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MATHEMATICAL MONOGRAPHS. 
 
 EDITED BV 
 
 MANSFIELD MERRIMAN and ROBERT S. WOODWARD. 
 
 No. 11 
 FUNCTIONS 
 
 OF A 
 
 COMPLEX VARIABLE 
 
 BY 
 
 THOMAS S. FISKE, 
 
 Professor of Mathematics in Columbia Univbrsitv* 
 
 FOURTH EDITION. 
 FIRST THOUSAND. 
 
 NEW YORK: 
 
 JOHN WILEY & SONS. 
 
 London: CHAPMAN & HALL, Limited. 
 1907 
 
 Goi 
 
 
ComuGHT, 1896, 
 
 BY 
 
 MANSFIELD MERRIMAN and ROBERT S. WOODWARD 
 
 UNDER THE TITLE 
 
 HIGHER MATHEMATICS. 
 
 First Edition, September, 1896. 
 Second Edition, January, 1898. 
 Third Edition, August, 1900. 
 Fourth Edition, November, 1906. 
 
 ItOnitltT DRUMMOND, PRINTER, NEW VORK* 
 
EDiTORS' PREFACE. 
 
 The volume called Higher Mathematics, the first edition 
 of which was published in 1896, contained eleven chapters by 
 eleven authors, each chapter being independent of the others, 
 but all supposing the reader to have at least a mathematical 
 training equivalent to that given in classical and engineering 
 colleges. