as defined by its power series. Hence the equation holds when x has any complex value. Next consider x as fixed at any desired complex value and let the two sides be con- sidered as functions of y regarded as complex. It follows that the equation must hold for any value of y. The equation is therefore true for any value of z and w. 179. Suppose that a function is analytic in all points of a region ex- cept at some one point within the region, and let it be assumed that COMPLEX VAEIABLE 479 the function ceases to be analytic at that point because it ceases to be continuous. The discontinuity may be either finite or infinite. In case the discontinuity is finite let [ f(z) | < G in the neighborhood of the point z = a of discontinuity. Cut the point out with a small circle and apply Cauchy's Integral to a ring surrounding the point. The integral is appli- cable because at all points on and within the ring the function is analytic. If the small circle be replaced by a smaller circle into which it may be shrunk, the value of the integral will not be changed. /(*) 1 2 17b m^m 1 2 Xow the integral about y,- which is constant can be made as small as desired by taking the circle small enough; for | f /"(£)|< G and \t — z\ > \a — z\ — Tj, where r { is the radius of the circle y,- and hence the integral is less than 2 7rr { G/[\z — a\— /•,•]. As the integral is con- stant, it must therefore be and may be omitted. The remaining inte- gral about C, however, defines a function which is analytic at z = a. Hence if f{«) be chosen as defined by this integral instead of the original definition, the discontinuity disappears. Finite discontinuities may therefore he considered as due to had judgment in defining a function at some point; and may therefore be disregarded. In the case of infinite discontinuities, the function may either heroine infinite for all methods of approach to the point of discontinuity, or it may heroine infinite for some methods of approach and remain finite for other methods. In the first case the function is said to have & pole at the point z = a of discontinuity; in the second case it is said to have an essential singularity. In the case of a pole consider the reciprocal function F(z) = m z =£ a, F(a) = 0. The function F(z~) is analytic at all points near z = a and remains finite, in fact approaches 0, as z approaches a. As F(«) = 0, it is seen that /•'(■-) has no finite discontinuity at z = a and is analytic also at z = a. Hence the Taylor expansion F(z) = a m (z - «)'" + a m+1 (z - «)" l+1 + • • ■ is proper. If E denotes a function neither zero nor infinite at z = a, the following transformations may be made. 480 THEORY OF FUNCTIONS F(z) = (z - aTE x (z), f(z) = (z- a)-E 2 (z), c_„ c_. +1 . . c m = 7r^ + ,~ "::;:-! +••• + (« — a) ra (s — a.) r "-* 2 — a + C o+ C x (z-a)+ C a (*-a) a +.... In other words, a function which has a pole at 3 = a may he written as the product of some power (z — a)~ m by an ^-function; and as the /^-function may be expanded, the function may be expanded into a power series which contains a certain number of negative powers of (z — a). The order m of the highest negative power is called the order of the pole. Compare Ex. 5, p. 449. If the function f(z) be integrated around a closed curve lying within the circle of convergence of the series C Q + C x (z — <') + ■■■, then r re jiz r c Az Jo or ff(z)dz = 2iriC_ 1 ; (4) Jo for the first m — 1 terms may be integrated and vanish, the term C_ x l(z — a) leads to the logarithm C_ x log (z — ' a) which is multiple valued and takes on the increment 2iriC_ 1 , and the last term vanishes because it is the integral of an analytic function. The total value of the integral of f(z) about a small circuit surrounding a pole is there- fore 2 7r!.C '_!• The value of the integral about any larger circuit within which the function is analytic except at z = <> and which may be shrunk into the small circuit, will also be the same quantity. The coefficient C_ 1 of the term (z — a)^ 1 is called the residue of the pole ; it cannot vanish if the pole is of the first order, but may if the pole is of higher order. The discussion of the behavior of a function f(z) when z becomes infinite may be carried on by making a, transformation. Let To large values of z correspond small values of z' ; if f(z) is analytic for all large values of z, then F(z') will be analytic for values of z' near the origin. At z' = the function F(z') may not be defined by (5) ; but if F(z') remains finite for small values of z', a definition may be given so that it is analytic also at ,-.■' = 0. In this case F(0) is said to be the COMPLEX VARIABLE 481 value of f(z) when z is infinite and the notation /(cc) = F(0) may be used. If F(z') does not remain finite but has a pole at z' = 0, then f(z) is said to have a pole of the same order at z = cc; and if F(z') has an essential singularity at z' = 0, then f(z) is said to have an essen- tial singularity at z = oo. Clearly if f(z) has a pole at z — cc, the value of f(z) must become indefinitely great no matter how z becomes infi- nite; but if f(z) has an essential singularity at z = cc, there will be some ways in which z may become infinite so that f(z) remains finite, while there are other ways so that f(z) becomes infinite. Strictly speaking there is no point of the g-plane which corresponds to z' = 0. Nevertheless it is convenient to speak as if there were such a point, to call it the point , (y) z/Q + z), (5) z/(z* + 1). 12. Show that if f(z) = (z - a)*E(z) with - 1 < k < 0. the integral of f(z) about an infinitesimal contour surrounding rem holds for an infinite contour ? = a is infinitesimal. What analogous theo- 180. Characterization of some functions. The study of the limita- tions which arc put upon a function when certain of its properties are known is important. For example, a function which is ana Lytic for all values of z including also z = oo is a constant. To show this, note that as the function nowhere becomes infinite, |/(V)| < G. Consider the dif- ference /(*-'„) — /(0) between the value at any point z = z Q and at the origin. Take a circle concentric with z = and of radius 11 > \z \. Then by Cauchy's Integral _1 2 iri ./K)- /(0) Jot-Zo Jo f - () 2 ™Jo f(t)dt Kt-*d \z I 2irRG G R - k, By taking R large enough the difference, which is constant, may be made as small as desired and hence must be zero; hence ,/'('-') = f(0). COMPLEX VARIABLE 483 Any rational function f(z) = P(z)/Q(z), where P(z) and Q(z) are polynomials in z and may be assumed to be devoid of common factors, can have as singularities merely poles. There will be a pole at each point at which the denominator vanishes; and if the degree of the numerator exceeds that of the denominator, there will be a pole at in- finity of order equal to the difference of those degrees. Conversely it may be shown that any function which has no other singularity than a pole of the mth order at infinity must be a polynomial of the with order ; that if the only singularities are a finite number of poles, whether at in- finity or at other jioints, the function is a rational function; and finally that the knowledge of the zeros and poles with the multiplicity or order of each is sufficient to determine the function exceptt for a constant multiplier. For, in the first place, if f(z) is analytic except for a pole of the mth order at infinity, the function may be expanded as f{z) = a- m z m + ■■■ + a-iz + a + c^ar 1 + (',z~ 2 + • • • , or f(z) - [a- m z m + ■■■ + a-iz] = a + a y z~^ + a. 2 z-° + ■■■. The function on the right is analytic at infinity, and so must its equal on the left be. The function on the left is the difference of a function which is analytic for all finite values of z and a polynomial which is also analytic for finite values. Hence the function on the left or its equal on the right is analytic for all values of z including z = x, and is a constant, namely « . Hence f(z) = a + (z) is a constant; if f(z) has a pole at z = oo, then <£ (z) is a polynomial in z and all of the polynomial except the constant term is the principal part of the pole at infinity. Hence if a function has no singularities except a finite number of poles, and the 'principal parts at these poles are known, the function is determined except for an additive constant. From the above considerations it appears that if a function has no other singularities than a finite number of poles, the function is ra- tional; and that, moreover, the function is determined in factored form, except for a constant multiplier, when the positions and orders of the finite poles and zeros are known ; or is determined, except for an addi- tive constant, in a development into partial fractions if the positions and principal parts of the poles arc known. All single valued functions other than rational functions must therefore; have either an infinite number of poles or some essential singularities. 181. The exponential function e z = r 7 '(eos y + i sin y) has no finite singularities and its singularity at infinity is necessarily essential. The function is periodic (§ 74) with the period 2 iri, and hence will take on all the different values which it can have, if z, instead of being allowed all values, is restricted to have its pure imagi- nary part ij between two limits y Q =§ y < y + 2 ir ; that is, to consider the values of e z it is merely ~ m \ necessary to consider the values in a strip of the s-plane parallel to the axis of reals and of breadth 2ir (but lacking one edge). For convenience the strip may be taken immediately above the axis of reals. The function c z becomes infinite as z moves out toward the right, and zero as z moves out toward the left in the strip. If c = a -f- hi is any number other than 0, there is one and only one point in the strip at which e z = c. For a . b z + 2ni r~r, 77, -, . ■ a V« + and cos y + i sin y = - - — \- i Va 2 + (r -y/ m =£ and will become infinite if b m = 0. When z moves off to the right, it (e z ) must become infinite if n > m, approach C if n = m, and approach if n < m. The denomi- nator may be factored into terms of the form (e z — a) k , and if the frac- tion is in its lowest terms each such factor will represent a pole of the /I'th order in the strip because e z — a = has just one simple root in the strip. Conversely it may be shown that: Any function f(z) which has the period 2 iri, which further has no singularities but a finite number of poles in each strip, and which either becomes infinite or ap- proaches a finite limit as z moves off to the right or to the left, must be f(z) = li(e z }, a rational function of e z . The proof of this theorem requires several steps. Let it first be assumed that/(z) remains finite at the ends of the strip and has no poles. Then f(z) is finite over all values of z, including z = co, and must be merely constant. Next let f(z) remain finite at the ends of the strip but let it have poles at some points in the strip. It will be shown that a rational function R (e z ) may be constructed such that f(z) — K (e z ) remains finite all over the strip, including the portions at infinity, and that there- fore /(z) = R(e z ) + C. For let the principal part of f(z) at any pole z = c be -r, , ^ C -k C-k+1 C_1 , C_l-e l ' C C-k P(z — c) = 1 ~ + - 1 + ; then = * f- • • ■ v (z - c)* (z - c)^- 1 z-c (e z - e c )* (z - c)* is a rational function of c z which remains finite at both ends of the strip and is such that the difference between it and P(z — c) or f(z) has a pole of not more than the (A; — l)st order at z = c. By subtracting a number of such terms from f(z) the pole at z = c may be eliminated without introducing any new pole. Thus all the poles may be eliminated, and the result is proved. Next consider the case where /(z) becomes infinite at one or at both ends of the strip. If f(z) happens to approach at one end, consider /(z) + C, which cannot approach at either end of the strip. Now if f(z) or f(z) + C, as the case may be, had an infinite number of zeros in the strip, these zeros would be confined within finite limits and would have a point of condensation and the function would vanish identically. It must therefore be that the function has only a finite number of zeros; its reciprocal will therefore have only a finite number of poles in the strip and will remain finite at the ends of the strips. Hence the reciprocal and conse- quently the function itself is a rational function of e z . The theorem is completely demonstrated. If the relation f(z -f- w) = f(z) is satisfied by a function, the func- tion is said to have the period w. The function f(2 iriz/oi) will then have the period 2 iri. Hence it follows that if f(z) has the period w, becomes infinite or remains finite at the ends of a strip of rector breadth 486 THEORY OF FUNCTIONS to, and has no singularities but a finite number of poles in the strip, the function is a rational function of i" n,z / uj . In particular if the period is 2 it, the function is rational in e' z , as is the case with sins and cos z; and if the period is 7r, the function is rational in e iz/ ", as is tan z. It thus appears that the single valued elemen- tary functions, namely, rational functions, and rational functions of the exponential or trigonometric functions, have simple general properties which are characteristic of these classes of functions. 182. Suppose a function /(.~) has two independent periods so that /(* + «,)=/(?), /(» + <■>') = /(*)• The function then has the same value at z and at any point of the form z + wico + nw ! , where m and n are positive or negative integers. The function takes on all the values of which it is capable in a parallel- ogram constructed on the vectors w and w'. Such , a function is called doubly periodic. As the values of the function are the same on opposite sides of the parallelogram, only two sides and the one in- cluded vertex are supposed to belong to the figure. It lias been seen that some doubly periodic func- tions exist (§ 177); but without reference to these special functions many important theorems concerning doubly periodic, functions may be proved, subject to a subsequent demonstration that the functions do exist. If a doubly periodic function has rm singularities la the parallelogram, if must be constant ; for the function will then have no singularities at all. If two periodic functions have the same periods end have the same poles anil zeros (each to the same order) In tin' jiarallelogram, the niio- tient of tit e functions is n constant; If they have the smite poles anil the same principal parts at the jjoles, their difference is a constant. In these theorems (and all those following) it is assumed that the functions have no essential singularity in the parallelogram. The proof of the theorems is left to the reader. If f(z) is doubly periodic, /*'(.*) is also doubly periodic. The integral of a doubly periodic function taken around any parallelogram equal and parallel to the parallelogram of periods is zero: for the function repeats itself on opposite sides of the figure while the differential dz changes sign. Hence in particular )dz = d, iz = 0, 'O f'(z)dz /v-; c = 0. COMPLEX VARIABLE 487 The first integral shows that the sum of the residues of the poles in the parallelogram is zero ; the second, that the number of zeros is equal to the number of poles provided multiplicities are taken into account; the third, that the number of zeros of f(z) — C is the same as the number of zeros or poles off(z), because the poles of f(z) and/(s) — C are the same. The common number m of poles of f(z) or of zeros of f(z) or of roots of /(.~) = C in any one parallelogram is called the order of the doubly periodic function. As the sum of the residues vanishes, it is impossible that there should be a single pole of the first order in the parallelogram. Hence there can be no functions of the first order and the simplest possible functions would be of the second order with the expansions : ~-r, + c + cfz -c)-\ or h c -\ and \- c' -\ (z — ay ° * z — <' l u z — a 2 in the neighborhood of a single pole at z = a of the second order or of the two poles of the first order at z — a 1 and z = a. y . Let it be assumed that when the periods w, w' are given, a doubly periodic function g (z, a) with these periods and with a double pole at z = a exists, and similarly that // (z, a } «.,) with simple poles at a x and a, 2 exists. Any doubly periodic function f(z) with the periods w, w' may be ex- pressed as a polynomial in the functions g (z, o) and h(z, a , a.,) of the second order. For in the first place if the function f(z) has a pole of even order 2h at z = a, then f(z) — C[g(z, a)f, where C is properly chosen, will have a pole of order less than 2 k at z = a and will have no other poles than /(.~). Hence the order of f(z) — C[g(z, «•)]* is less than that of /(.-). And if f(z) has a pole of odd order 2 k + 1 at z = a, the function f(z) — C[g(z, a)y'h(z, a, b), with the proper choice of C, will have a pole of order 2 k or less at ,~ = a and will gain a simple pole at z = b. Thus although / — Cg k h will generally not be of lower order than /, it will have a complex pole of odd order split into a pole of even order and a pole of the first order ; the order of the former may be reduced as before and pairs of the latter may be removed. By repeated applications of the process a function may be obtained which has no poles and must be constant. The theorem is therefore proved. With the aid of series it is possible to write down some doubly peri- odic functions. In particular consider the series i>(")= h+X and *■<*)= -2£ JTZ^, (z — hum — nw')' 2 (inw -\- nwj 1 (6) 488 THEORY OF FUNCTIONS where the second 2 denotes summation extended over all values of m, n, whether positive or negative or zero, and 2' denotes summation extended over all these values except the pair m — n = 0. As the sum- mations extend over all possible values for m, ?i, the series constructed for z + w and lor z + «/ must have the same terms as those for z, the only difference being a different arrangement of the terms. If, there- fore, the series are absolutely convergent so that the order of the terms is immaterial, the functions must have the periods w, w'. Consider first the convergence of the series p'{z). For z = mu + nu\ that is, at the vertices of the net of parallelograms one term of the series becomes infinite and the series cannot converge. But if z he restricted to a finite region R about 2 = 0, there will be only a finite number of terms which can become infinite. Let a parallelogram P large enough to surround the region be drawn, and consider only the vertices which lie outside this par- allelogram. For convenience of computation let the points z = mu + nu' outside P be considered as ar- ranged on successive parallelograms P 1 , P , •••, 1\-. • • • . If the number of vertices on P be v, the number on P x is v + 8 and on P k is v + 8k. The shortest vector z — mu — nu' from z to any vertex of P l is longer than a, where a is the least altitude of the parallelogram of periods. The total contribution of P 1 to p'(z) is therefore less than (v + 8)n-° and the value contributed by all the vertices on successive parallelograms will be less than , v+8 J/+8-2 v + 8 ■ 3 v+ 8-k S = _ — + — 1- - - - -f + — I 1 . a 3 (2r() 3 (3 a) 8 {kuf This series of positive terms converges. Hence the infinite series for p'(z), when the first terms corresponding to the vert ires within P, are disregarded, converges absolutely and even uniformly so that it represents an analytic function. The whole series for p'(z) therefore represents a doubly periodic function of the third order analytic everywhere except at the vertices of the parallelograms where it has a pole of the third order. As the part of the series p'(z) contributed by ver- tices outside 7' is uniformly convergent, it may be integrated from to z to give the corresponding terms in p (z) which will also be absolutely convergent because the terms, grouped as for )>'(z), will be less than the terms of IS where I is the length of the path of integration from to z. The other terms of p'(z), thus far disregarded, may be integrated at sight to obtain the corresponding terms of p(z). Hence p'(z) is really the derivative of p(z); and ttxp(z) converges absolutely ex- cept for the vertices of the parallelograms, it is clearly doubly periodic of the second order with the periods a>, a/, for the same reason thatp'(z) is periodic. It has therefore been shown that doubly periodic functions exist, and hence the theorems deduced for such functions are valid. Some further important theorems are indicated among the exercises. They lead to the inference that any doubly periodic function which has Hie COMPLEX VARIABLE 489 periods w, w' and has no other singularities than poles may be expressed as a rational function of p(z) and 7/(2), or as an irrational function of 2>(z) alone, the only irrationalities being square roots. Thus by em- ploying only the general methods of the theory of functions of a complex variable an entirely new category of functions has been char- acterized and its essential properties have been proved. EXERCISES 1. Find the principal parts at z = for the functions of Ex. 4, p. 481. 2. Prove by Ex. 6, p. 482, that e" — c = has only one root in the strip. 3. How does e^) behave as z becomes infinite in the strip? 4. If the values B(e z ) approaches whenz becomes infinite in the strip are called exceptional values, show that H(e z ) takes on every value other than the excep- tional values k times in the strip, k being the greater of the two numbers n, m. 5. Show by Ex. i). p. 482, that in any parallelogram of periods the sum of the positions of the roots less the sum of the positions of the poles of a doubly peri- odic function is 7nw + not', where m and ?i are integers. 6. Show that the terms of p'(z) may be associated in such a way as to prove thatp'(— .?) = — p'(z), and hence infer that the expansions are p'(z) = — 2 z~ 3 + 2 c x z + 4 c 2 z 3 + • • ■ , only odd powers, and p (z) = z~' 2 + c x z 2 + c.-^z* + • • • , only even powers. 7. Examine the series (6) forp'(z) to show thatp'O w) =p'(\ w') =P'(l w + J w') =0. Why canp'(z) not vanish for any other points in the parallelogram? 8. Let p(},u) — e. p(hw') — t'. p (| w + \ <*>') = e". Prove the identity of the doubly periodic functions [p'(z)]' 2 anu - ^[P (~) — e ] [ V ( z ) — e '] [P ( z ) ~ <-'"]• 9. By examining the series defining p(z) show that any two points z = a and z = a' such that p (a) =p(n') are symmetrically situated in the parallelogram with respect to the center z = \ (w + w'). How could this be inferred from Ex. 5 '.' 10. "With the notations g(z, (z) + p>(a) l >_ _ KY ' ilp(z)-p(a)j *\) v\* t *\ -t Ip{z) — P(a) J 11. Demonstrate the final theorem of the text of §182. 490 THEORY OF FUNCTIONS 12. By combining the power series forp(z) and p'(z) show [p\z)] 2 — 4 O(z)] 3 + 20 c 1 p(z) + 28 c 2 = Az' 2 + higher powers. Hence infer that the right-hand side must be identically zero. 13. Combine Ex. 12 with Ex. 8 to prove e + e' -f e" = 0. 14. With the notations g 2 = 20 r l and ) — f(z) and f (z + «') — f (z) must be merely constants 77 and rj'. 183. Conformal representation. The transformation (§ 178) v:=f(z) or u + iv = u(x,y) + io(x,y) is conformal between the planes of -„• and w at all points .? at which f'(z) =£ 0. The correspondence between the planes may be represented by ruling the s-plane and drawing the corresponding rulings in the w-plane. If in particular the rulings in the 2-plane be the lines x = const., y = const., parallel to the axes, those in the jf-plane must be two sets of curves which are also orthogonal; in like manner if the -.'-plane be ruled by circles concentric with the origin and rays issuing from the origin, the w-plane must also be ruled orthogonally: for in both cases the angles between curves must be preserved. It is usually most convenient to consider the ?/• -plane as ruled with the lines u = const., r = const., and hence to have a set of rulings u (x, y) = c v v(x, y) = c in the s-plane. The figures represent several different cases arising from the functions • w— plane (1) z~plane (1) w = az = (a, + aj) (x + iy), u = (2) w = log z — log V./'- + //-'+ I tan -1 - > u = log V.'-- + y'\ v = tan -1 - ■ Consider w = .-:'"', and apply polar coordinates so that ir — J,' (cos 4> 4- i sin 3>) = /•-(cos 2 -f- / sin 2 ), R = /•", <& = 2 COMPLEX VARIABLE 491 To any point (;•, ) in the s-plane corresponds (II = r 1 , <& = 2 <£) in the ^•-plane ; circles about z = become circles about w = and rays is- suing from z = become rays issuing from w = at twice the angle. (A figure to scale should be supplied by the reader.) The derivative w' = 1z vanishes at z — only. The transformation is conformal for all points except z = 0. At z = it is clear that the angle between two curves in the s-plane is doubled on passing to the corresponding curves in the w-plane ; hence at ;: — the transformation is not con- formal. Similar results would be obtained from w = z m except that the angle between rays issuing from w = would be m times the angle between the rays at z = 0. A point in the neighborhood of which a function w = f(z) is ana- lytic but has a vanishing derivative /'(*-') is called a critical point of f(z) ; if the derivative f'(z) has a root of multiplicity k at any point, that point is called a critical point of order k. Let z — z be a critical point of order k. Expand f'(z) as f(z) = a k (z - z o y + a k+x (z - z )*+* + a, + 2 (z - * )* + 2 + ■ • • j then /(*) = f(z Q ) + J^ l (z- * )* +1 + ^ (* - ^ ) t + 2 + • • • , w = Wq + (z-zJ'+*E(z) or )r - % = (z-z ( /^E(z), (7) where £ is a function that does not vanish at z Q . The point z = z Q goes into w = w Q . For a sufficiently small region about z () the transforma- tion (7) is sufficiently represented as iv-w =C(z-z Q )*+\ C = E(z ). On comparison with the case a- = z"\ it appears that the angle between two curves meeting at z will be multiplied by k + 1 on passing to the corresponding curves meeting at w . Hence at a critical point of the kth order the transformation is not conformal hut angles are multiplied hi/ k + 1 on passing from the z-plane to the ic-plane. Consider the transformation v = z 1 more in detail. To each point z corresponds one and only one point w. To the points z in the first quadrant correspond the points of the first two quadrants in the v- plane, and to the upper half of the 2-plane corresponds the whole ?r-plane. In like manner the lower half of the 2-plane will be mapped upon the whole ?r-plane. Thus in finding the points in the w-plane which cor- respond to all the points of the .--plane, the ?/>plane is covered twice. This double counting of the ?r-plane may be obviated by a simple de- vice. Instead of having one sheet of paper to represent the ?/>plane, 492 THEORY OF FUNCTIONS let two sheets be superposed, and let the points corresponding to the upper half of the 2-plane be considered as in the upper sheet, while those corresponding to the lower half are considered as in the lower sheet. Now consider the path traced upon the double w-plane when z traces a path in the 2-plane. Every time z crosses from the second to n u-^% n w— surface w— surface z— "plane the third quadrant, w passes from the fourth quadrant of the upper sheet into the first of the lower. When z passes from the fourth to the first quadrants, w comes from the fourth quadrant of the lower sheet into the first of the upper. It is convenient to join the two sheets into a single surface so that a continuous path on the ."-plane is pictured as a continuous path on the ff-surface. This may be done (as indicated at the right of the middle figure) by regarding the lower half of the upper sheet as con- nected to the upper half of the lower, and the lower half of the lower as connected to the upper half of the upper. The surface therefore (aits through itself along the positive axis of reals, as in the sketch on the left*; the line is called the junction line of the surface. The point ir = which corresponds to the critical point z = is called the branch 2>oint of the surface. Now not only does one point of the £-plane go over into a single point of the //--surface, but to each point of the sur- face corresponds a single point z; although any two points of the //•- surface which arc superposed have the same value of //•, they correspond to different values of z except in the case of the branch point. 184. The //•-surface, which has been obtained as a mere convenience in mapping the 2-plane on the //--plane, is of particular value in study- ing the 1 , inverse function z = v //•. For v //• is a multiple valued func- tion and to each value of //■ correspond two values of z; but if //• be * Practically this may lie accomplished for two sheets of paper by pastim strips to the sheets which are to he connected across the cut. 'ummed COMPLEX VARIABLE 493 regarded as on the ?r-surface instead of merely in the ?#-plane, there is only one value of z corresponding to a point w upon the surface. Th us the function ~\w which is double valued over the ic-plane heroines single valued over t/te w-surface. The ^--surface is called the Riemann surface of the function z = ~vw. The construction of Riemann surfaces is im- portant in the study of multiple valued functions because the surface keeps the different values apart, so that to each point of the surface corresponds only one value of the function. Consider some surfaces. (The student should make a paper model by following the steps as indicated.) Let w = z s — 3z and plot the u>-surface. First solve /'(z) = to find the critical points z and substitute to find the branch points w. Now if the branch points be considered as removed from the wj-plane, the plane is no longer simply connected. It must be made simply connected by drawing proper lines in the figure. This may be accomplished by drawing a line from each branch point to infinity or by con- necting the successive branch points to each other and connecting the last one to the point at infinity. These lines are the junction lines. In this particular case the critical points are z = + 1. — 1 and the branch points are w = — 2, +2. and the junction lines may be taken as the straight lines joining w = — 2 and w = + 2 to I , II , III I II 1 1 ii in m A\ dr •>e iv =o uj/ 11 i ii III' k b / i'ii'iii \\tl\to! iv -surface z- plane infinity and lying along the axis of reals as in the figure. Next spread the requi- site number of sheets over the u'-plane and cut them along the junction lines. As w = z z — 3z is a cubic in z, and to each value of w, except the branch values, there correspond three values of z. three sheets are needed. Now find in the z-plane the image of the junction lines. The junction lines are represented by v = ; but f = Sx->/ — y s — 3 //. and hence the line y = and the hyperbola Sx 2 — if- — 3 will be the images desired. The z-plane is divided into six pieces which will be seen to correspond to the six half sheets over the u'-plane. Next z will be made to trace out the images of the junction lines and to turn about the critical points so that w will trace out the junction lines and turn about the branch points in such a manner that the connections between the different sheets may be made. It will be convenient to regard z and w as persons walking along their respective paths so that the terms "right " and "left" have a meaning. 494 THEORY OF FUNCTIONS Let z start at z = and move forward to z = 1 ; then, as/'(z) is negative, w starts at w = and moves back to w = — 2. Moreover if z turns to the right as at P, so must w turn to the right through the same angle, owing to the conf ormal property. Thus it appears that not only is OA mapped on oa, but the region 1' just above OA is mapped on the region I' just below oa ; in like manner OB is mapped on ob. As ab is not a junction line and the sheets have not been cut through along it, the regions 1, 1' should be assumed to be mapped on the same sheet, say, the upper- most, I, I'. As any point Q in the whole infinite region 1' may be reached from without crossing any image of ab, it is clear that the whole infinite region 1' should be considered as mapped on T ; and similarly 1 on I. The converse is also evident, for the same reason. If, on reaching A, the point z turns to the left through 90° and moves along AC. then w will make a turn to the left of 180°, that is, will keep straight along ac ; a turn as at R into 1' will correspond to a turn as at r into T. This checks with the statement that all 1' is mapped on all I'. Suppose that z described a small circuit about + 1. When z reaches D, w reaches d ; whenz reaches E, w reaches c. But when w crossed ac, it could not have crossed into I, and when it readies c it cannot be in I ; for the points of I are already accounted for as corresponding to points in 1. Hence in crossing ac. w must drop into one of the lower sheets, say the middle, II; and on reaching e it is still in II. It is thus seen that II corre- sponds to 2. Let z continue around its circuit; then II' and 2' correspond. When 2 crosses AC from 2' and moves into 1, the point w crosses ac' and moves from II' up into I. In fact the upper two sheets are connected along ac just as the two sheets of the surface for w = z- were connected along their junction. In like manner suppose that z moves from to — 1 and takes a turn about B so that w moves from to 2 and takes a turn about b. When z crosses BF from 1' to 3, w crosses //from I' into the upper half of some sheet, and this must be III for the reason that I and II are already mapped on 1 and 2. Hence Y and III are con- nected, and so are I and III'. This leaves II which has been cut along bf, and III cut along ac, which may be reconnected as if they had never been cut. The reason for this appears forcibly if all the points z which correspond to the branch points are added to the diagram. When w = 2, the values of z are the critical value — 1 (double) and the ordinary value z = 2 ; similarly, w = — 2 corresponds to z = — 2. Hence if z describe the half circuit AE so that iv gets around to e in II. then if z moves out to z = 2. v: will move out to w = 2, passing by w = in the sheet II as z passes through z = v3 ; but as z = 2 is not a critical point, w = 2 in II cannot be a branch point, and the cut in II may be reconnected. The i/>surface thus constructed for ic =f(z) = 2 3 — Zz is the Riemann surface for the inverse function z =f- l (v:). of which the explicit form cannot be given without solving a cubic. To each point of the surface corresponds one value of z, and to the three superposed values of w correspond three different values of z ex- cept at the branch points where two of the sheets come together and give only one value of z while the third sheet gives one other. The Riemann surface could equally well have been constructed by joining the two branch points and then connecting one of them to x. The image of v = would not have been change:!. The connections of the sheets could lie established as before, but would be dif- ferent. If the junction line be — 2. 2. + x. the point w = 2 has two junctions running into it. and the connections of the sheets on opposite sides of the point are not independent. It is advisable to arrange the work so that the first branch point n COMPLEX VARIABLE 495 which is encircled shall have only one junction running from it. This may be done by taking a very large circuit in z so that w will describe a large circuit and hence cut only one junction line, namely, from 2 to oo, or by taking a small circuit about z = 1 so that w will take a small turn about w = — 2. Let the latter method be chosen. Let z start from z = at and move to z = 1 at A ; then w starts at w = and moves to w — — 2. The correspondence between Y and Y is thus established. Let z turn about A ; then iv turns about w — — 2 at a. As the line — 2 to — co or ac is not now a junction line, w moves from Y into the upper half I, and the region across *w 5 11 l \ l I ™ AC from Y should be labeled 1 to corre- - \„ spond. Then 2', 2 and II', II may be filled in. The connections of I— II' and II-T are indicated and III-III' is reconnected, as the w— surface z— plane branch point is of the first order and only two sheets are involved. Xow let z move from z = 0toz = — 1 and take a turn about B ; then w moves from w = to w — 2 and takes a turn about b. The region next Lis marked 3 and Y is connected to III. Passing from 3 to 3' for z is equivalent to passing from III to III' for w between and b where these sheets are connected. From 3' into 2 for z indicates III' to II across the junction from w = 2 to co. This leaves I and II' to be connected across this junction. The connections are com- plete. They may be checked by allowing z to describe a large circuit so that the regions 1, 1', 3, 3', 2, 2', 1 are successively traversed. That I, I', III, III', II, II', I is the corresponding succession of sheets is clear from the connections between w = 2 and co and the fact that from w = — 2 to — co there is no junction. Consider the function w = z 6 — 3z 4 + 3z 2 . The critical points are z = 0, 1, 1, — 1,-1 and the corresponding branch points are w — 0, 1, 1, 1, 1. Draw the junc- tion lines from vi = to — co and from w = 1 to + co along the axis of reals. To find the image of v = on the z-plane, polar coordinates may be used. 2 = r(cos

    i — 3 r 4 e 4 *' + 3r 2 e 2 * 1 '. v = = r 2 [r* sin 6

    [) A {S - 4 sin 2 0) - 6 r 2 cos + 3] . The equation v ~ therefore breaks up into the equation sin 2 = and 3 cos 2 ± V5sin2 V3 sin (60 ± 2 0) \Z~3 V — 3-4 sin 2 20 2 sin (60 + 20) sin (60 - 2 0) 2 sin (60 ±20) Hence the axes = 0° and = 90° and the two rectangular hyperbolas inclined at angles of ± 15° are the images of v = 0. The z-plane is thus divided into six por- tions. The function w is of the sixth order and six sheets must be spread over the tc-plane and cut along the junction lines. To connect up the sheets it is merely necessary to get a start. The line w = to w = 1 is not a junction line and the sheets have not been cut through along it. But when z is small, real, and increasing, w is also small, real, and increasing. Hence to OA corresponds oa in any sheet desired. Moreover the region above OA will correspond to the upper half of the sheet and the region below OA to the lower half. Let the sheet be chosen as III and place the numbers 3 and 3' so as to correspond with III and III'. Fill in the numbers 4 and 4' around z = 0. When 496 THEORY OF FUNCTIONS z turns about the critical point z — 0, iv turns about w ?= 0, but as angles are doubled it must go around twice and the connections 1II-IV, IV-III' must be made. Fill in more numbers about the critical point z = 1 of the second order where angles are tripled. On the w-sur- face there will be a triple connection III'- I— VI II, II'-I, F-III. In like manner the criti- W ' /// cal point z = — \ may be treated. The sur- face is complete except for reconnecting sheets I, II, V, VI along w - to iv — — oo as if they had never been cut. ic— surface z— plane ~\"\"7?/ Ill iVvi' iv— surface EXERCISES 1. Plot the corresponding lines for : (a) w = (1 + 2 i)z, (/3) w — (1— \ i)z. 2. Solve for x and y in (1) and (2) of the text and plot the corresponding lines. 3. Plot the corresponding orthogonal systems of curves in these cases: (a) w = - , (j3) w = 1 + z 2 , (y) w = cos z. 2 4. Study the correspondence between z and iv near the critical points: (a) iv = 2 3 , (/3) w — 1 — z' 2 , (7) w = s\nz. 5. Upon the 10-surface for iv = z- plot the points corresponding to z = 1, 1 + i. 2 (. _ 1 + I VSi, — I, — I V3 — I t, — i, I — i i. And inthe 2-plane plot the points corresponding to iv = V2 + "\ 2 i, i, —4, — J — J V 3 i, 1 — i, whether in the upper or lower sheet. 6. Construct the ic-surface for these functions: (a) w = z 3 , (j8) w = z- '-'. (7) 10 = 1 + z-, (5) 10 = (z - l) 3 . In (|3) the singular point 2 = should be joined by a cut to 2 = co. 7. Construct the Riemann surfaces for these functions: (a) xv = z 4 - 2 z 2 , (3) « = 2+ 1 03) iv ( e ) W = gS + + 4z. 1 (7) 10 = 2: 5 z z , (f) w z 3 + V3 z 2 2" ' v 3 z 2 + 1 185. Integrals and their inversion. Consider the function r w dic . , . defined by an integral, and let the methods of the theory of functions he applied to the study of the function and its inverse. If iv describes a path surrounding the origin, the integral need not vanish; for the plane ic— plane COMPLEX VARIABLE 497 integrand is not analytic at w = 0. Let a cut be drawn from 10 = to w — — cc. The integral is then a single valued function of w provided the path of integration does not cross the cut. Moreover, it is analytic except at w = 0, where the derivative, which is the integrand 1/iv, ceases to be continuous. Let the w-plane as cut be mapped on the s-plane by allowing w to trace the path labcdefyhil, by computing the value of z sufficiently to draw the image, and by applying the principles of conformal representation. When w starts from w = 1 and traces la, z starts from z — and becomes nega- tively very large. When w turns to the left to trace ab, z will turn also through 90° to the left. As the integrand along ab is idfji, z must be changing by an amount which is pure imaginary and must reach B when w reaches b. When w traces be, both w and div are negative and z must be increasing by real positive quantities, that is, z must trace EC. When v moves along cdefff the same reasoning as for the path ab will show that z moves along CDEFG. The remainder of the path may be completed by the reader. It is now clear that the whole ?t*-plane lying between the infinitesimal and infinite circles and bounded by the two edges of the cut is mapped on a strip of width 2 iri bounded upon the right and left by two infi- nitely distant vertical lines. If w had made a complete turn in the posi- tive direction about w = and returned to its starting point, z would have received the increment 2 iri. That is to say, the values of z which correspond to the same point w reached by a direct path and by a path which makes k turns about w = will differ by 2 hrri. Hence when w is regarded inversely as a function of ,~, the function will be periodic with the period 2 iri. It has been seen from the correspondence of cdefg to CDEFG that w becomes infinite when z moves off indefinitely to the right in the strip, and from the correspondence of BA III with balh that ic becomes when z moves off to the left. Hence w must be a rational function of e z . As w neither becomes infinite nor vanishes for any finite point of the strip, it must reduce merely to Ce kz with /.• integral. As w has no smaller period than 2 nri, it follows that k = 1. To determine C, compare the derivative div/dz = Ce z at z = with its reciprocal dz/dir = w~ x at the corresponding point w = 1; then C = l The inverse function In -1 ,-; is therefore completely determined as e z . 498 THEORY OF FUNCTIONS In like manner consider the integral duo Jo 1 + w 2 /(«>), *(z)=f-Hz). B AK J Here the points w = ± i must be eliminated from the w-plane and the plane ren- dered simply connected by the proper cuts, say, as in the figure. The tracing of the figure may be left to the reader. The chief difficulty may be to show that the integrals along oa and be are so nearly equal that C lies close to the real axis; no com- putation is really necessary inasmuch as the integral along oc' would be real and hence C must lie on the axis. The image of the cut M>-plane is a strip of width tt. Circuits around either -f i or — i add tt to z, and hence w as a function of z has the period tt. At the ends of the strip, w approaches the finite values + I and — i. The function w = '- - e- + 1 i e iz + e- = tan z, tan -1 tw. 186. As a third example consider the integral C" die Vl — ir- :/(„■), lV== ( Z - )= /"!(,). (8) Here the integrand is double valued in w and consequently there is liable to be confusion of the two values in attempting to follow a path in the /r-plane. Hence a two-leaved surface for the integrand will lie constructed and the path of integration will be considered to be on the surface. Then to each point of the path there will correspond only one value of the integrand, although to each value of //• there correspond two superimposed points in the two sheets of the surface. As the radical \ 1 — v: 2 vanishes at w = ± 1 and takes on only the single value instead of two equal and opposite values, the points u: = ± 1 are branch points on the surface and they are the only finite branch points. Spread two sheets over the ic-plane, mark the branch points v: = ± 1, ami draw the junction line between them and continue it (provisionally) to y = ac. At !/• = — ] the function Vl — w' 2 may be written Vl + w E(iv), where K denotes a function which does not vanish at w = — \. Hence in the neighborhood of (/• = — ] the surface looks like that for Vic near w = 0. This may be accomplished by making the connections across the COMPLEX VARIABLE -LOO junction line. At the point w = + 1 the surface must cut through itself in a similar manner. This will be so provided that the sheets are reconnected across loo as if never cut ; if the sheets had been cross-connected along 1 cc, each sheet would have been separate, though crossed, over 1, and the branch point would have disappeared. It is noteworthy that if w describes a large I n circuit including both branch points, the values of Vl — w' 2 are not interchanged; the circuit closes in each sheet without pass- ing into the other. This could be expressed by saying that w — oo is not a branch point of the function. Xow let w trace out various paths on the surface in the attempt to map the sur- face on the z-plane by aid of the integral (8). To avoid any difficulties in the way of double or multiple values for z which might arise if w turned about a branch point w = ± 1, let the surface be marked in each sheet over the axis of reals from — co to + 1. Let each of the four half planes be treated separately. Let w start, at vo = in the upper half plane of the tipper sheet and let the value of Vl — w' 2 at this point be + 1 ; the values of Vl — w' 2 near w — in II' will then be near + 1 and will be sharply distinguished from the values near — 1 which are supposed to correspond to points in I', II. As w traces oa, the integral z increases from to a definite positive number a. The value of the integral from a to b is infinitesimal. Inasmuch as w = 1 is a branch point where two sheets connect, it is natural to assume that as w passes 1 and leaves it on the right, z will turn through half a straight angle. In other words the integral from b to c is naturally presumed to be a large pure imaginary affected with a positive sign. (This fact -5 — <2 Q G_ D may easily be checked by exam- ining the change in Vl — w 2 when v: describes a small circle about w = 1. In fact if the E- function Vl + w be discarded and if 1 — w be written as re*% then Vres* 1 is that value of the radical which is positive when 1 — io is positive. Now when w describes the small semicircle, changes from 0° to — 180° and hence the value of the radical along be becomes — t Vr and the integrand is a positive pure imaginary.) Hence when w traces be, z traces BC. At c there is a right-angle turn to the left, and as the value of the integral over the infinite quadrant cc' is I 7r, the point z will move back through the distance \ it. That the point (" thus reached must lie on the pure imaginary axis is seen by noting that the integral taken directly along oc' would be pure imagi- nary. This shows that a = \ tr without any necessity of computing the integral over the interval oa. The rest of the map of I may be filled in at once by symmetry. To map the rest of the to-surface is now relatively simple. For I' let w trace cc"d' ; then z will start at C and trace CD' = tt. When w comes in along the lower side of the cut d'e' in the upper sheet I', the value of the integrand is identical with the value when this line de regarded as belonging to the upper half plane was de- scribed, for the line is not a junction line of the surface. The trace of z is there- fore D'E'. When w traces fo' it must be remembered that V joins on to II and hence that the values of the integrand are the negative of those along fo. This z— plane w — surface 500 THEORY OF FUNCTIONS makes z describe the segment F'O' = — a = — \ ir. The turn at F'F' checks with the straight angle at the branch point — 1. It is further noteworthy that when w returns to o' on I', z does not return to but takes the value ir. This is no contra- diction; the one-to-one correspondence which is being established by the integral is between points on the lo-surface and points in a certain region of the z-plane, and as there are two points on the surface to each value of w, there will be two points z to each iv. Thus far the sheet I has been mapped on the z-plane. To map II let the point w start at o' and drop into the lower sheet and then trace in this sheet the path which lies directly under the path it has traced in I. The integrand now takes on values which are the negatives of those it had previously, and the image on the z-plane is readily sketched in. The figure is self-explanatory. Thus the complete surface is mapped on a strip of width 2 ir. To treat the different values which z may have for the same value of w, and in particular to determine the periods of w as the inverse function of z, it is necessary to study the value of the integral along different sorts of paths on the surface. Paths on the surface may be divided into two classes, closed paths and those not closed. A closed path is one which returns to the same point on the surface from which it started ; it is not sufficient that it return to the same value of w. Of paths which are not closed on the surface, those which close in w, that is, which return to a point superimposed upon the starting point but in a different sheet, are the most important. These paths, on the particular surface here studied, may be fur- ther classified. A path which closes on the surface may either include neither branch point, or may include both branch points or may wind twice around one of the points. A path which closes in w but not on the surface may wind once about one of the branch points. Each of these types will be discussed. If a closed path contains neither branch point, there is no danger of confusing the two values of the function, the projection of the path on the w-plane gives a region over which the integrand may be considered as single valued and analytic, and hence the value of the circuit integral is 0. If the path surrounds both branch points, there is again no danger of confusing the values of the function, but the projection of the path on the 10-plane gives a region at two points of which, namely, the branch points, the integrand ceases to be analytic. The inference is that the value of the integral may not be zero and in fact will not be zero unless the in- tegral around a circuit shrunk close up to the branch points or expanded out to infinity is zero. The integral around cc'dc"c is here equal to 2v; the value of the integral around any path which incloses both branch points once and only once is therefore 2 -k or — 2 7r ac- cording as the path lies in the upper or lower sheet ; if the path surrounded the points k times, the value of the integral would be 2kir. It thus appears that w re- garded as a function of z has a period 2 it. If a path closes in v) but not on the surface, let the point where it crosses the junction line be held fast (figure) while the path is shrunk down to vobaa'b'w. The value of the integral will not change during this shrinking of the path, for the. new and old paths may together be regarded as closed and of the first case considered. Along the paths wba and a'b'w the integrand has opposite signs, but so has dw ; around the small circuit the value of the integral is infini- tesimal. Hence the value of the integral around the path which closes in w is 2 I or — 2 I if I is the value from the point a where the path crosses the junction line * — COMPLEX VARIABLE 501 to the point iv. The same conclusion would follow if the path were considered to shrink down around the other branch point. Thus far the possibilities for z corre- sponding to any given w are z + 2k-7r and2imr — z. Suppose finally that a path turns twice around one of the branch points and closes on the surface. By shrink- ing the path, a new equivalent path is formed along which the integral cancels out term for term except for the small double circuit around ± 1 along which the value of the integral is infinitesimal. Hence the values z + 2 kir and 2nnr — z are the only values z can have for any given value of w if z be a particular possible value. This makes two and only two values of z in each strip for each value of w, and the function is of the second order. It thus appears that w, as a function of z, has the period 2tt, is single valued, becomes infinite at both ends of the strip, has no singularities within the strip, and has two simple zeros at z = and z = ir. Hence w is a rational function of e iz with the numerator e 2iz — 1 and the denominator c- '~- + 1. In fact w= C 1 e h - — e" e lz + e~ i e iz + e~ sin z. The function, as in the previous cases, has been wholly determined by the general methods of the theory of functions without even computing a. One more function will be studied in brief. Let a>0, z=f{w), w = -surface upon the z-plane, and second upon the simplicity of the map, which was such as to indi- cate that the inverse function was a single valued periodic function. It should be 502 THEORY OF FUNCTIONS realized that if an attempt were made to apply the methods to integrands which appear equally simple, say to 2 = | -\ a- — w 2 du\ z = I (a — w)dw/vw, the method would lead only with great difficulty, if at all, to the relation between 2 and w ; for the functional relation between z and w is indeed not simple. There is, however, one class of integrals of great importance, namely, dw f V (w — a,)(w.' — a. 2 ) ■ • • (w — a„) for which this treatment is suggestive and useful. EXERCISES 1. Discuss by the method of the theory of functions these integrals and inverses f iv dw , , r w 2 dw , , r"' dw a I --> (/3) I : > (7) / :,• J i 2 u> Jn 1 — w Jo 1 — r- (S) I • (0 / . 1 (f) / — = Jo Vh,- 2 — 1 Jo Vi/j 2 + 1 "^ wViv 2 - + 1 * / * w vm 2 + a (w + The results may be checked in each case by actual integration. dw . r"' dv: r"' dv: ,„. r"' dw , . r" : dw (v) I - iO) I ■ • (0 I Jr w V w' 2 — a' 1 J{ ' \ 2 aw — u> 2 Jl (w + l)\ w' 2 — 1 X"' dw r w dw and I (§ 182. and Ex. 10. p. 489). - \ w(\ — it") (1 + w) J[ > \'l- ir 4 CHAPTER XIX ELLIPTIC FUNCTIONS AND INTEGRALS 187. Legendre's integral I and its inversion. Consider I die 0 2~ - = sin \tt; it is seen that the curve sn x has ordinates numerically greater than sin (ttx/2 A). As en x = V 1 — sir x, dn x = V 1 — Ir sir x, (5) the curves y = en x, y = dn x, may readily be sketched in. It may be noted that as sn (x + K ) =£ en x, the curves for sn x and en x cannot be superposed as in the case of the trigonometric functions. The segment 0, iK' of the pure imaginary axis fur z corresponds to the whole upper half of the pure imaginary axis for //•. Hence sn ix with x real is pure imaginary and — £ sn ix is real and positive for s ,/• < A' ' and becomes infinite for ,/■ = A''. Hence — £ sn ix looks in general like tan (ttx/2 A'). By (5) it is seen that the curves for y = en ix, ij = dn ix look much like see (irx/2 K') and that en ix lies above dn ix. These functions are real for pure imaginary values. It was seen that when /.• and k' interchanged, A' and A'' also inter- changed. It is therefore natural to look for a relation between the ellip- tic functions sn (.~, /.-), en (.~, k), dn (.?, k) formed with the modulus k 506 THEORY OF FUNCTIONS and the functions sn (z, k'), en (z, k'), dn (z, k') formed with the com- plementary modulus k' It will be shown that . sn (z, k') ,sn(iz,k') sn (iz, k) = i — ) — -f i sn (z, k) = — t — y - — -^ , en (z, k) J en (iz, k) 1 1 en (iz, k) = — - — jj- 5 en (z, k) = en (*, k') ^' ' en (iz, k') dn (.?, A-') dn (fe, £') dn (ts, k) = ; — ~ > dn (z, k) = 7^ — —£■ • v ' J en (z, k') v ' ' en (iz, k') Consider sn (iz, k). This function is periodic with the periods 4 A" and 2 IK' if /.~ be the variable, and hence with periods 4 iK and 2 A'' if z be the variable. With z as variable it has zeros at 0, 2 iK, and poles at A"', 2 t'A" + A"'. These are precisely the positions of the zeros and poles of the quotient H(z, q'^/H^z, q'), where the theta functions are con- structed with q' instead of y. As this quotient and sn (iz, k) are of the second order and have the same periods, Bn(ix y H(z,g<) = ^(z,k<) , ( ' )_ H x (z,q>) C ^n(z,k>) The constant C\ may be determined as C l = i by comparing the deriva- tives of the two sides at z = 0. The other five relations may be proved in the same way or by transformation. The theta series converge with extreme rapidity if q is tolerably small, but if q is somewhat larger, they converge rather poorly. The relations just obtained allow the series with q to be replaced by series with q' and one of these quantities is surely less than 1/20. In fact if v = ttx/2 K and v' = irx/2 A'', then V7 2 sin v — 2 7- sin 3 v + 2 a 6 sin 5 v — • ■ • sn ( c k) = — = ' VA- 1 - - 7 c ' os - v + 2 y 4 cos 4 v - 2 y 9 cos 6 v H 1 sinh v' — '/'" sinh 3 v + y' 6 sinh 5 v' — • • • V/.- cosh v + 7'" cosh 3 v + y' 6 cosh 5 v + • • • The second scries has the disadvantage that the hyperbolic functions increase rapidly, and hence if the convergence is to be as good as for the first series, the value of q' must lie considerably less than that of q, that is, A' must be considerably less than A". This can readily be arranged for work to four or five places. For q m = e K ', cosh 5 v' = £ \e 2K ' + e 2K ') , ^ x =2 A', where owing to the periodicity of the functions it is never necessary to take x > A''. The term in y' 6 is therefore less than \ y' 3 -. If the term ELLIPTIC FUNCTIONS 507 in q m is to be equally negligible with that in '/', 2 '/ = 1 f['* with log q log q' = 7T 2 , from which '/ is determined as about = y = sn x, <£ = am x, cos <£ = Vl — sn 2 x = en x \cf> = Vl - /.•-/ = Vl - /.- sin 2 cf> = dn x, k' 2 = 1 - A' 2 , .r = sn- 1 ^, /.-) = cn-^Vl-y 2 , /.-) = dn'^Vl _ khf, k). The angle <£ is called the amplitude of ,r : the functions sn x, en a', dnx are the sine-amplitude, cosine-amplitude, delta-amplitude of a;. The half periods are then X Vl-/.- u silr0 \2 /' and are known as the complete elliptic integrals of the first kind. 189. The elliptic functions and integrals often arise in problems that call for a numerical answer. Here k 2 is given and the complete integral K or the value of the elliptic functions or of the elliptic inte- gral F((j>, k) are desired for some assigned argument. The values of K and F(cf>. /.•) in terms of sin -1 /.- are found in tables (B. O. Peirce, pp. 117-119), and may be obtained therefrom. The tables may be used by inversion to find the values of the function sn x, en x, dn x when x is given ; for sn x = sn F(, k) = sin <£. and if x =F is given, <£ may be found in the table, and then sn x = sin <£. It is, however, easy to compute the desired values directly, owing to the extreme rapidity of the convergence of the series. Thus l2K \2 Kk' 1 + V/c' ,— 1 ^|— = ©/o), ^— = ©(O), -±=- Va- = -(© l( o) + 0(0)), V/v= r ^(l + 2^+...) = > J-flog,' (9) = V-21og ? ' (1 2 h + ...n l + Va 508 THEORY OF FUNCTIONS The elliptic functions are computed from (6) or analogous series. To compute the value of the elliptic integral F (<£, k), note that if dn x 1 + 2 n cos 2 v + 2 y- 4 cos 4 v -\ cot A = —7=- = ^ „ n — , o 4 ^ ; ' (!") VP 1 — 2q cos 2 v + 2 y 4 cos 4 y + • • • v J /l \ cot X — 1 _ cos 2 v + y R cos 6 v H tan - 7r — A = . = 2 q 4 - ; \4 / cot A + 1 1 + 2 y 4 cos 4 v -\ 2 ^ cos 2 v and tan Q ir - A) = 2 « cos 2 v or tan (1 tt - A) = z ~ : — (10') V4 ' V4 J 1 + 2 y 4 cos 4 v v y are two approximate equations from which cos 2 v may be obtained ; the first neglects y 4 and is generally sufficient, but the second neglects only q s . If U 2 is near 1, the proper approximations are 1 dn(.x, k )_ dn(lx, 7c') _ 1 + 2 g' cosh 2 „' + •■■ Vfc en (x, k) ~ s/Tc l-2q' cosh 2 v' + • • • ' (11) tan (i tt - A) = 2 q' cosh 2 v' or tan (J tt - A) = ± ^ ^J^ y , ' C 11 ') Here y' 8 cosh 8 v' < y' 4 is neglected in the second, but y' 4 cosh 4 v < with sn x = sin or if y = sn x is given, dn x = V 1 — k 2 sn' 2 x and en x = V 1 — sir a; are readily computed. r dff As an example take I , and find K, sn I /v, F(l 7r. I). As /i;" 2 = 4 ' 1 Jo Vl- |sin 2 3 ' u ' 2; 4 and V&'>0.9, the first term of (37), p. 472, gives 7 accurately to five places. Compute in the form : (Lg = log 10 ) Lg k' 2 = 0.8750(5 Lg (l - VF) = 8.84136 Lg 2 tt = 0. 7082 Lg VF = 9.96876 Lg (l + VF) = 0.28560 2 Lg (l + VF) = 0.571 4 VF = 0.03060 Lg 2r/ = 8.55567 Lg K = 0.2268 1 - VF = 0.06040 2 q = 0.03505 K = 1 .080 1+VF= 1.03060 g = 0.01797 Check with table. 2 _ „ Vy sin I tt — q 2 sin 7r 4- • • • „ V7 A V3 sn - 1\ = 2 — - = 2 — - • 3 v /fc 1 — 2 7 cos I tt + • • • Vj ; ! + <1 2 , V6 Vq \ L .^ r ° = 0.38008 Lg sn \ K = 0.0450 S " 3 V ~ 1.01797 I Lg 7 = 0.563(56 sn f if = 0.8810. - Lg 1.018 = 0.00226 A = dn x = Vl — I sin- J tt — Vl — \ sin \ tt Vl 4- | sin J 7r. ELLIPTIC FUNCTIONS 509 I sin \ tt = 0.19134 X = 43° 28' 28" Lg 42.20 = 1.6253 1 - \ sin 1 7T = 0.80866 \ ir - X = 1° 31' 32" Lg A" = 0.2268 1+ isin \ tt = 1.19134 Lg tan = 8.42540 - Lg 180 = 7.7447 I Lg (1 - | sin I 7t) = 9.95388 Lg 2 g = 8.55567 Lgx = 9.5968 \ Lg (1 + \ sin | tt) = 0.03802 Lg cos 2 v = 9.86973 x = 0.3952 - Lg VA? = 0.03124 2 » = 42° 12' Check with table. Lg cot X = 0.02314 180 x = A (42.20) As a second example consider a pendulum of length a oscillating through an arc of 300°. Find the period, the time when the pendulum is horizontal, and its position after dropping for a third of the time required for the whole descent. Let x 2 + y 2 — 2 ay be the equation of the path and h = a(l + | v3) the greatest height. When y = h, the energy is wholly potential and equals mgh; and mgy is the general value of the potential energy. The kinetic energy is m /ds\ 2 I ma 2 /dy\ 2 , \ma 2 (dy\ 2 , , — — ) = — — ) and — : — ) + may = mgh 2\dt) 2ay-y 2 \dt) 2ay-y 2 \dtJ is the equation of motion by the principle of energy. Hence t=r ^ =J* r dw «>=*,*»= a, J ° V2gV{h — y)(2ay-y*) \gJ° V(l - w 2 ) (1 - k 2 w 2 ) h 2a Vg/at = sn-^te, k), 10 = sn (Vg/at, k), y = 7isn 2 (\ g/at, k), are the integrated results. The quarter period, from highest to lowest point, is A Va/g ; the horizontal position is y = a, at which t is desired ; and the position for Vg/at = § A is the third thing required. k 2 = 0.93301, 2 q' = 1 — ^ , A = - — log q' = 1+Vk * M(l+Vkf Lg k 2 = 9.96988 Lg (l - Vk) = 8.23553 Lg 2 = 0.3010 Lg Vic = 9.99247 - Lg (l + Vk) = 9.70272 Lg 2 g'-i = 0.3734 Vic = 0.98280 - Lg 2 = 9.69897 - Lg M = 0.3622 1 - Vic = 0.01 720 Lg q' = 7.63722 - 2 Lg (l + Vk) = 9.4034 1 + Vk = 1.98280 q' = 0.00434 Lg A = 0.4420. Hence K = 2.708 and the complete periodic time is 4 A' Va/g. y — a, w 2 = - , en iv — V 1 — a/h, dn w = V 1 — k 2 a/h. h 1 dn w 4 ('4~ /l \ n , , „ , a , vK \g t — r= A - fc 2 = cot X, tan ( - 7r — X =2q' cosh 2 v\ 2 / = — — - a - - V'fc en w \3 \4 / A \« £ Lg A: 2 = 9.96988 X = 43° 26' 12" 2/ =1.813 Lg 4 = 0.60206 \ tt - X = 1° 33' 48" Lg 2 / = 0.2584 - Lg 3 = 9.52288 Lg tan = 8.43603 - Lg 2 g'-i = 9.6266 Lg cot 4 X = 0.09482 Lg 2 g' = 9.93825 Lg M = 9.6378 Lg cot X = 0.02370 Lg cosh 2 / = 0.49778 Lg \ /- - - = 9.5228. V a A 510 THEORY OF FUNCTIONS Hence the time for y — a is t = 0.3333 K \ a/g = J whole time of ascent. , iff 2 „ la h /si nh ttK/3 K' — «' 2 sinh ttK /A'\ 2 y = h sn 2 a l -A -\ ,'- = - ( — ) Y a 3 Y g k \cosh ttA/3 A" + r/ 2 cosh irK/K'J V/' s + r/.i + ry /2 (r/'-l + r/')/ \o' 3 + ' s + q' ALgg' = 9.21241 o'i = 0.1631 a , /5.9645\ 2 2/ = 2 ak ( — — ) • - I LgY = 0.78750 q'~ i = 0.1319 \6.2993/ This gives y = 1.732 a. which is very near the top at h = 1.866a. In fact starting at 30° from the vertical the pendulum reaches 43° in a third and 90° in another third of the total time of descent. As sn \ A is (1 + k')~ 2 it is easy to calculate the position of the pendulum at half the total time of descent. EXERCISES 1. Discuss these integrals by the method (if mapping: ((f) z = f - , a > b > 0. w = b sn az, k = - , J ^ ( U 2 _ y.2) (ip. _ w 2) a u)z= r / dw = , w=^(Uk\ ,=2 8n -i(vi 1 t) 1 Jo A >r(l - w) (1 - k-ir) \- / (7) z=f m ' h '- • w = -«" \ Z ■ ;•» = tn (, k). z = tn-i («,, *). 2. Establish these Maclaurin developments with the aid of § 177: (a) sn z = z - (1 + fr 2 ) f- + (1 + HA: 2 + *4) f! , 3 ! o . (/3) cn z = 1- *"■ + (1 + 4/,--) ■ - - (1 + 44/r- + 16 fc*) |- + • • • , '1 '. 4 1 fi '. (7) dn z = 1- k- — + fr 2 (4 + fc 2 ) A--(l<> + 44 t 2 + /,- 4 ) :=_ + .... 2 ! 4 ! ! 3. Prove I — — — — - | — = , / > 1. sin 2 ^ = Psin 2 0. «/o Vi_ Z 2 sin 2 ^" a 1 — /- -sin--^ 4. Carry out the computations in these cases : (a) f* — = — — to find A', sn - A", Ft -ir, -4= \ Jo \ 1- 0.1 sin- H 3 \ 8 a 10/ 1, in terms of elliptic functions. dd x a — cos I 11. A ladder stands on a smooth floor and rests at an angle of 30° against a smooth wall. Discuss the descent of the ladder after its release from this position. Find the time which elapses before the ladder leaves the wall. 12. A rod is placed in a smooth hemispherical bowl and reaches from the bot- tom of the bowl to the edge. Find the time of oscillation when the rod is released. 190. Legendre's Integrals II and III. The treatment of I, • v » - '*■* , r <■-■■->'■■ ,„, Vl - a-' 1 J« V(l - ir 1 ) (1 - k*ic*) by the method of conformal mapping to determine the function and its inverse does not give satisfactory results, for the map of the Riemann surface on the .-.-plane is not a simple region. But the integral may be treated by a change of variable and be reduced to the integral of an elliptic function. Lor with ic = sn h, i> = sn -1 //•, C" ( 1 — Irir 1 )

    ™* W = b(u-iK + -' Hence k* sn 2 u = - Z\u) + Z'(0), Z'(0) = 0"(O)/®(O), sn 2 u dw = — Z (w) + wZ'(0), o (1 - k 2 sn 2 ») e?w = u (1 - Z'(0)) + Z («). (14) '0 The derivation of the expansions of Z'(m) and sn 2 u about u = iK' are easy. 0(u) = CTt(i - q 2n+1 e ±K "), loge(w) = V log(l - q 2n+i e ± K u ) + \ og C ( -*"A log 6 («) = log \1 — qe A J + function analytic near u = iK'. in Q'(u) __ iirqe K iirq K\l-qe & ) K\e« - q) f(u) = e« " =f{iK') + (u - iK')f'(lK') +... = q + (u - iK') l -^q + . . . , e'(w) _ + i d e'(u) _ - 1 Q(u) u — iK' du Q(u) (u — iK') 2 sn (u + tit') = , sn 2 (w + iK') = , k sn u A: 2 sn 2 u f(u) = sn u = u/'(0) + I a 3 /'"(0) + ... = « + cm 3 + • • • , sn 2 (u + ^) = I si ^ = I(l- C u + ...) 2 =l(l-2 C + ...), 1 / 1 „ \ sn 2 u — — ( 2 c + • • • ) . A; 2 \(it - iA") 2 / In a similar manner may he treated the integral f H^=— = f ^- an) Jo (tc 2 - a) V(l - w; 2 ) (1 - //-'«••-) Jo s,r " ~ a Let a be so chosen that sn 2 a = a. The integral becomes JC U (hi 1 r 2 sn ff en « dn n , — ^ = o i I ^ a du - ( 15 ) B sn - u — sn" a 2 sn a en a dn a J sir w — sn a ' ELLIPTIC FUNCTIONS 513 The integrand is a function with periods 2 K, 2 iK' and with simple poles at u = ± a. To find the residues at these poles note .. u^fa .. 1 ±1 Inn — r — = Inn : a sir u — sn 2 a u =± a 2 sn u en u dn w 2 sn a en a dn a The coefficient of (m =F ( ')~ l in expanding about ± a is therefore ± 1. Such a function niay be written down. In fact 2 sn a en a dn a _ //'(?£ — a) H'(u + a) sri 2 u — sir a 11 (h — a) H(ii -f a) = Z t (u - a) - Z x (w + a) + C, if Z = IV J II. The verification is as above. To determine C let u = 0. 2 en a dn a n , N , 1 7/(?0 Then C = h 2 Z,(a), but sn it = — = — f~{ , sn a 1V y VA- ®(") c/ . en u dn ?< „ , . „ . . and — log sn ?< = = Z.(u) — Z(ii). dn an k IV ' v J Hence C reduces to 2Z(") and the integral is dn 1 i/0 sir u — sn'- m 2 sn a en a dn )VTr fi(w) + /«'(».•) \ Tr fi (">) Vik A 7 ir where B means not always the same function. The integral of B (ic, V W) is thus reduced to the integral of i? t (w) which is a rational fraction, plus the inte- gral of wR. ? (uf-)/\'\V which by the substitution w 2 = u reduces to an integral of R (m, V (1 — i/)(l — k-u) and may be considered as belonging to elementary calculus, plus finally rRJw 2 ) r I — L -—aw = I R 3 (sn 2 u)du, w = snw. By the met Ik id of partial fractions R., may be resolved and /p du sn 2n udu 7i SO, I >i>0 J (sn- u — a) n are the types of integrals which must be evaluated to finish the integration of the given R(w, Vw), An integration by parts (B. 0. Peirce, No. -507) shows that for 514 THEORY OF FUNCTIONS the first type ?i may be lowered if positive and raised if negative until the integral is expressed in terms of the integrals of sn' 2 x and sn° x = 1, of which the first is integrated above. The second type for any value of n may be obtained from the integral for n = 1 given above by differentiating with respect to a under_the sign of integration. Hence the whole problem of the integration of R(w, V W) may be regarded as solved. 191. With the substitution w = sin cf>. the integral II becomes Vl - k* sin- Odd = / dw / 17) o Jo V 1 — w 2 = m(1-Z''(0))+Z(u), u = F(,k). In particular A'( }, 7r. /.•) is called the complete integral of the second kind and is generally denoted by E. When — \ it, the integral u = F(, k) becomes the complete integral K. Then E = K (1 - Z'(0)) + Z (A) = A (1 - Z'(0)), (18) and E(, /.) = EF(, k)/K + Z ('//). (19) The problem of computing E(cf>. k) thus reduces to that of computing K, E, E((f>, /.■) = u, and Z(u). The methods of obtaining A" and /"(.<£. k) have been given. The series for Z(«) converges rapidly. The value of E may be found by computing A'(l — Z'(0)). For the convenience of logarithmic computation note that A' - E -w-^->fe-^-^+»*'-o K v ' 0(0) or A' - £ = i tt/ VZ^ • (2 ir/Ky q (1 - 4 q 3 + • • •)■ (20) Q'(u) 2, y = 1) cos . ds = \<7<£ ; tin 1 eccentricity e may be as high as 0.99 without invalidating the approximate formulas. An example follows. Let it be required to determine the length of the quadrant of an ellipse of eccentricity e = 0.0 and also the length of the portion over half the semiaxis major. Here the series in q' converge better than those in q, but as the proper ELLIPTIC FUNCTIONS 515 expression to replace Z (m) has not been found, it will be more convenient to use the series in q and take an additional term or two. As k = 0.9, k" 2 = 0.19. Lgfc" 2 = 9.27875 Lg Vk' = 9.81969 Vk' = 0.66022 1 - Vk' = 0.33978 1 + VV = 1.66022 Lgg = 9.0101 3Lgr/ = 7.0303 4Lgf/ = 6.0404 g 8 = 0.0011 c/ 4 = 0.0001 Lg(l — Vk') = 9.53120 Lg(l + VP) =0.22017 cliff. = 9.31103 Lg2 = 0.30103 Lg term 1 = 9.01000 Lg2 7r = 0.7982 2Lg(l+ Vk') =9.5597 Lg(l+ 2q*) = 0.0001 LgK = 0.3580 K = 2.280 5 dill = 6.55515 LglG = 1.20412 Lg term 2= 5.35103 terml = 0.10233 term 2 = 0.00002 q = 0.10235. Lg \ tt/VP = 0.3764 | log 2 ir/ K = 0.6603 Lgg = 9.0101 Lg(l-4g 3 ) = 9.9981 ~Lg{K-E) =0.0449. Hence K — E = 1.109 and E = 1.171. The quadrant is 1.171a. Th e point cor- responding to x = i a is given by $ LgdnF= 9.9509 Lg Vk' = 9.8197 LgcotX = 0.1312 X = 36° 28£' Now 180 F = K (42.92). The computation for F, Z, E{\ir) is then i = 30°. Then dn F = VI - 0.2025. Ltt-X = 8°31i' cos 2 j/ = 0.7323 Lg tan = 9.1758 Hence 4 » near 90° Lg 2 q = 9.3111 1+ 2? 4 cos4i> = 1.0000 ' cos 2 v = 9.8647 2 v = 42° 55'. LgA' = 0.3580 Lg 42.92 = 1.6326 -Lgl80= 7.7447 LgF= 9.7353 F = 0.5436 Lg 2 tt/K = 0.4402 Lgg = 9.0101 Lgsin2i> = 9.8331 - Lg (l-2q cos 2 x) = 0.0705 Lsr Z = 9.3539 ~LgE/K = 9.7106 LgF= 9.7353 .EF/A = 0.2792 Z = 0.2256* E(lir) = 0.5048. The value of Z marked * is corrected for the term — 2c/ 3 sin 4 v. The part of the quadrant over the first half of the axis is therefore 0.5048 a and 0.666 a over the second half. To insure complete four-figure accuracy in the result, five places should have been carried in the work, but the values here found cheek with the table except for one or two units in the last place. EXERCISES 1. Prove the following relations for Z(u) and 7j x {u). Z (- w) = - Z (w), Z (u + 2 K ) = Z (u), Z (u + 2 IK') = Z (w) - iir/K. If W = flog//<«) = "^ du II (u) 1 sn' 2 u Z 1 (u + iK') = Z(u)- — du Z[(u) + Z'(0), f _f- = - Z^h) + «Z'(0), J sn 2 u 7,(0) = oo. d , cn« dn » Z, (u) — Z (m) = — log sn u = du ' sn u 516 THEORY OF FUNCTIONS 2. An elliptic function with periods 2 K, 2 IK' and simple poles at a^ a 2 , • ■ • , a n with residues c 1 , c 2 , ■ • ■ , c„. Zc = 0, may be written f(u) = qZ^u — a,) + c 2 Z 1 (m — a 2 ) + ■ • • + c„Z,(u — a„) 4- const. „ i 2 snacnadn«sn 2 ii 1 . 1 . . _,. . 3 — , — r^ — i i — = o Z (" ~ a ) _ o Z <" + a > + Z < a )' 1 — A; 2 sn- a sir u 2 2 r" sn 2 wdn 1. 8(a — u) _., % fc 2 snacnadna f — ; — = -log— - + u7*'{a). Jo 1 — k' 2 sn 2 « sn'- 2 a 2 6 (« + m) r \du ,,, >~ r /' \ i- en V Xw dn \ Xu 4. (a) I — = XuZ'(O) - a XZ(VXu) - V X — - + C J sn- \ Xu sn \ Xu r-_/ . - \ -en Vxu dn a Xu „ = Xw — V\E( = sin- 1 sua Xu)— VX h C, sn a Xu % r k" 2 du r, ., , .., snucnu . ,„ snu cum (/3) I = l dn 2 udu — k 2 - — = E( = sin- 1 snu) — k 2 , J dn- u J dn u dn u . , r cn 2 udu . _,. . , , en m „ , , „ » (7) I = u — 2E((p = sin- 1 sn u) -f (1 — 2 dn 2 u). J sn- u dn- u sn u dn u 5. Find the length of the quadrant and of the portion of it cut off by the latus rectum in ellipses of eccentricity e = 0.1, 0.5, 0.75. 0.95. 6. If e is the eccentricity of the hyperbola x 2 /a 2 — i/' 2 /b' 2 = 1, show that b- r * sec' 2 0c70 ae ,1 s = — I — — , where — y = tan 0, A; = - , aeJo Vl — A; 2 sin 2 ^' 2 e /j 2 = — F(0, /l) — acE( b > c, find the area of one octant. 1 ,, irub r c 2 _, _ it' 2 — r' 2 _, , N ~1 c - 7TC 2 + -- - F (0, ft) + — £ (0, fc) , COS = - , A-- 4 4 sin L« "" « 11. Compute the area of the ellipsoid with axes 3, 2, 1. a 2 - <■■■ 7/ 2 sin 2 1 12. A hole of radius b is bored through a cylinder of radius a>b centrally and perpendicularly to the axis. Find the volume cut out. 13. Find the area of a right elliptic cone, and compute the area if the altitude is 3 and the semiaxes of the base are 1J and 1. ELLIPTIC FUNCTIONS 517 192. Weierstrass's integral and its inversion. In studying the general theory of doubly periodic functions (§ 182), the two special functions p(u), l'\ u ) Avere constructed and discussed. It was seen that z -s: W-9* (22) V4(it'-« 1 )(w-e J )(w-eJ where the fixed limit x lias been added to the integral to make w = x and z = correspond and where the roots have been called e , e 2 , e . Conversely this integral could be studied in detail by the method of mapping ; but the method to be followed is to make only cursory use of the conformal map sufficient to give a hint as to how the function p(z) may be expressed in terms of the functions sn z and en -.-. The discussion will be restricted to the case which arises in practice, namely, w, _i, 1 J-?+~- ^-l"— when ^ and g are real quantities. 2' 2 -co" cTt^o e x r '+co There are two cases to consider, one when all three roots are real, the other when one is real and the other two are conjugate imaginary. The root e x will be taken as the largest real root, and e„ as the smallest root if all three are real. Note that the sum of the three is zero. In the case of three real roots the Eiemann surface may be drawn with junction lines e. 2 , e,, and e v x. The details of the map may readily be filled in, but the observation is sufficient that there are only two essentially different paths closed on the surface, namely, about e. 2 , e z (which by deformation is equivalent to one about e , cc) and about e 3 , e x (which is equivalent to one about e , — x). The integral about e 2 , e a is real and will be denoted by 2 w , that about e 3 , e x is pure imaginary and will be denoted by 2 w.,. If the function p (z) be constructed as in § 182 with a) = 2 <«) , w' = 2 w., the function will have as always a double pole at z = 0. As the periods are real and pure imaginary, it is natural to try to express p (,~) in terms of sn z. Asp(.~) depends on two constants (I.,, fj , whereas sn z depends on only the one /.-, the function p (z) will be expressed in terms of sn ( Va.v. k), where the two constants A, /.• are to be determined so as to fulfill the identity p n = 4y/ 3 — g^p — g r In particular try p(z) = A H — ? -1, A, /.• constants. sn 2 (VA.", h) 518 THEORY OF FUNCTIONS This form surely gives a double pole at z — with the expansion 1/z 2 . The determination is relegated to the small text. The result is *(*) = «,+ e, — e„ "<1, (23) 5ii 2 ( Vxz, k) 0, WjVX = A', a> 2 Va = a'A*'. In the case of one real and two conjugate imaginary roots, the Riemann surface may be drawn in a similar manner. There are again two independent closed paths, one about e.„ e s and another about e 3 , e.. Let the integrals about these paths be respectively 2 w 1 and 2 w,. That 2 Wj is real may be seen by deforming the path until it consists of a very distant portion along which the integral is infinitesimal and a path in and out along e v cc, which gives a real value to the integral. As 2 w, is not known to be pure imaginary and may indeed be shown to be complex, it is natural to try to express p (-) in terms of en z of which one period is real and the other complex. Try l + cn(2 VjLs, k) p(z)=A +/* l-cn(2 -\Zfxz, k) This form surely gives a double pole at z = with the expansion 1/z 2 The determination is relegated to the small text. The result is P 0) = «i + V- l + cn(2Vu*, k k 1 < 1, l-cn(2 vV», k) fS = (e x - i-(A-MA'). (23') p {z)=A + 4\» sn 2 (V\z, fc) (1- s?i 2 )(l- fc 2 sn 2 ) p'(z) = 2X2 Sir : (V\z, k) cn(Vxz, &)dn(Vxz, fc), 4/13 + 3 /i 2 X 3 .IX 2 X 3 sir sn' sn- Equate coefficients of corresponding powers of sn' 2 . Hence tlie equations 4 .4 3 - g 2 A -g a = 0, 4 X% 2 = 12 A 2 - g,,\, - X (1 + k 2 ) - 3 A. ELLIPTIC FUNCTIONS 519 The first shows that A is a root e. Let A = e.,. Note — g. z = e 1 e 2 + e 1 e 3 + e 2 e 3 . X • \Jc~ — 3 e 2 2 + t^c + CjCy + e 2 e 3 = (e l — e 2 )(e 3 — e 2 ), X + XA:- = — 3 e, = e 1 — e 2 + e 3 — e 2 , by virtue of the relation e Y + e., + e 3 = 0. The solution is immediate as given. To verify the second determination, the substitution is similar. l+cn2v^ //N 4/izsndn p(z) =A+fi [i>'(2)]2 = 16 M 8 1 — en 2 V/xz (1 + cn)(fc' 2 + A;' 2 en 2 ) p'(z) (1 - en) 2 = 4 M 3[i 3 + 2(l-2k 2 )t- + t] (1 - en) 3 where t = (1 + en)/(l — en). The identity p' 2 = 4p 3 — g n p — g s is therefore 4 M 3 ^3 + 2 (1 _ 2 A; 2 ) I- + t] = 4 (.43 + 3 4 V + 3 yM 2 + /* 8 * 3 ) -<7 2 ^ - g,fxt - g 3 . 4 .4 3 - g„A -g,, = 0, 4 M 2 = 12.4 - - -<•/.,. 2 m (1-2 A; 2 ) = 3 J.. Here let A — e,. The solution then appears at once from the forms f = 3 e 2 + e,e, + e,e 3 + e 2 e 3 = (e, - e 2 )(e, - e 3 ), m(1 - 2 A- 2 ) = 3 A/2. The expression of the function p in terms of the functions already studied permits the determination of the value of the function, and by inversion permits the solution of the equation p> (z) = c. The function p(z) may readily be expressed directly in terms of the theta series. In fact the periodic properties of the function and the corresponding- properties of the quotients of theta series allow such a representation *e 2 p>e 2 2> P >0 £ 3 £ .§ ^ e. P'<° B ■2 2 2r«3, to be made from the work of § 175, provided the series be allowed com- plex values for q. But for practical purposes it is desirable to have the expression in terms of real quantities only, and this is the reason for a different expression in the two different cases here treated.* The values of z for which 2 } ( z ) 1 3 rea l ma y be read off from (23) and (23') or from the correspondence between the ^--surface and the s-plane. They are indicated on the figures. The functions/; and/?' may be used to express parametrically the curve V' = -i ■'■ 3 - g& - g 3 by y = p'(z), x=p (*)• * It is. however, possible, if desired, to transform the given cubic 4 w 3 — g. z iv — g 3 with two complex roots into ;i similar cubic with all three roots real and thus avoid the dupli- cate forms. The transformation is not given here. 520 THEORY OF FUNCTIONS z=w„i-u .e 3 e i (z=u 1 'Z=l>3\\f- Wo 'Z=U),+ W^rU The figures indicate in the two cases the shape of the curves and the range of values of the parameter. As the function p is of the second order, the equation p (z) = c has just two roots in the parallelogram, and as p (z) is an even function, they will be of the form z — a and z = 2 — a and be symmetri- cally situated with respect to the cen- ter of the figure except in case a lies on the sides of the parallelogram so that 2 Wj + 2 oj., — a would lie on one of the excluded sides. The value of the odd function j?' at these two points is equal and opposite. This corresponds precisely to the fact that to one value x = c of x there are two equal and opposite values of y on the curve if = 4 x 3 — tj 2 x — g 3 . Conversely to each point of the parallelo- gram corresponds one point of the curve and to points symmetrically situated with respect to the center correspond points of the curve sym- metrically situated with respect to the a^-axis. Unless z is such as to make both p {£) and _£>'(;«:) real, the point on the curve will be imaginary. 193. The curve y 2 = 4x 3 — g x — g 3 may be studied by means of the properties of doubly periodic functions. For instance Ax + By + C = Ap'(z) + Bp(z) + C = is the condition that the parameter z should be such that its representative point shall lie on the line Ax + By +0 = 0. But the function Ap'(z) + Bp(z) + C is doubly periodic with a pole of the third order ; the function is therefore of the third order and there are just three points z v z 2 , z 3 in the parallelogram for which the function vanishes. These values of z correspond to the three intersections of the line with the cubic curve. Now the roots of the doubly periodic function sat- isfy the relation z i + z 2 + ~ :! — 3 x = 2 ??t 1 w 1 + 2 wi 2 w 2 . It may be observed that neither ?», nor »i„ can be as great as 3. If conversely z v z n , z 3 are three values of z which satisfy the relation z 1 + z. 2 + z 3 = 2 m i u 1 + 2»i 2 w 2 , the three corresponding points of the cubic will lie on a line. For if z 3 be the point in which a line through z v z, cuts the curve, Z l + Z 1 + Z '-A — - m i w l + - m 2 W 2' Z 3 — Z 'i = ~ ('"l — m 'l) W \ + " ( ?W 2 ~~ m '-l) W -2» and hence z.,, z' 3 are identical except for the addition of periods and must therefore be the same point on the parallelogram. One application of this condition is to find the tangents to the curve from any point of the curve. Let z lie the point from which and z' that to which the tangent is drawn. The condition then is z + 2z' = 2 m.u. + 2 m.,u.,, and hence z' = — \ z, z' = — \ Z + Wj , z' — — \ z + w„, z' = — I Z + Wj + w 2 are the four different possibilities for z' corresponding to m l = m„ = : ?« t = 1, m., = ; //(j = 0, »*., = 1 ; m i = 1, ?»., = 1. To give other values to >n l or ///., would ELLIPTIC FUNCTIONS 521 merely reproduce one of the four points except for the addition of complete periods. Hence there are four tangents to the curve from any point of the curve. The question of the reality of these tangents may readily be treated. Suppose z denotes a real point of the curve. If the point lies on the infinite portion, < z < 2 w v and the first two points z' will also satisfy the conditions < z' < 2 u 1 except for the possible addition of 2u>j. Hence there are always two real tangents to the curve from any point of the infinite branch. In case the roots e v e„, e 3 are all real, the last two points z' will correspond to real points of the oval portion and all four tangents are real ; in the case of two imaginary roots these values of z' give imag- inary points of the curve and there are only two real tangents. If the three roots are real and z corresponds to a point of the oval, z is of the form w., + u and all four values of z' are complex, and none of the tangents can be real. The discussion is complete. As an inflection point is a point at which a line may cut a curve in three coin- cident points, the condition 3z = 2??i,w 1 + 2??i.,w., holds for the parameter z of such points. The possible different combinations for z are nine : z = § « 2 | w 2 f w l f w l + f w 2 f W l+.f W 2 f«l t W l+f W 2 |«l + f w 2- Of these nine inflections only the three in the first column are real. When any two inflections are given a third can be found so that z x + z., + z 3 is a complete period, and hence the inflections lie three by three on twelve lines. If p and ]/ be substituted in Ax 2 + Hxy + ('//- + l)x + Ey + F, the result is a doubly periodic function of order 6 with a pole of the (3th order at the origin. The function then has zeros in the parallelogram connected by the relation z t + z. 2 + z s + z 4 + z. + z = 2 m 1 w 1 + 2 m 2 w 2 , and this is the condition which connects the parameters of the points in which the cubic is cut by the conic Ax 2 + Bxy + Cy 2 + Bx + Ey + F= 0. One applica- tion of interest is to the discussion of the conies which may be tangent to the cubic at three points z t , ?.,, z.,. The condition then reduces to z x + 2., + z., = fftjW, + m 2 w . If m v m. 2 are or any even numbers, this condition expresses the fact that the three points lie on a line and is therefore of little interest. The other possibilities, apart from the addition of complete periods, are z x + z., + z ?i = «,, z x + z 2 + z 3 = w 2 , z x + z. 2 + z 3 = u> y + w 2 . In any of the three cases two points may be chosen at random on the cubic and the third point is then fixed. Hence there are three conies which are tangent to the cubic at any two assigned points and at some other point. Another application of interest is to the conies which have contact of the 5th order with the cubic. The condition is then 6z = 2m 1 w ] + 2m.,w.,. As m t , m may have any of the values from to 5, there are 30 points on the cubic at which a conic may have contact of the 5th order. Among these points, however, are the nine inflections obtained by giving m x . m 2 even values, and these are of little interest because the conic reduces to the inflectional tangent taken twice. There remain 27 points at which a conic may have contact of the 5th order with the cubic. 522 THEORY OF FUNCTIONS EXERCISES 1. The function f (z) is denned by the equation -f'(2) = p(z) or f(z) = - fi)(z)dz = ---c 1 2 3 + .... •^ Z 6 Show by Ex. 4, p. 516, that the value of f in the two cases is en VXzdn VXz f(z) = - e x z + V\E(4>, k) + v X sn VX; t(z) = -( fJ L+e l )z + 2V^E ( is positive and less than 180 D . 13. Let a(z) = cA ** 6(z) - , ^(z) = e^> " ^ (z) . O'(0) ' a (O) Prove a(z + 2w x ) = — e 2l, i(* + <,, i)„, with a pole I/2- of the second order at z = and with no constant term in its development. State why this identifies p (2) with the function of the text. CHAPTER XX FUNCTIONS OF REAL VARIABLES 194. Partial differential equations of physics. In the solution of physical problems partial differential equations of higher order, partic- ularly the second, frequently arise. With very few exceptions these equations are linear, and if they are solved at all, are solved by assum- ing the solution as a product of functions each of which contains only one of the variables. The determination of such a solution offers only a particular solution of the problem, but the combination of different particular solutions often suffices to give a suitably general solution. For instance c 2 V c 2 V c 2 V lcV 1 c-Y ■7r-; + ^—;= Or -r-H —+——-= (1) ex cy or v cr r~ C(f>- is Laplace's equation in rectangular and polar coordinates. For a solu- tion in rectangular coordinates the assumption V = X(x) Y(y) would be made, and the assumption I' = R (/•)(<£) for a solution in polar coor- dinates. The equations would then become X" Y" r-R" R' $" — + — = or +,. — + — = 0. (2) A Y R R <5 Now each equation as written is a sum of functions of a single variable. P>ut a function of x cannot equal a function of y and a function of r cannot equal a function of <£ unless the functions are constant and have the same value. Hence A" — = — m~, Y" Y = + m , or r 2 R" R' -ir+'-ji (20 + VI'. These are ordinary equations of the second order and may be solved as such. The second case will be treated in detail. The solution corresponding to any value of m is <£> = a m cos i/xf> + b m sin m, R = A w r'" 4- R m r~ '" and V = R(p = (A m r m 4- B m r" '")(",„ cos )»cf> 4- b m sin m + h ™ sin m *)" ( 3 ) That ain- number of solutions corresponding to different values of m may be added together to give another solution is due to the linearity of the given equation (§ 96). It may be that a single term will suffice as a solution of a given problem. But it may be seen in general that : A solution for V may be found in the form of a Fourier series which shall give V any assigned values on a unit circle and either be conver- gent for all values within the circle or be convergent for all values outside the circle. In fact let f(4>) be the values of V on the unit circle. Expand f(4>) into its Fourier series f() = \ a {) + "V (n )tl cos w + b„, sin ///<£) (3') m will be a solution of the equation which reduces to /'(<£) on the circle and, as it is a power series in r, converges at every point within the circle. In like manner a solution convergent outside the circle is V=\ a + V /-"' (",„ cos m$ + b m sin m). (3") The infinite series for V have been called solutions of Laplace's equation. As a matter of fact they have not been proved to be solutions. The finite sum obtained by taking any number of terms of the scries would surely be a solution ; but the limit of that sum when the series becomes infinite is not thereby proved to be a solu- tion even if the series is convergent. For theoretical purposes it would be necessary tn give the proof, but the matter will be passed over here as having a negligible bearing on the practical solution of many problems. For in practice the values of f() on the circle could not be exactly known and could therefore be adequately represented by a finite and in general not very large number of terms of the de- velopment of /(0), and these terms would give only a finite series for the desired function V. In some problems it is better to keep the particular solutions sepa- rate, discuss each possible particular solution, and then imagine them compounded physically. Thus in the motion of a drumhead, the most general solution obtainable is not so instructive as the particular solution corresponding to particular notes; and in the motion of the surface of the ocean it is preferable to discuss individual types of waves and com- pound them according to the law of superposition of small vibrations (p. 22G). For example if 526 THEORY OF FUNCTIONS be taken as the equation of motion of a rectangular drumhead, T _ fsin c Va'- -f- fiH I cos c Vrr -f fiH fsin ax, .. _ fsin fix, Loos ax, I cos fix, are particular solutions which may be combined in any way desired As the edges of the drumhead are supposed to be fixed at all times, z = if oe = 0, x = a, y = 0, y = 5, £ = anything, where the dimensions of the head are « by h. Then the solution W77-.'' . viri/ \ur n z = X YT = sm — — sm —f- cos cir \ — r + — z 1 (J) is a possible type of vibration satisfying the given conditions at the perimeter of the head for any integral values of m, n. The solution is periodic in t and represents a particular not; 1 which may be omitted. A sum of such expressions multiplied by any constants would also be a solution and would represent a possible mode of motion, but would not be periodic in 1 and would represent no note. 195. For three dimensions Laplace's equation becomes cr \ cr 1 C 2 V 1 d ■ «0V siu-0 c in polar coordinates. Substitute V = 11 (r)®(6)) ; then Id/ .,ie \ 6 - ' Here the first term involves r alone and no other term involves r Hence the first term must be a constant, say. n (n -\- 1). Then d I , (111 • ICrX Vr (» + 1)72 = 0, 72 =Ar" + Br Next consider the last term after multiplying through by siir#. It ap pears that ~ ! " is a constant, say, — nr. Hence " = — nr = a m cos in (ft -f- h m sin ni. Moreover the equation for now reduces to the simple form d (I CO (I — cos 2 0) d® d cos e n(n +1) = 0. The problem is now separated into that of the integration of three differential equations of which the first two are readily integrable. The third equation is a generalization of Legendre's (Exs. 13 - 17, p. 252). REAL VARIABLES 527 and in case n, m are positive integers the solution may be expressed in terms of polynomials P n m (cos 0) in cos 6. Any expression 2 (A J* + B n r ~ n ~ 1 )( a m C0S m + h m sin '" <£) P n, m (COS 6) is therefore a solution of Laplace's equation, and it may be shown that by combining such solutions into infinite series, a solution may be obtained which takes on any desired values on the unit sphere and converges for all points within or outside. Of particular simplicity and importance is the case in which V is sup- posed independent of so that m = and the equation for © is soluble in terms of Legendre's polynomials P n (cos 6) if n is integral. As the potential V of any distribution of matter attracting according to the in- verse square of the distance satisfies Laplace's equation at all points exterior to the mass (§ 201), the potential of any mass symmetric with respect to revolution about the polar axis = may be expressed if its expression for points on the axis is known. For instance, the poten- tial of a mass M distributed along a circular wire of radius a is r Mi 1 /•- 1-3 >' 4 1-3-5 j fi M l7( 1 -2^ + ^4^"in7o^ + -'-' r < a > V = , = A 1 -3-o /' r = < ! M (a 1 a 3 1 3 « 5 1-3-5 ) so that V shall reduce to assigned values on certain lines. In fact (p. 466) f(x) = - I f f(\) cos to (X - x) d\dm. (7) Hence if the limits for to be and co and if the choice i /" +co i r +o ° « (to) = — I ,/'(^) (!0S mXd\, b (m) — — | /(^) sm ?»A^/A is taken for a (in), b(nt), the expression (6) for F becomes V = - f \ e- m,J f(\) cos m(X — x)dXdm (8) and reduces to /(.>•) when // = 0. Hence a solution V is found which takes on any assigned values f(x) along the ;r-axis. This solution clearly becomes zero when y becomes infinite. When f(x) is given it is some- times possible to perform one or more of the integrations and thus simplify the expression for V. For instance if f(x) = 1 when x > and f(x) — when x < 0, the integral from — co to drops out and V = f f v->»y • 1 • cos m (X — x) d\dm = f f e~ '"■'' cos m (X — x) dmd\ TT J J TT J J 1 r ' ?/dX \ lir , x \ 1 , y = I — •-- = + Ian- 1 ) = 1 tan- 1 -. tt J o if 1 -f (X — x)'- 7r \2 y/irx REAL VARIABLES 529 It may readily be shown that when y > the reversal of the order of integration is permissible ; but as V is determined completely, it is simpler to substitute the value as found in the equation and see that V' x ' x + V' y ' y = 0, and to check the fact that V reduces to f(x) when y = 0. It may perhaps be superfluous to state that the proved correctness of an answer does not show the justification of the steps by which that answer is found ; but on the other hand as those steps were taken solely to obtain the answer, there is no practical need of justifying them if the answer is clearly right. EXERCISES 1. Find the indicated particular solutions of these equations : >-V _c 2 V ct CX 2 (a) c' 2 — = — - , V = 2_, A»fi~ '""' («m cos cmx + b m sin cmx), rt, fix' 2 *~l (/J) = . V = ^ (A m cos cmt + B m sin cmt)(a m cos mx + b m sin mx), c' 2 ct' 2 dx 2 ^-4 (y) C 2?X = &V PV t x= (smcax Y= (sin eft, T=e -V + M ct dx 2 cy 2 icos cax, [cos eft/, 2. Determine the solutions of Laplace's equation in the plane that have V =1 for < < 7r and V = — 1 for tt <

    art; known to be ex 2 c?/ 2 Lff/ J .'/ = -'' L<^ 2 <"2/J?/ = o Find and combine particular solutions to show that 4> may have the form (p = A cosh k(tj + h) cos (fcc — ?ti), n 2 = t/A: tanh /c/i. 9. Obtain the solutions or types of solutions for these equations. , . c 2 V c 2 V 1 cV 1 d 2 V n , A , fcosmtffj _ .. . x ; c2 2 rr 2 r cr r 2 d 2 L' S1U Vl( t> J (|8) — - H ■ h —„ - — r + F = 0, J.ns. V (amCOsm^ + bnjSinm^Jn^r), er 2 »' cr r- ?<£- ~4 (7) ^ + S + S+ ^= ' ^«- r " ij .+ lW P --( C08 ^ X {a n ,m cos ?/i0 + 6„, m sin m0), ex' 2 S//'- C2" 2 r jr f 2 F . . i 3 2 v c 2 v c 2 v c 2 v (-2 = , (e) -= h 1 Zi 2 Zt dx 2 c 2 U 2 ex 2 dy 2 cz 2 10. Find the potential of a homogeneous circular disk as (Ex. 22, p. 68 ; Ex. 23, p. 332) 2 M Via 1.1 a 3 1 ■ 1 ■ 3 a 5 1 • 1 • 3 • 5 « 7 r > a, a \2 r 2 ■ 4 r 3 ' 2 • 4 • r 5 4 2 • 4 • 6 • 8 r 7 2 M |\ r _ lr 2 , Mr* 1 • 1 • 3 ?•« 1 = lT-?,+ ?o P 4 + - — P fi , r < a, a L a 2 a 2 2 2 • 4 a 4 4 2 • 4 ■ 6 a 6 6 J where the negative sign before P 1 holds for 6 < l tt and the positive for > \ir. 11. Find the potential of a homogeneous hemispherical shell. 12. Find the potential of (a) a homogeneous hemisphere at all points outside the hemisphere, and (/3) a homogeneous circular cylinder at all external points. 13. Assume — - cos -1 — — — is the potential at a point of the axis of a conduct- 2 a x' 2 + a 2 ing disk of radius a charged with Q units of electricity. Find the potential anywhere. 196. Harmonic functions ; general theorems. A function which satisfies Laplace's equation F^+ r^ = 0or 1",',+ '^+ f^ = 0, whether in the plane or in space, is called a harmonic function. It is assumed that the first and second partial derivatives of a harmonic function are continuous except at specified points called singular points. There are many similarities between harmonic functions in the plane and har- monic functions in space, and some differences. The fundamental theo- rem is that: If a function is harmonic and has no singularities vjion or within a simple closed curve (or surface}, the line integral of its nor- mal derivative along the curve (respectively, surface} vanishes', and con- versely if a function !*(,/■, //), or V (x, y, z\ has continuous first and second REAL VARIABLES 531 partial derivatives and the line integral (or surface integral) along every closed curve (or surface) in a region vanis/ies, the function is harmonic. For by Green's Formula, in the respective cases of plane and space (Ex. 10, p. 349), r dv , rev . dv 7 rr/c-v d 2 r\ , 7 f -r~ as = j -7T- (hi — -77— dx = II ( -rr-r + ~~ ^7 I dxdii, Jo dn Jo Cx h J J J \ Cx C 'J'I (9) Now if the function is harmonic, the right-hand side vanishes and so must the left ; and conversely if the left-hand side vanishes for all closed curves (or surfaces), the right-hand side must vanish for every region, and hence the integrand must vanish. If in particular the curve or surface be taken as a circle or sphere of radius a and polar coordinates be taken at the center, the normal de- rivative becomes dV/dr and the result is \ -^4 = or f f -f- sin 6d0dcf> = 0, Jo cr Jo Jo cr where the constant a or or the element of surface a 2 sin 6. If these equations be inte- grated with respect to r from to a, the integrals may be evaluated In- reversing the order of integration. Thus =P h f '£«+ =f£ d i ■"■"* =f " (r « - "«> "*■ and I V a d = r d, or V a = F , (10) Jo Jo where V a is the value of V on the circle of radius a and l r is the value at the center and V a is the average value along the perimeter of the circle. Similar analysis would hold in space. The result states the important theorem: The average value of a harmonic function over a circle (or sphere) is equal to the value at the center. This theorem has immediate corollaries of importance. A harmonic function which has no singularities within a region cannot become maxi- mum or minimum at any point within the region. For if the function were a maximum at any point, that point could be surrounded by a circle or sphere so small that the value of the function at every point of the contour would be less than at the assumed maximum and hence the average value on the contour could not be the value at the center. 532 THEORY OF FUNCTIONS A harmonic function which has no singularities within a region and is constant on the boundary is constant throughout the region. For the maximum and minimum values must be on the boundary, and if these have the same value, the function must have that same value through- out the included region. Two harmonic functions which hare identical values upon a closed contour and have no singularities within, are iden- tical throughout the included region. For their difference is harmonic and has the constant value on the boundary and hence throughout the region. These theorems are equally true if the region is allowed to grow until it is infinite, provided the values which the function takes on at infinity are taken into consideration. Thus, if two harmonic functions have no singularities in a certain infinite region, take on the same values at all points of the boundary of the region, and approach the same values as the point (,r, y) or (./■, y, z) in any manner recedes indefinitely in the region, the two functions are identical. If Green's Formula be applied to a product UdV/dn, then r dv 7 r dv , dv 7 / U — ds= I U-rdy—U-r-dx Jo dn Jo dx d 'J =jJr()Z+v;;,,)d.rdy + jJ(r; : v;.+ r;v; /) d.rdy, or CudS»VV = /''V-VlWr + ( VT-VlWr (11) in the plane or in space. In this relation let V be harmonic without singularities within and upon the contour, and let U = V. The first inte- gral on the right vanishes and the second is necessarily positive unless the relations V' x = \" v = or \" x = \",, = \" z = 0, which is equivalent to V V = 0, are fulfilled at all points of the included region. Suppose further that the normal derivative dV/dn is zero over the entire bound- ary. The integral on the left will then vanish and that on the right must vanish. Hence 1' contains none of the variables and is constant. /g' the normal derivative of a function harmonic and devoid of singular- ities at all points on and within a given contour vanishes Identically upon the contour, the function Is constant. As a corollary: If two functions are harmonic and devoid of singularities upon and within a given contour, and if their normal derivatives are identically equal upon the contour, the functions differ at most by an additive constant. In other words, a harmonic function without singularities not only Is determined l>y Its values on a contour hut also (except for an additive constant) by the values of Its normal derivative upon a contour. REAL VARIABLES 533 Laplace's equation arises directly upon the statement of some problems in physics in mathematical form. In the first place consider the flow of heat or of electricity in a conducting body. The physical law is that heat flows along the direction of most rapid decrease of temperature T, and that the amount of the flow is proportional to the rate of decrease. As — V7 1 gives the direction and magni- tude of the most rapid decrease of temperature, the flow of heat may be represented by — kVT, where A; is a constant. The rate of flow in any direction is the compo- nent of this vector in that direction. The rate of flow across any boundary is therefore the integral along the boundary of the normal derivative of T. Now the flow is said to be steady if there is no increase or decrease of heat within any closed boundary, that is ~ kl dS«Vr=0 or T is harmonic. Hence the problem of the distribution of the temperature in a body supporting a steady flow of heat is the problem of integrating Laplace's equation. In like manner, the laws of the flow of electricity being identical with those for the flow of heat except that the potential V replaces the temperature T, the problem of the distribution of potential in a body supporting a steady flow of electricity will also be that of solving Laplace's equation. Another problem which gives rise to Laplace's equation is that of the irrotational motion of an incompressible fluid. If y is the velocity of the fluid, the motion is called irrotational when Vxv = 0, that is, when the line integral of the velocity about any closed curve is zero. In this case the negative of the line integral from a fixed limit to a variable limit defines a function <£(./:. y, z) called the velocity potential, and the velocity may be expressed as v =— V4>. As the fluid is incom- pressible, the flow across any closed boundary is necessarily zero. Hence f(ZS.V = or fv.V is a harmonic function. Both these problems may be stated without vector notation by carrying out the ideas involved with the aid of ordinary coordinates. The problems may also be solved for the plane instead of for space in a precisely analogous manner. 197. The conception of the flow of electricity will be advantageous in discussing the singularities of harmonic functions and a more gen- eral conception of steady flow. Suppose an electrode is set down on a sheet of zinc of which the perimeter is grounded. The equipotential lines and the lines of flow which are orthogonal to them may be sketched in. Electricity passes steadily from the electrode to the rim of the sheet and off to the ground. Across any circuit which does not surround the electrode the flow of electricity is zero as the flow is stead}", but across any circuit surrounding the electrode there will be a certain definite flow ; the circuit integral of the normal derivative of the potential V around such 534 THEORY OF FUNCTIONS a circuit is not zero. This may be compared with the fact that the circuit integral of a function of a complex variable is not necessarily zero about a singularity, although it is zero if the circuit contains no singularity. Or the electrode may not be considered as corresponding to a singularity but to a portion cut out from the sheet so that the sheet is no longer simply connected, and the comparison would then be with a circuit which could not be shrunk to nothing. Concerning this latter interpretation little need be said ; the facts are readily seen. It is the former conception which is interesting. For mathematical purposes the electrode will be idealized by assum- ing its diameter to shrink down to a point. It is physically clear that the smaller the electrode, the higher must be the potential at the elec- trode to force a given flow of electricity into the plate. Indeed it may be seen that 1' must become infinite as — C log r, where r is the distance from the point electrode. For note in the first place that lug /• is a solu- tion of Laplace's equation in the plane; and let U— V + C log r or V = U — C log /•, where U is a harmonic function which remains finite at the electrode. The flow across any small circle concentric with the electrode is f> n -T?v ^-2-^ r — | — /v/0 = — / — rd + 2 irC = 2 ir< ', Jo Cr Jo cr and is finite. The constant C is called the strength of the source situ- ated at the point electrode. A similar discussion for space would show that the potential in the neighborhood of a source would become infinite as C Jr. The particular solutions — log r and 1/V of Laplace's equation in the respective cases may be called the fundamental solutions. The physical analogy will also suggest a method of obtaining higher singular- ities by combining fundamental singularities. Fur suppose that a powerful positive electrode is placed near an equally powerful negative electrode. Unit is. suppose a strong source and a strong sink near together. The greater part of the flow will be nearly in a straight line from the source to the sink, but some part of it will spread out over the sheet. The value of V obtained by adding together the two values for source and sink is V = -l C log (;•- + /'- - 2 rl cos . REAL VARIABLES 535 It was seen that a harmonic function which had no singularities on or within a given contour was determined by its values on the contour and determined except for an additive constant by the values of its normal derivative upon the contour. If now there be actually within the contour certain singularities at which the function becomes infinite as certain particular solutions V v V„, • • • , the function U = V — I \ — I'., — • • • is har- monic without singularities and may be determined as before. Moreover, the values of V v V 2 , • • ■ or their normal derivatives may be considered as known upon the contour inasmuch as these are definite particular solu- tions. Hence it appears, as before, that the harmonic function V is deter- mined by its values on the boundary of the region or {except for an additive constant) by the values of its normal derivative on the boundary, provided tin' singularities are sjiccifed in position and their modi: of becoming infin- ite is given in each case as some paH'icular solution of Laplace 's equation. Consider again the conducting sheet with its perimeter grounded and with a single electrode of strength unity at some interior point of the sheet. The potential thus set up has the properties that : 1° the poten- tial is zero along the perimeter because the perimeter is grounded ; 2° at the position P of the electrode the potential becomes infinite as — log r; and 3° at any other point of the sheet the potential is regular and sat- isfies Laplace's equation. This particular distribution of potential is denoted by G(P) and is called the Green Function of the sheet relative to I'. In space the Green Function of a region would still satisfy 1° and 3°, but in 2° the fundamental solution — log r would have to be replaced by the corresponding fundamental solution 1/r. It should be noted that the Green Function is really a function G(P) = G(a, b; x, y) or G(P) = G(a, b, c; x, y, z) of four or six variables if the position P(a, b) or l'(u, b, <■) of the elec- trode is considered as variable. The function is considered as known only when it is known for any position of /•". If now the symmetrical form of Green's Formula - JJ(»A. - ri. W +£[u f n - . £)* = 0, (12) where A denotes the sum of the second derivatives, be applied to the entire sheet with the exception of a small circle concentric with P and if the choice a = G and r = }' be made, then as G and 1' are harmonic the double integral drops out and C dG , r 2n dV r 27r dG (13) 536 THEORY OF FUNCTIONS Now let the radius /• of the small circle approach 0. Under the assump- tion that V is devoid of singularities and that G becomes infinite as — log r, the middle integral approaches because its integrand does, and the final integral approaches 2 7rJ r (P). Hence This formula expresses the values of 1 ' at any interior point of the sheet in terms of the values of V upon the contour and of the normal deriva- tive of G along the contour. It appears, therefore, that the determination of the value of a harmonic function devoid of singularities within and upon a contour may be made in terms of the values on the contour pro- vided the Green Function of the region is known. Hence the particular importance of the problem of determining the Green Function for a given region. This theorem is analogous to Cauchy's Integral (§ 126). EXERCISES 1. Show that any linear function ax + by + cz + d = is harmonic. Find the conditions that a quadratic function be harmonic. 2. Show that the real and imaginary parts of any function of a complex vari- able are each harmonic functions of (x, y). 3. Why is the sum or difference of any two harmonic functions multiplied by any constants itself harmonic ? Is the power of a harmonic function harmonic ? 4. Show that the product UV of two harmonic functions is harmonic when and only when U' X V' X + U'V' tl = or VU-VV = 0. In this case the two functions are called conjugate or orthogonal. What is the significance of this condition geometrically ? 5. Prove the average value theorem for space as for the plane. 6. Show for the plane that if V is harmonic, then U = I -7— f7.s = I dy — — dx J d)L J ex cy is independent of the path and is the conjugate or orthogonal function to V, and that U is devoid of singularities over any region over which 1" is devoid of them. Show that, 1' + ill is a function of z = x -f iy. 7. State the problems of the steady flow of heat or electricity in terms of ordi- nary coordinates for the case of the plane. 8. Discuss for space the problem of the source, showing that C/r gives a finite flow 4ttC. where C is called the strength of the source. Note the presence of the factor 4 7r in the place of '2 tv as found in two dimensions. 9. Derive the solution M)—- cos for the source-sink combination in space. REAL VARIABLES 537 10. Discuss the problem of the small magnet or the electric doublet in view of Ex. 9. Note that as the attraction is inversely as the square of the distance, the potential of the force satisfies Laplace's equation in space. 11. Let equal infinite sources and sinks be located alternately at the vertices of an infinitesimal square. Find the corresponding particular solution (a) in the case of the plane, and (£J) in the case of space. What combination of magnets does this represent if the point of view of Ex. 10 be taken, and for what purpose is the combination used ? 12. Express V(P) in terms of G(P) and the boundary values of V in space. 13. If an analytic function has no singularities within or on a contour, Cauchy's Integral gives the value at any interior point. If there are within the contour cer- tain poles, what must be known in addition to the boundary values to determine the function ? Compare with the analogous theorem for harmonic functions. 14. Why were the solutions in § 104 as series the only possible solutions provided they were really solutions? Is there any difficulty in making the same inference relative to the problem of the potential of a circular wire in § 195? 15. Let G(P) and G(Q) be the Green Functions for the same sheet but relative to two different points P and Q. Apply Green's symmetric theorem to the sheet from which two small circles about P and Q have been removed, making the choice u = G(P) and v = G(Q). Hence show that G (P) at Q is equal to G(Q) at P. This may be written as G(a, h; x, y) = G(x, y ; a, h) or G (a, b, c; x, y, z) = G(x, y, z ; u. b, c). 16. Test these functions for the harmonic property, determine the conjugate functions and the allied functions of a complex variable: (a) xy, (fl) x 2 y - § y 3 , (y) \ log (x- + y 2 ), (5) e*sinz, (e) sin x cosh ?/, (f) tan _1 (cot« tanh y). 198. Harmonic functions ; special theorems. For the purposes of the next paragraphs it is necessary to study the properties of the geo- metric transformation known as Inversion. The definition of inversion will be given so as to be applicable either to space or to the plane. The transformation which replaces each point P by a point P' such that OP- OP' = lr where is a given fixed point, /.• a constant, and P' is mi the line OP, is called inversion with tin- center <> end the radius /.'. Note that if ]' is thus carried into P', then P' will be carried into P ; and hence if any geometrical configuration is carried into another, that other will be carried into the first. Points very near to are carried off to a great distance; for the point itself the definition breaks down and corresponds to no point of space. If desired, one may add to space a fictitious point called the point at infinity and may then say that the center of the inversion corresponds to the point at infinity (p. 481). A pair of points P, P' which go over into each other, and another pair Q, Q' satisfy the equation OP- OP' = OQ • OQ'. 538 THEORY OF FUNCTIONS A curve which cuts the line <>P at an angle t is carried into a curve which cuts the line at the angle t' = it — t. For by the relation OP- OP' = OQ- OQ', the triangles OPQ, OQ'P' are similar and Z OPQ = Z OQ'P' = ir — Z — Z OP'Q'. Noav if Q = P and Q' = P', then Z = 0, Z OPQ = T , Z OP'Q' = t and it is seen that t = tv — t' or t' — ir — r. An immediate extension of the argument will show that the magnitude of the angle between two intersecting curves will be unchanged by the transformation; the transformation is therefore conformed. (In the plane where it is possible to distinguish between positive and neg- ative angles, the sign of the angle is reversed by the transformation.) If polar coordinates relative to the point be introduced, the equations of the transformation are simply /■/•' = Ir with the understanding that the angle <£ in the plane or the angles c£, 6 in space are unchanged. The locus r = k, which is a circle in the plane or a sphere in space, becomes /•' = Jc and is therefore unchanged. This is called the circle or the sphere of inversion. Relative to this locus a simple construction for a pair of inverse points P and P' may be made as indicated in the figure. The locus r + Jr — 2 Vrr -f Irr cos becomes Jr + /•'- = 2 Vcr + /„•->•' cos <£ and is therefore unchanged as a whole. This locus represents a circle or a sphere of radius a orthogonal to the circle or sphere of inversion. A construction may now be made for finding an inversion which car- ries a given circle into itself and the center P of the circle into any assigned point P' of the circle ; the construction holds for space by re- volting the figure about the line OP. To find what figure a line in the plane or a plane in space becomes on inversion, let the polar axis = or 6 = be taken perpendicular to the line or plane as the case may be. Then r = p sec eft. r' sec <£ = l?/p or /• = p sec 0, r' sec 8 = /■'"//> are the equations of the line or plane and the inverse locus. The locus is seen to be a circle or sphere through the center of hiversion. This may also be seen directly by applying the geometric definition of in- version. In a similar manner, or analytically, it may be shown that any circle in the plane or any sphere in space inverts into a circle or into a sphere, unless it passes through the center of inversion and becomes a line or a plane. REAL VARIABLES 539 If d be the distance of P from the circle or sphere of inversion, the distance of P from the center is k — d, the distance of P' from the center is k-/(k — d), and from the circle or sphere it is d' — dk/(k — d). Now if the radius k is very large in comparison with d. the ratio k/(k — d) is nearly 1 and d' is nearly equal to d. If k is allowed to become infinite so that the center of inversion recedes indefinitely and the circle or sphere of inversion approaches a line or plane, the distance d' approaches d as a limit. As the transformation which replaces each point by a point equidistant from a given line or plane and perpendicularly opposite to the point is the ordinary inversion or reflection in the line or plane such as is familiar in optics, it appears that reflection in a line or plane may be regarded as the limit- ing case of inversion in a circle or sphere. The importance of inversion in the study of harmonic functions lies in two theorems applicable respectively to the plane and to space. First, if V is harmonic over any region of tit e plane and if that region be inverted in any circle, the function V'(P'\ = V(P) formed by assign- ing the same value at P' in the new region as the function had at the point P which inverted into P' is also harmonic. Second, if V is har- monic over any region in space, and if that region be inverted in a sphere of radius k, the function F'(P') = kV(P)/r' formed by assigning at P' the value the function had at P multiplied by k and divided by the dis- tance OP' = /•' of P' from the renter of inversion is also harmonic. The significance of these theorems lies in the fact that if one distribution of potential is known, another may he derived from it by inversion; and conversely it is often possible to determine a distribution of poten- tial by inverting an unknown case into one that is known. The proof of the theorems consists merely in making the changes of variable r = k 2 /r' or /•' = Jr/r, <£' = <£, 6' = in the polar forms of Laplace's equation (Exs. 21, 22, p. 112). The method of using inversion to determine distribution of potential in electro- statics is often called the method of electric images. As a charge e located at a point exerts on other point charges a force proportional to the inverse square of the distance, the potential due to e is as l/p. where p is the distance from the charge (with the proper units it may be taken as e/p), and satisfies Laplace's equation. The potential due to any number of point charges is the sum of the individual potentials due to the charges. Thus far the theory is essentially the same as if the charges were attracting particles of matter. In electricity, however, the question of the distribution of potential is further complicated when there are in the neighborhood of the charges certain conducting surfaces. For 1° a conduct- ing surface in an electrostatic field must everywhere be at a constant potential or there would be a component force along the surface and the electricity upon it would move, and 2 : there is the phenomenon of induced electricity whereby a variable surface charge is induced upon the conductor by other charges in the neighborhood. If the potential V(P) due to any distribution <>f charges be inverted in any sphere, the new potential is k\'{l')//. As the potential V(P) 540 THEORY OF FUNCTIONS becomes infinite as c/p at the point charges e, the potential kV(P)/r' will become infinite at the inverted positions of the charges. As the ratio ds' : ds of the in- verted and original elements of length is r" 2 /k' 2 ,. the potential kV(P)/r' will become infinite as k/r' ■ c/p' ■ r" 2 /k 2 , that is, as r'c/kp'. Hence it appears that the charge c inverts into a charge e' — r'c/k ; the charge — e' is called the electric image of e. As the new potential is kY(P)/r' instead of V(P). it appears that an equipoten- tial surface V = const, will not invert into an equipotential surface V'(P') — const, unless V — or r' is constant. But if to the inverted system there be added the charge c = — kV at the center O of inversion, the inverted equipotential surface becomes a surface of zero potential. With these preliminaries, consider the question of the distribution of potential due to an external charge e at a distance r from the center of a conducting spheri- cal surface of radius k which has been grounded so as to be maintained at zero potential. If the system be inverted with respect to the sphere of radius k, the potential of the spherical surface remains zero and the charge e goes over into a charge c' = r'c/k at the inverse point. Now if p, p' are the distances from e, e' to the sphere, it is a fact of elementary geometry that p : p' — const. = r' : k. Hence the potential kp' - r'p kp' V = kpp' due to the charge e and to its image — e', actually vanishes upon the sphere ; and as it is harmonic and has only the singularity c/p outside the sphere (which is the same as the singularity due to e), this value of V throughout all space must be precisely the value due to the charge and the grounded sphere. The distribution of potential in the given system is therefore determined. The potential outside the sphere is as if the sphere were removed and the two charges e, — e' left alone. By Gauss's Integral (lis.. 8, p. 348) the charge within any region may be evaluated by a surface integral around the region. This integral over a surface surrounding the sphere is the same as if over a surface shrunk flown around the charge — c', and hence the total charge induced on the sphere is — e' — — r'c/k. 199. Inversion will transform the average value theorem 1 nr) = T. Vil into V'(P') 2^1 V'df, (14) a form applicable to determine the value of 1" at any point of a circle in terms of the value upon the circumference. For suppose the circle with (.•enter at /' and with the set of radii spaced at angles , as implied in the computation of the average value, he inverted upon an orthogonal circle so chosen that P shall go over into 1''. The given circle goes over into itself and the series of lines goes over into a series of circles through P' and the center f) of inversion. (The figures are drawn separately instead of superposed.) From the conformal property REAL VARIABLES 541 the angles between the circles of the series are equal to the angles be- tween the radii, and the circles cut the given circle orthogonally just as the radii did Let V along the arcs 1', 2', 3', ■ • ■ be equal to V along the corresponding arcs 1, 2, 3, • • • and let V(P) = V'(P') as required by the theorem on inversion of harmonic functions. Then the two inte- grals are equal element for element and their values V(P) and F'(P') are equal. Hence the desired form follows from the given form as stated. (It may be observed that d and d\\i, strictly speaking, have opposite signs, but in determining the average value V'(P'), d\j/ is taken positively.) The derived form of integral may be written as a line integral along the arc of the circle. If P' is at the distance r from the center, and if a be the radius, the center of inversion is at the distance a 2 /r from the center of the circle, and the value of k is seen to be k~ = (a 2 — ) fi )a 2 /r. Then, if Q and Q' be points on the circle, 7 ^Q' 2 ^O" - 2 « 3 ''" 1 C0S ' + « 4 ''~ 2 ) 7 ds = as —-r- = — -, t- — '- add). k~ (a- — r)ar Xow dif//ds' may be obtained, because of the equality of d\\i and d, and ds' may be written as add)'. Hence dd)'- ^ > 2ttJ d 2 -2ar COS d) + V- Finally the primes may be dropped from V' and P', the position of P' may be expressed in terms of its coordinates (?*, ^>), and is the expression of V in terms of its boundary values. The integral (15) is called Poisson's Integral. It should be noted par- ticularly that the form of Poisson's Integral first obtained by inversion represents the average value of V along the circumference, provided that average be computed for each point by considering the values along the circumference as distributed relative to the angle \p as independent vari- able. That V as defined by the integral actually approaches the value on the circumference when the point approaches the circumference is clear from the figure, which shows that all except an infinitesimal fraction of the orthogonal circles cut the circle within infinitesimal limits when the point is infinitely near to the circumference. Poisson's Integral may be 542 THEORY OF FUNCTIONS obtained in another way. For if P and P' are now two inverse points relative to the circle, the equation of the circle may be written as p/p' = const. = r/a, and G (P) = — log p + log p + log (r/a) (16) is then the Green Function of the circular sheet because it vanishes along the circumference, is harmonic owing to the fact that the logarithm of the distance from a point is a solution of Laplace's equation, and becomes infinite at P as — log p. Hence It is not difficult to reduce this form of the integral to (15). If a harmonic function is defined in a region abutting upon a segment of a straight line or an arc of a circle, and if the function vanishes along the segment or arc, the function may be extended across the segment or arc by assigning to the inverse point P' the value V(P') =— V(P), which is the negative of the value at P; the conjugate function U = = C< 1 f ds +C=C d ^ d y- d ^ dx+ C (17) J dn J ex J ci, takes on the same values at P and P'. It will be sufficient to prove this theorem in the case of the straight line because, by the theorem on inversion, the arc may be inverted into a line by taking the center of inversion at any point of the arc or the arc produced. As the Laplace operator Z)| + D* is independent of the axes (Ex. 25, p. 112), the line may be taken as the ir-axis without restricting the conclusion. Xow the extended function V (P') satisfies Laplace's equation since -!'(/>') c 2 V(P') c 2 V(P) B*V(P) 0. cy - ex- cy- Therefore V(P r ) is harmonic. By the definition F(P') — — }'(P) and the assumption that V vanishes along the segment it appears that the function 1' on the two sides of the line pieces on to itself in a continuous manner, and it remains merely to show thatit pieces on to itself in a harmonic manner, that is, that the function V and its extension form a function harmonic at points of the line. This follows from Poisson's Integral applied to a circle centered on the line. For let II (x. y) = f~" Vdf ; then II (x. 0) = because V takes on equal and opposite values on the upper and lower semicircum- ferences. Hence II = Y(I') = V(P') = along the axis. But // = V(P) along the upper arc and II = Y(P) along the lower arc because Poisson*s Integral takes on the boundary values as a limit when the, point approaches the boundary. >«'ow as If is harmonic and agrees with !"(/') upon the whole perimeter of the upper semi- circle it must be identical with V(P) throughout that semicircle. Ju like manner REAL VARIABLES 543 it is identical with V(P') throughout the lower semicircle. As the functions V(P) and F(P') are identical with the single harmonic function H, they must piece together harmonically across the axis. The theorem is thus completely proved. The statement about the conjugate function may be verified by taking the integral along paths symmetric with respect to the axis. 200. If a function to = f(z) = u + iv of a complex variable becomes real along the segment of a line or the arc of a circle, the function may be extended analytically across the segment or arc by assigning to the inverse -point P' the value w = u — iv conjugate to that at P. This is merely a corollary of the preceding theorem. For if w be real, the harmonic function v vanishes on the line and may be assigned equal and opposite values on the opposite sides of the line ; the conjugate function u then takes on equal values on the opposite sides of the line. The case of the circular arc would again follow from inversion as before. The method employed to identify functions in §§ 185-187 was to map the halves of the ?r-plane, or rather the several repetitions of these halves which were required to complete the map of the w-surface, on a region of the -.'-plane. By virtue of the theorem just obtained the con- verse process may often be carried out and the function w =f(z) which maps a given region of the £-plane upon the half of the w-plane may be obtained. The method will apply only to regions of the «-plane which are bounded by rectilinear segments and circular arcs ; for it is only for such that the theorems on inversion and the theorem on the extension of harmonic functions have been proved. To identify the function it is necessary to extend the given region of the s-plane by inversions across its boundaries until the ^-surface is completed. The method is not satisfactory if the successive extensions of the region in the s-plane result in overlapping. The method will be applied to determining the function (a) which maps the first quadrant of the unit circle in the s-plane upon the upper half of the «--plane, and ((3) which maps a 30°-60°-90° triangle upon the upper half of the et'-plane. Sup- pose the sector ABC mapped on t|L tg' the ^r-half-plane so that the perim- 'Wc\ ' lM :"^\ eter ABC corresponds to the ^ WMb '< ■ , : > , i , , , , : ,$^A ' real axis abc. When the perime- ter is described in the order written and the interior is on the left, the real axis must, by the principle of conformality, be described in such an order that the upper half-plane which is to correspond to the interior shall also lie on the left. The points a, b, c correspond to points 544 THEORY OF FUNCTIONS A , Ji, < '. At these points the correspondence required is such that the conformably must break down. As angles are doubled, each of the points A, ]'>, C must be a critical point of the first order for w=f(z) and a, b, c must be branch points. To map the triangle, similar con- siderations apply except that whereas C" is a critical point of the first order, the points A', Ji' are critical of orders 5, 2 respectively. Each case may now be treated separately in detail. Let it be assumed that the three vertices A, B, C of the sector go into the points* w = 0, 1, x. As the perimeter of the sector is mapped on the real axis, the function w=f(z) takes on real values for points z along the perimeter. Hence if the sector be inverted over any of its sides, the point P' which corre- sponds to P may be given a value conjugate to w at P, and the image of P' in the w-plane is symmetrical to the image of P with respect to the real axis. The three regions 1', 2', 3' of the z-plane correspond to the lower half of the ic-plane ; and the perimeters of these regions correspond also to the real axis. These regions may now be inverted across their boundaries and give rise to the regions 2, 3, 4 which must correspond to the upper half of the w>-plane. Finally by inversion from one of these regions the region 4' may be obtained as corresponding to the lower half of the 10-plane. In this manner the inver- sion has been carried on until the entire z-plane is covered. Moreover there is no overlapping of the regions and the figure may be inverted in any of its lines with- out producing any overlapping ; it will merely invert into itself. If a Riemann sur- face were to be constructed over the w-plane, it would clearly require four sheets. The surface could be connected up by studying the correspondence ; but this is not necessary. Note merely that the function /(z) becomes infinite at C when z— i by hypothesis and at (" when z = — i by inversion ; and at no other point. The values ± i will therefore be taken as poles of f(z) and as poles of the second order because angles are doubled. Note again that the function /(z) vanishes at A when z = by hypothesis and at z = x by inversion. These will be assumed to be zeros of the second order because the points are critical points at which angles are doubled. The function w =/(z) = Cz°-(z - I)- Hz + fr 2 = Cz 2 (z 2 + lr- has the above zeros and poles and must be identical with the desired function when the constant C is properly chosen. As the correspondence is such that/(l) = 1 by hypothesis, the constant C is 4. The determination of the function is complete as given. Consider next the case of the triangle. The same process of inversion ami re- peated inversion may be followed, and never results in overlapping except as one * It may be observed that the linear transformation (yii; + 5) v;' — a"' + /3 (Ex. 15, p. 157) has three arbitrary constants a: (3: y: 5, and that by such a transformation any three points of the tr-plane may be carried into any three points of the e/'-plane. It is therefore a proper ami trivial restriction to assume that 0, 1, x are the points of the w-plane which correspond to .1. Ji, V. REAL VARIABLES 545 2iK' region falls into absolute coincidence with one previously obtained. To cover the whole z-plane the inversion would have to be continued indefinitely ; but it may be observed that the rectangle inclosed by the heavy line is repeated indefinitely. Hence w = f(z) is a doubly periodic function with the periods 2 K, 2 iK' if 2 K, 2 K' be the length and breadth of the rectangle. The function has a pole of the second order at C or z = and at the points, marked with circles, into which the origin is carried by the successive inversions. As there are six poles of the second order, the function is of order twelve. "When z = K at A or z = iK' at A' the function vanishes and each of these zeros is of the sixth order because angles are increased G-fold. Again it appears that the function is of order 12. It is very simple to write the function down in terms of the theta functions constructed with the periods 2 K,2 iK'. ■■m = c H*(z)&(z) II\z) e;(z) HHz - a) el(z - a) R\z - /3)»e?(z - /3) For this function is really doubly periodic, it vanishes to the sixth order at K, iK', and has poles of the second order at the points 0, K + iK', a = \ K + \ iK' , a + K + iK', /3 = 2 K — [ H*(z + a), Q^z - £ As /3 = 2 K — ex the reduction H 2 (z — /3) be made. w=f(z)=C p + K + iK'. e x (z + a) may H 1 6 (z)e 6 (z) H*(z)ef(z)II*{z - a)ir 2 (z + a)Q°-(z - a)9&z + a) The constant C may be determined, and the expression for f(z) may be reduced further by means of identities; it might be expressed in terms of sn (z, k) and en (z, k), with properly chosen k, or in terms of p(z) and p'{z). For the purposes of computations that might be involved in carrying out the details of the map, it would probably be better to leave the expression of f(z) in terms of the theta functions, as the value of q is about 0.01. EXERCISES 1. Show geometrically that a plane inverts into a sphere through the center of inversion, and a line into a circle through the center of inversion. 2. Show geometrically or analytically that in the plane a circle inverts into a circle and that in space a sphere inverts into a sphere. 3. Show that in the plane angles are reversed in sign by inversion. Show that in space the magnitude of an angle between two curves is unchanged. 4. If ds, dS, dv are elements of arc, surface, and volume, show that ds' . ds = — ds, dS' = — dS = — dS, k* f 1 A- 4 dv' — dv. k G Note that in the plane an area and its inverted area are of opposite sign, and that the same is true of volumes in space. 546 THEORY OF FUNCTIONS 5. Show that the system of circles through any point and its inverse with respect to a given circle cut that circle orthogonally. Hence show that if two points are in- verse with respect to any circle, they are carried into points inverse with respect to the inverted position of the circle if the circle be inverted in any manner. In par- ticular show that if a circle be inverted with respect to an orthogonal circle, its cen- ter is carried into the point which is inverse with respect to the center of inversion. 6. Obtain Poisson's Integral (15) from the form (16'). Note that dG _ cos (p, n) cos (p\ n) _ a 2 — r 2 r 2 = p 2 + a 2 — 2 ap cos (p, n), dn p p' a 2 p 2 7. From the equation p/p' = const. = r/a of the sphere obtain G(P) = !_?I, v = -±- f !'(**->*) as P r P' 4 m J [a 2 + r 2 - 2 ar cos (r, a)]§ the Green Function and Poisson's Integral for the sphere. 8. Obtain Poisson's Integral in space by the method of inversion. 9. Find the potential due to an insulated spherical conductor and an external charge (by placing at the center of the sphere a charge equal to the negative of that induced on the grounded sphere). 10. If two spheres intersect at right angles, and charges proportional to the diameters are placed at their centers with an opposite charge proportional to the diameter of the common circle at the center of the circle, then the potential over the two spheres is constant. Hence determine the effect throughout external space of two orthogonal conducting spheres maintained at a given potential. 11. A charge is placed at a distance h from an infinite conducting plane. Determine the potential on the supposition that the plane is insulated with no charge or maintained at zero potential. 12. Map the quadrantal sector on the upper half-plane so that the vertices C. A, B correspond to 1, oo, 0. 13. Determine the constant C occurring in the map of the triangle on the plane. Find the point into which the median point of the triangle is carried. 14. With various selections of correspondences of the vertices to the three points 0, 1, oo of the w-plane, map the following configurations upon the upper half-plane : (a) a sector of 60°, (/3) an isosceles right triangle, (7) a sector of 45°, (5) an equilateral triangle. 201. The potential integrals. If p (./-, >/, ,~) is a function defined at different points of a region of space, the integral evaluated over that region is called the potential of p at the point (£, 77. £). The significance of the integral may be seen by considering the attraction and the potential energy at the point (t, rj. £j due to a EEAL VARIABLES 547 distribution of matter of density p (x, y, z) in some region of space. If fi be a mass at (£, 77, £) and m a mass at (x, y, z), the component forces exerted by m upon /x are A = c /A?>i x — £ y = c fxm y — rj Z = c fim z — t, and F = c jxm. (19) V = -ep-+C (M,n are respectively the total force on /x and the potential energy of the two masses. The potential energy may be considered as the work done by F or A", }', Z on /x in bringing the mass /x from a fixed point to the point (£, rj, £) under the action of m at (a*, ?/, .~) or it may be regarded as the function such that the nega- tive of the derivatives of V by x, y, z give the forces A, )', Z, or in vector notation F = — V V. Hence if the units be so chosen that c = 1, and if the forces and potential at (£. 77, £) be measured per unit mass by dividing by /x, the results are (after dis- regarding the arbitral'}- constant ( ') m x — £ in 1 1 — 77 in z — £ in H X/~ A' = V Z (19') Now if there be a region of matter of density p(x, y, z), the forces and potential energy at ($, 77. £) measured per unit mass there located may be obtained by summation or integration and are A p(.r. //, z)(x-£ £)JxJydz r = _ rpdv (19") l($-xy + (r,- y y + (L-z)-y It therefore appears that the potential U defined by (18) is the negative of the potential energy V due to the distribution of matter.* Xote fur- ther that in evaluating the integrals to determine A', Y, Z, and U = — V, the variables x, y, z with respect to which the integrations are per- formed will drop out on substituting the limits which determine the region, and will therefore leave X, Y, Z, V as functions of the param- eters 4-. 77. £ which appear in the integrand. And finally (20) A = «tf Z Crj n *In electric and magnetic theory, where like repels like, the potential and potential energy have the same sign. 548 THEORY OF FUNCTIONS are consequences either of differentiating U under the sign of integration or of integrating the expressions (19') for X, Y, Z expressed in terms of the derivatives of U, over the whole region. Theorem. The potential integral U satisfies the equations c 2 U c 2 U PU A d*U , c-r d 2 U /OHX ie + h' + W m Te + ij + W^-^ (21) known respectively as Laplace's and Poisson's Equations, according as the point (£, 77, £) lies outside or within the body of density p (x, y, z). In case (£, rj, £) lies outside the body, the proof is very simple. For the second derivatives of U may be obtained by differentiating with respect to £, 77, £ under the sign of integration, and the sum of the results is then zero. In case (£, rj, £) lies within the body, the value for r vanishes when (£, rj, £) coincides with (x, y, z) during the integra- tion, and hence the integrals for U, X, Y, Z become infinite integrals for which differentiation under the sign is not permissible without jus- tification. Suppose therefore that a small sphere of radius r concentric with (£, r], £) be cut out of the body, and the contributions F' of this sphere and F* of the remainder of the body to the force F be considered separately. For convenience suppose the origin moved up to the point (£, v , 0- Then F = VU = F* + F' = f P V - du + F'. Xow as the sphere is small and the density p is supposed continuous, the attraction F' of the sphere at any point of its surface may be taken as + Tri^pjr, the quotient of the mass by the square of the distance to the center, where p is the density at the center. The force F' then reduces to — -* 7Tp o r in magnitude and direction. Hence v-F = v.vr = v-F* + T.F' = / pV.V - dc + V.F ' = jfpv.vl The integral vanishes as in the first case, and V«F' = — 4 irp.. Hence if the suffix be now dropped, V.VT =—4 irp, and Poisson's Equation is proved. Gauss's Integral (p. 348) affords a similar proof. A rigorous treatment of the potential U and the forces X, Y, Z and their de- rivatives requires the discussion of convergence and allied topics. A detailed treat- ment will not be given, but a few of the most important fads may be pointed out. ( '(insider the ordinary case where the volume density p remains finite and the body itself does not extend to infinity. The integrand p/r becomes infinite when r = 0. But as ih is an infinitesimal of the third order around the point where r = 0, the term pih/r in the integral U will be infinitesimal, may be disregarded, and the integral U converges. In like manner the integrals for A", Y, Z will converge REAL VARIABLES 549 because p (£ — x)/r 3 , etc.. become infinite at r = to only the second order. If cX/li, were obtained by differentiation under the sign, the expressions p/i- 3 and p(£ — x) 2 /r> would become infinite to the third order, and the integrals f f*-dv= f f f ^r"sh\6drd(pd0, etc., as expressed in polar coordinates with origin at r = 0, are seen to diverge. Hence the derivatives of the forces and the second derivatives of the potential, as ob- tained by differentiating under the sign, are valueless. Consider therefore the following device : t 1 8 1 ell c c 1 , r c 1 , = , — = I p dv = — J p dv, • c£ r ex r c£ J c| r J ex r ex r ex r ex r J dx r J r ex J ex r The last integral may be transformed into a surface integral so that eU _ r 1 ep r p _ rrr 1 ep dv. 8* CiPdc-f? cos adS = CCC - j£ dxdydz - ff- dydz. (22) It should be remembered, however, that if r = within the body, the transforma- tion can only be made after cutting out the singularity r = 0, and the surface inte- gral must extend over the surface of the excised region as well as over the surface of the body. But in this case, as dS is of the second order of infinitesimals while r is of the first order, the integral over the surface of the excised region vanishes when r = and the equation is valid for the whole region. In vectors VJ7= f^dv- fP-dS. (22') It is noteworthy that the first integral gives the potential of Vp, that is, the inte- gral is formed for Yp just as (18) was from p. As Vp is a vector, the summation is vector addition. It is further noteworthy that in Vp the differentiation is with respect to x, y. z, whereas in VL T it is with respect to f, 77. 'c. Now differentiate (22) under the sign. (Distinguish V as formed for f, 77. f and x, y, z by V^ and V^..) or again V^ U = - f Y x - .V x pdv + f pV x - -dS. (23) This result is valid for the whole region. Now by Green's Formula (Ex. 10. p. 349) / pV,V, 1 ft + f V, I .V, pllc = / V,. ( p V. 1) A = / pV, L,S = / p £ 1 dS. Here the small region about ?• = must again be excised and the surface integral must extend over its surface. If the region be taken as a sphere, the normal du, being exterior to the body, is directed along — dr. Thus for the sphere f p -- - 'IS = ffp— r 2 sin dddS + 4 irp, where the volume integrals extend over the whole volume and the surface integral extends like that of (23) over the surface of the body but not over the small sphere. Hence (23) reduces to V.V[' = — 4 irp. Throughout this discussion it has been assumed that p and its derivatives are continuous throughout the body. In practice it frequently happens that a body consists really of several, say two, bodies of different nature (separated by a bound- ing surface S 12 ) in each of which p and its derivatives are continuous. Let the suffixes 1, 2 serve to distinguish the bodies. Then • u=fPidv 1 + f^dv 2 = je.dv. The discontinuity in p along a surface S 12 does not affect a triple integral. VU^f^ldv.-f^dS^ + f^dv.-f^dS^n. Here the first surface integral extends over the boundary of the region 1 which includes the surface S 12 between the regions. For the interface S 12 the direction of dS is from 1 into 2 in the first case, but from 2 into 1 in the second. Hence "=/>-/?»-/ ?±— & dS, It may be noted that the first and second surface integrals are entirely analogous because the first may be regarded as extended over the surface separating a body of density p from one of density 0. Now V«Vt7 may be found, and if the proper modifications be introduced in Green's Formula, it is seen that V»VL"= — 4 Trp still holds provided the point lies entirely within either body. The fact that p comes from the average value p upon the surface of an infinitesimal sphere shows that if the point lies on the interface S JO at a regular point, V-VT = — ±7r(lp l + \ p.,). The application of Green's Formula in its symmetric form (Ex. 10. p. 340) to the two functions r~ l and U, and the calculation of the integral over the infini- tesimal sphere about r = 0, gives /e r) J \r dn dn rj (24) where 2 extends over all the surfaces of discontinuity, including the boundary of the whole body where the density changes to 0. Now V«VZ7 = — 4 7rp and if the definitions be given that («!)-(<£)=-*„, „,-*, = *„, \d)L /i v dn l-z LEAL VARIABLES 551 then U = f ^dv + f -dS + fr—-dS, (25) J r J r J dn r where the surface integrals extend over all surfaces of discontinuity. This form of U appeal's more general than the initial form (18), and indeed it is more general, for it takes into account the discontinuities of U and its derivative, which cannot arise when p is an ordinary continuous function representing a volume distribution of matter. The two surface integrals may be interpreted as due to surface distribu- tions. For suppose that along some surface there is a surface density a of matter. Then the first surface integral represents the potential of the matter in the surface. Strictly speaking, a surface distribution of matter with a units of matter per unit surface is a physical impossibility, but it is none the less a convenient mathemati- cal fiction when dealing with thin sheets of matter or with the charge of electricity upon a conducting surface. The surface distribution may be regarded as a limit- ing case of volume distribution where p becomes infinite and the volume through- out which it is spread becomes infinitely thin. In fact if dn be the thickness of the sheet of matter pdndS = \ r„ V\ 2 rj dn r dn r Hence if adn = r. the potential takes the form rdr~ 1 /dndS. Just this sort of dis- tribution of magnetism arises in the case of a magnetic shell, that is. a surface covered on one side with positive poles and on the other with negative poles. The three integrals in (25) are known respectively as volume potential, surface poten- tial, and double surface potential. 202. The potentials may be used to obtain particular integrals of some differential equations. In the first place the equation c-r c-r c-u N . — l r fdv &? + T j + -I -/(*, », «) has '' = ^JV as its solution, when the integral is extended over the region through- out which /'is defined. To this particular solution for C may be added any solution of Laplace's equation, but the particular solution is fre- quently precisely that particular solution which is desired. If the functions U and f were vector functions so that U = H\ + jr., + kU 3 , and f = i/j + j/', + k/ 3 , the results would be t-^ + -7T-; + -T--7 = f (x, //, z) and U = - — \ > Cj- cij- cz- v *" \ir J r where the integration denotes vector summation, as may be seen by adding the results for T.VT\ = f v V.Vf'., =/,, V*VU 3 = f % after multi- plication by i, j, k. If it is desired to indicate the vectorial nature of U and f. the potential U may be called a vector potential. 552 THEORY OF FUNCTIONS In evaluating the potential and the forces at (£, rj, £) due to an ele- ment dm at (x,, y, z), it has been assumed that the action depends solely on the distance r. Now suppose that the distribution p (x, y, z, t) is a function of the time and that the action of the element pdv at (x, y, z) does not make its effect felt instantly at (£, w, £) but is propagated toward (£, rj, £) from (x, y, z) at a velocity 1/a so as to arrive at the time (t + ar). The potential and the forces at (£,77, £) as calculated by (18) will then be those there transpiring at the time t + ar instead of at the time t. To obtain the effect at the time t it would therefore be necessary to calculate the potential from the distribution p (x, y, z, t — ar) at the time t — ar. The potential /p (x, y, z, t — ar) dxdydz — = fe& dv+ f p<£>?>&0 or (27) where for brevity the variables x, y, z have been dropped in the second form, is called a retarded potential as the time has been set back from t to t — ar. The retarded potential satisfies the equation c i U c-U dHJ_ 2 cPU W + ly + W ~ a w according as (f, -q, £) lies within or outside the distribution p. There is really no need of the alternative statements because if (£, -q, £) is out- side, p vanishes. Hence a solution of the equation c 1 U c i U d-U c 2 U te + W + 'd*-" w =f( ?> *>*>*> 1 it J r The proof of tlie equation (27) is relatively simple. For in vector notation, V.V [7 = V.V f eM dv + V.V f Pit-ar)-p(t) d}) 4 7T/3 + V.V f p(t-ar)-p(t) dv. The first reduction is made by Poisson's Equation. The second expression may be evaluated by differentiation under the sign. For it should be remarked that p(t—ar)—p(t) vanishes when r — 0, and hence the order of the infinite in the integrand before and after differentiation is less by unity than it was in the cor- responding steps of § 201. Then rp(t-ar)-p { t )dv= r | (-^(* - «r) V _ 1| ^ J r J I r r J REAL VARIABLES 553 V f .V f f PV-^-pW fo = f r (~ a)VV|r.Vgr + (-a)p'V f .V f r + (- «)/ 3 ' v f' v f- + (~ a)p'V^r.V f - + [>(t- ar)-p(0]V r V f -|.do. But V^ = — V x and Vr = r/r and Vr~ * = — r/r 3 and V.Vj— 1 = 0. Hence V^r.V^r = 1, V^r.V^r- 1 =— r— 2 , V^-V f r = 2r- 1 and v.v f p«-»»-)-p(o dp = r ^ de = r * **'-«) fy = a 2^E. «/ r J r J r ct 2 ct 2 It was seen (p. 345) that if F is a vector function with no curl, that is, if Y*F = 0, then Fv/r is an exact differential ; and F may be ex- pressed as the gradient of , that is, as F = V<£. This problem may also be solved by potentials. For suppose -If V.F F = Y<£, then V-F = V«V0, d> = - - \ dv. (28) 4 7rJ r ' It appears therefore that <£ may be expressed as a potential. This solu- tion for <£ is less general than the former because it depends on the fact that the potential integral of Y^F shall converge. Moreover as the value of thus found is only a particular solution of V«F = V«Y<£, it should be proved that for this <£ the relation F = Y<£ is actually sat- isfied. The proof will be given below. A similar method may now be employed to show that if F is a vector function with no divergence, that is, if V«F = 0, then F may be written as the curl of a vector function G, that is, as F = V X G. For suppose F = V*G, then Y*F = YxY*G = VY«G - Y«YG. As G is to be determined, let it be supposed that Y«G = 0. Then F = Y>• (29) Here again the solution is valid only when the vector potential integral of Y X F converges, and it is further necessary to show that F = Y*G. The conditions of convergence are, however, satisfied for the functions that usually arise in physics. To amplify the treatment of (28) and (29), let it be shown that 1 r V'F 1 r VxF V0 = V / dv = F, VxG = — Vx | — dv = F. 4 7r J r 4 7T J r By use of (22) it is possible to pass the differentiations under the sign of integra- tion and apply them to the functions V.F and VxF, instead of to 1/r as would be required by Leibniz's Rule (§ 110). Then 1 r W 'F , 1 r V.F 70 V0 = - — - | dv + — I dS. 4 tt J r 4 tv J r 554 THEORY OF FUNCTIONS The surface integral extends over the surfaces of discontinuity of V«F, over a large (infinite) surface, and over an infinitesimal sphere surrounding r = 0. It will be assumed that V«F is such that the surface integral is infinitesimal. Now as VxF = 0, VxVxF = and VV.F = V.VF. Hence if F and its derivatives are continuous, a reference to (24) shows that V.VF 7 - dv = F. i r 4 tv J r 4 7r J r 4 tt J r In like manner V.VF dv Questions of continuity and the significance of the vanishing of the neglected sur- face integrals will not be further examined. The elementary facts concerning potentials are necessary knowledge for students of physics (especially electro- magnetism) ; the detailed discussion of the subject, whether from its physical or mathematical side, may well be left to special treatises. EXERCISES 1. Discuss the potential U and its derivative VU for the case of a uniform sphere, both at external and internal points, and upon the surface. 2. Discuss the second derivatives of the potential, that is, the derivatives of the forces, at a surface of discontinuity of density. 3. If a distribution of matter is external to a sphere, the average value of the potential on the spherical surface is the value at the center ; if it is internal, the average value is the value obtained by concentrating all the mass at the center. 4. What density of distribution is indicated by the potential er r " ? What den- sity of distribution gives a potential proportional to itself '.' 5. In a space free of matter the determination of a potential which shall take assigned values on the boundary is equivalent to the problem of minimizing iffim+m+m}^- u vu.vudv. 6. For Laplace's equation in the plane and for the logarithmic potential — log r, develop the theory of potential integrals analogously to the work of § 201 for Laplace's equation in space and for the fundamental solution !/;•. BOOK LIST A short list of typical books with brief comments is given to aid the student of this text in selecting material for collateral reading or for more advanced study. 1. Some standard elementary differential and integral calculus. For reference the book with which the student is familiar is probably preferable. It may be added that if the student has had the misfortune to take his calculus under a teacher who has not led him to acquire an easy formal knowledge of the subject, he will save a great deal of time in the long run if he makes up the deficiency soon and thoroughly ; practice on the exercises in Granville's Calculus (Ginn and Com- pany), or Osborne's Calculus (Heath & Co.), is especially recommended. 2. B. 0. Peirce, Table of Liter/ mis (new edition). Ginn and Company. This table is frequently cited in the text and is well-nigh indispensable to the student for constant reference. 3. Jahxke-Emde, Funktionentafeln m It Formeln und Kurven. Teubner. A very useful table for any one who has numerical results to obtain from the analysis of advanced calculus. There is very little duplication between this table and the previous one. 4. Woods and Bailey, Course hi Mathematics. Ginn and Company. 5. Byerly, Differential Calculus and Integral Calculus. Ginn and Company. 6 Tooiiuxter, Differential Calculus and Integral Calculus. Mac- millan. 7. Williamsox, Differential Calculus and Integral Calculus. Long- mans. These are standard works in two volumes on elementary and advanced calculus. As sources for additional problems and for comparison with the methods of the text they will prove useful for reference. 8. C. J. de la Vallee-Poussix, Cours d' analyse. Gauthier-Villars. There are a few books which inspire a positive affection for their style and beauty in addition to respect for their contents, and this is one of those few. My Advanced Calculus is necessarily under considerable obligation to de la Vallee- Poussin's Cours d' analyse, because I taught the subject out of that book for several years and esteem the work more highly than any of its compeers in any language. 555 556 BOOK LIST 9. Goursat, Cours d? analyse. Gauthier-Villars. 10. Goursat-Hedrick, Mathematical Analysis. Ginn and Company. The latter is a translation of the first of the two volumes of the former. These, like the preceding five works, will be useful for collateral reading. 11. Bertrand, Calcul differentiel and Calcul integral. Tins older French work marks in a certain sense the acme of calculus as a means of obtaining formal and numerical results. Methods of calculation are not now so prominent, and methods of the theory of functions are coming more to the fore. Whether this tendency lasts or does not, Bertrand's Calculus will remain an inspiration to all who consult it. 12. Forsyth, Treatise on Differential Equations. Macmillan. As a text on the solution of differential equations Forsyth's is probably the best. It may be used for work complementary and supplementary to Chapters VIII-X of this text. 13. Piekpoxt, Theory of Funetions of Ileal Variables. Ginn and Company. In some parts very advanced and difficult, but in others quite elementary and readable, this work on rigorous analysis will be found useful in connection with Chapter II and other theoretical portions of our text. 14. Gibbs-Wilsox, Veetor Analysis. Seribners. Herein will be found a detailed and connected treatment of vector methods mentioned here and there in this text and of fundamental importance to the mathematical physicist. 15. B. O. Peirce, Xetctonian Potential Function. Ginn and Company. A text on the use of the potential in a wide range of physical problems. Like the following two works, it is adapted, and practically indispensable, to all who study higher mathematics for the use they may make of it in practical problems. 16. Byerly, Fourier Scries and Spherical Harmonics. Ginn and Company. of international repute, this book presents the methods of analysis employed in the solution of the differential equations of physics. Like the foregoing, it gives an extended development of some questions briefly treated in our Chapter XX. 17. "Whittakek, Modern Analysis. Cambridge University Press. This is probably the only book in any language which develops and applies the methods of the theory of functions for the purpose of deriving anil studying the formal properties of the most important functions other than elementary which occur in analysis directed toward the needs of the applied mathematician. IS. Osgood, Le/n-burh der FunhtionentJieorie. Teubner. For the pure mathematician this work, written with a grace comparable only to that of de la Vallee-Foussin"s Calculus, will be as useful as it is charming. INDEX (The numbers refer to pages) a x , a z , 4, 45, 162 Abel's theorem on uniformity, 438 Absolute convergence, of integrals, 357, 369 ; of series, 422, 441 Absolute value, of complex numbers, 154 ; of reals, 35 ; sum of, 36 Acceleration, in a line, 13 ; in general, 174 ; problems on. 186 Addition, of complex numbers, 154 ; of operators, 151 ; of vectors, 154, 163 Adjoint equation, 240 Algebra, fundamental theorem of, 159, 306, 482 ; laws of, 153 Alternating series, 39, 420, 452 am = sin -1 sn, 507 Ampere" s Law, 350 Amplitude, function, 507; of complex numbers, 154 ; of harmonic motion, 188 Analytic continuation, 444, 543 Analytic function, 304, 435. ..See Func- tions of a complex variable Angle, as a line integral, 297, 308 ; at critical points, 491 ; between curves, 9 ; in space. 81 ; of a complex number, 154; solid. 347 Angular velocity, 178, 346 Approximate formulas. 60, 77, 101, 383 Approximations, 59, 195; successive. 198. See Computation Arc. differential of, 78. 80. 131; of ellipse, 77, 514 ; of hyperbola. 516. .See Length Area, 8. 10. 25, 67, 77; as a line integral, 288; by double integration, 324, 329 ; directed, 167; element of, 80, 131, 175, 340, 342 : general idea, 311; of a sur- face. 339 Areal velocity, 175 Argument of a complex number, 154 Associative law, of addition, 153, 163 ; of multiplication. 150, 153 Asymptotic expansion, 390, 397, 456 Asymptotic expression for n\, 383 Asymptotic lines and directions, 144 Asvmptotic series. 390 Attraction. 31. 68. 308, 332, 348, 547; Law of Nature. 31, 307; motion under, 190. 264. See Central Force and Po- tential Average value, 333 ; of functions. 333 ; of a harmonic function, 531; over a surface, 340 Axes, right- or left-handed, 84, 167 Axiom of continuity, 34 B. .See Bernoulli numbers, Beta function Bernoulli's equation, 205, 210 Bernoulli's numbers, 448, 456 Bernoulli's polynomials, 451 Bessel's equation, 248 Bessel's functions, 248, 393 Beta function, 378 Binomial theorem, finite remainder in, 60 ; infinite series, 423, 425 Binomial, 83 Boundary of a region, 87. 308, 311 Boundary values, 304, 541 Brachistochrone. 404 Branch of a function, of one variable, 40 ; of two variables, 90 ; of a com- plex variable, 492 Branch point, 492 C„. .See Cylinder functions Calculation. .See Computation, Evalua- tion, etc. Calculus of variations, 400-418 Cartesian expression of vectors. 167 Catenary, 78, 190 ; revolved, 404, 408 Cauchy's Formula, 30, 49. 61 Cauchy's Integral, 304, 477 Cauchy's Integral test, 421, 427 Caustic, 142 Center, instantaneous, 74, 178; of in- version. 538 Center of gravity or mass, motion of the, 176; of areas or laminas, 317. 324; of points or masses, 168 ; of volumes, 328 Central force, 175, 264 Centrode, fixed or moving, 74 Chain, equilibrium of, 185, 190, 409; motion of, 415 Change of variable, in derivatives. 12, 14,' 67, 98, 103. 106: in differential equations. 204. 235. 245; in integrals, 16, 21. 54. 65, 328, 330 Characteristic curves. 140, 267 Characteristic strip, 279 557 558 INDEX Charge, electric, 539 Charpit's method, 274 Circle, of curvature, 72 ; of convergence, 433, 437; of inversion, 538 Circuit, 89 ; equivalent, irreducible, re- ducible, 91 Circuit integrals, 294 Circulation, 345 Clairaut's equation, 230; extended, 273 Closed curve, 308; area of, 289, 311; integral about a, 295, 341, 360, 477, 536; Stokes's formula, 345 Closed surface, exterior normal is posi- tive, 167, 341; Gauss's formula, 342; Green's formula, 349, 531 ; integral over a, 341, 536 ; vector area vanishes, 167 en, 471, 505, 518 Commutative law, 149, 165 Comparison test, for integrals, 357 ; for series, 420 Complanarity, condition of, 169 Complementary function, 218, 243 Complete elliptic integral, 507, 514, 77 Complete equation, 240 Complete solution, 270 Complex function, 157, 292 Complex numbers, 153 Complex plane, 157, 302, 360, 433 Complex variable. See Functions of a Components, 163, 167, 174, 301, 342, 507 Computation, 59 ; of a definite integral, 77; of Bernoulli's numbers, 447; of elliptic functions and integrals, 475, 507, 514, 522; of logarithms, 59; of the solution of a differential equation, 195. See Approximations, Errors, etc Concave, up or down, 12, 143 Condensation point, 38, 40 Condition, for an exact differential, 105 ; of complanarity, 169 ; of integrability, 255 ; of parallelism, 166 ; of perpendic- ularity, 81, 165. See Initial Conformal representation, 490 Conformal transformation, 132, 477, 538 Congruence of curves, 141 Conjugate functions, 536 Conjugate imaginaries, 156, 543 Connected, simply or multiply, 89 Consecutive points, 72 Conservation of energy, 301 Conservative force or system, 224, 307 Constant, Eider's, 385 Constant function, 482 Constants, of integration, 15. 183; phys- ical, 183 ; variation of. 243 Constrained maxima and minima. 120, 404 Contact, of curves, 71 ; order of, 72 ; of conies with cubic, 521 ; of plane and curve, 82 Continuation, 444, 478, 542 Continuity, axiom of, 34 ; equation of, 350 ; generalized, 44 ; of functions, 41, 88, 476; of integrals, 52, 281, 368; of series, 430 ; uniform, 42, 92, 476 Contour line or surface, 87 Convergence, absolute, 357, 422, 429 ; asymptotic, 456 ; circle of, 433, 437 ; of infinite integrals, 352 ; of products, 429; of series, 419; of suites of num- bers, 39 ; of suites of functions, 430 ; nonuniform, 431 ; radius of, 433 ; uni- form, 368, 431 Coordinates, curvilinear, 131 ; cylindri- cal, 79;' polar, 14; spherical, 79 cos, cos- 1 , 155, 161, 393. 456 cosh, cosh-i, 5, 6, 16. 22 Cosine amplitude, 507. See en Cosines, direction, 81. 109; series of , 460 cot, coth, 447, 450, 454 Critical points, 477, 491 ; order of, 491 esc, 550, 557 Cubic curves, 519 Curl, Vx, 345, 349, 418, 553 Curvature of a curve. 82 ; as a vector, 171; circle and radius of, 73, 198; problems on, 181 Curvature of asurface, 144 ; lines of, 14'! ; mean and total, 148; principal radii, 144 Curve, 308 ; area of, 311 ; intrinsic equa- tion of, 240 ; of limited variation, 309 ; quadrature of, 313; rectifiable, 311. See Curvature, Length, Torsion, etc., and various special curves Cui'vilinear coordinates, 131 Curvilinear integral. See Line Cuspidal edge, 142 Cuts, 90, 302. 362, 497 Cycloid, 76, 404 Cylinder functions, 247. See Bessel Cylindrical coordinates, 79, 328 D, symbolic use, 152, 214, 279 Darboux's Theorem, 51 Definite integrals, 24, 52 ; change of variable, 54, 65; computation of, 77; Duhamel's Theorem. 63 ; for a series, 451; infinite, 352 ; Osgood's Theorem, 54, 65 ; Theorem of the Mean, 25, 29, 52, 359. See Double, etc., Functions, Infinite, Cauchy's, etc. Degree of differential equations, 228 Del, V, 172. 260, 343, 345. 349 Delta amplitude. 507. See dn De Moivre's Theorem, 155 Dense set. 39. 44. 50 Density, linear, 28; surface. 315; vol- ume, 110, 326 Dependence, functional. 129: linear, 245 Derivative, directional. U~. 172; geo- metric properties of, 7: infinite, 46; IXDEX 559 logarithmic, 5; normal, 97, 137, 172; of higher order, 11, 07, 102, 197; of integrals, 27, 52, 288, 370 ; of products, 11, 14, 48 ; of series term by term, 430 ; of vectors, 170; ordinary, 1, 45, 158; partial, 93, 99 ; right or left, 46; The- orem of the Mean, 8, 10, 46, 94. See Change of variable, Functions, etc. Derived units, 109 Determinants, functional, 129 ; "Wron- skian, 241 Developable surface, 141, 143, 148, 279 Differences, 49, 462 Differentiable function, 45 Differential, 17, 64 ; exact, 106, 254, 300 ; of arc, 70. 80. 131 ; of area, 80, 131 ; of heat, 107, 294 ; of higher order, 67, 104; of surface. 340; of volume, 81, 330; of work, 107, 292; partial, 95, 104 ; total, 95, 98, 105, 208, 295 ; vec- tor, 171, 293, 342 Differential equations, 180, 267; degree of, 228 ; order of, 180 ; solution or integration of, 180 ; complete solution, 270; general solution, 201, 230, 269; infinite solution, 230 ; particular solu- tion, 230; singular solution, 231, 271. See Ordinary, Partial, etc. Differential equations, of electric cir- cuits, 222, 226 ; of mechanics, 186, 263 ; Hamilton's, 112 ; Lagrange's, 112, 224, 413; of media, 417; of physics, 524; of strings, 185 Differential geometry, 78. 131, 143, 412 Differentiation, 1; logarithmic, 5; of implicit functions, 117; of integrals, 27,283; partial, 03 ; total, 95; under the sign, 281 ; vector, 170 Dimensions, higher, 335; physical, 109 Direction cosines, 81, 169 ; of a line, 81 ; of a normal, 83; of a tangent, 81 Directional derivative, 97, 172 Discontinuity, amount of, 41, 462 ; finite or infinite, 479 Dissipative function, 225, 307 Distance, shortest, 404, 414 Distributive law, 151, 165 Divergence, formula of, 342 ; of an inte- gral, 352 ; of a series, 419 ; of a vector, 343, 553 Double integrals, 80, 131, 313, 315, 372 Double integration, 32, 285, 319 Double limits, 89, 430 Double points, 119 Double sums, 315 Double surface potential. 551 Doubly periodic functions, 417, 486, 504, 517; order of, 487. See p, sn, en. dn Duhamel's Theorem. 28, 63 Dupin's indicatrix, 145 e = 2.718-.., 5, 437 E, complete elliptic integral, 77, 514 ^-function, 62, 353, 479 E (2 Surface integral, 340, 347 Symbolic methods, 172, 214, 223, 260, 275, 447 Systems, conservative, 301; dynamical, 413 Systems of differential equations, 223, 260 tan, tan- 1 , 3, 21, 307, 450, 457, 498 Tangent line, 8, 81, 84 Tangent plane, 96, 170 tanh, tank- 1 , 5, 6, 450, 501 Taylor's Formula, 55, 112, 152, 305, 477 Taylor's Series, 197, 435, 477 Taylor's Theorem, 49 Test, Cauchy's, 421; comparison, 420; Raabe's, 424; ratio, 422; YVeierstrass's M-, 432, 455 Test function. 355 Theorem of the Mean, for derivatives. 8, 10, 46, 94 ; for integrals, 25, 29, 52 359 Thermodynamics, 106, 294 Theta functions, 77, II l .Q, Q v as Fourier's series, 467; as products, 471 ; define elliptic functions, 471, 504; logarith- mic derivative, 474, 512 ; periods and half periods, 408 ; relations between squares, 472 ; small thetas, 0, 6 a , 523 ; zeros, 469 Torsion, 83; radius of, 83, 175 Total curvature, 148 Total differential, 95, 98, 105, 209, 295 Total differential equation, 254 Total differentiation, 99 Trajectory, 196; orthogonal, 194, 234, 260 Transformation, conformal, 132, 476; Filler's, 449; of inversion, 537; orthog- onal, 100; of a plane, 131; to polars, 14, 79 Trigonometric functions, 3, 161, 453 Trigonometric series, 458, 465, 525 Triple integrals, 326 ; element of, 80 Umbilic, 148 Undetermined coefficients, 199 Undetermined multiplier, 120, 126, 406, 411 Uniform continuity, 42. 92,. 476 Uniform convergence, 369, 431 Units, fundamental and derived, 109; dimensions of, 109 Unity, roots of, 156 Unlimited set or suite, 38 Vall(5e-Poussin, de la, 373, 555 Value. See Absolute, Average, Mean Variable, complex, 157; equicrescent, 48 ; real. 35. See Change of, Functions Variable limits for integrals, 27, 404 Variables, separable, 179, 203. See Functions Variation, 179 ; of a function, 3, 10, 54; limited. 54. 309; of constants, 243 Variations, calculus of, 401 ; of integrals, 401, 410 Vector, 154, 163; acceleration, 174; area, 167, 290; components of a, 163, 167, 174. 342; curvature, 171; moment. 176; moment of momentum, 1 76 ; momentum, 173 ; torsion, 83, 171 ; velocity, 173 Vector addition, 154, 163 Vector differentiation, 170, 260, 342, 345; force, 173 Vector functions, 200, 293, 300, 342, 345, 551 Vector operator 7. see Del Vector product. 165, 108. 345 Vectors, addition of, 154, 163; coin- planar, 169; multiplication of, 155, 163 ; parallel, 166 ; perpendicular, 165 ; products of, 164, 165, 168, 345; pro- jections of, 164, 167, 342 566 INDEX Velocity, 13, 173 ; angular, 346 ; areal, 175 ; of molecules, 392 Vibrations, small, 224, 526; superposi- tion of, 226, 524 Volume, center of gravity of, 328 ; ele- ment of, 80 ; of parallelepiped, 169 ; of revolution, 10 ; under surfaces, 32, 317, 381 ; with parallel bases, 10 Volume integral, 341 Wave equation, 276 Waves on water, 529 Weierstrass's integral, 517 Weierstrass"s Jf-test, 432 Weights, 333 Work, 107, 224, 292, 301 ; and energy, 293, 412 Wronskian determinant, 241 z-plane, 157, 302, 360, 433; mapping the, 490. 497, 503, 517, 543 Zeta functions, Z, 512 ; f, 522 Zonal harmonies. See Legendre's poly- nomials THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. UUN6 1936 ">^ r> «*^ JII7 1084 34 JJETD0CT5 198710 t- f CI 1 f #ODSEP2fJ198828 r: Series 9482 3 1205 00413 5495 1>£ UC SOUTHERN REGIONAL LIBRARY FACILITY 8 850 7 AA 000 1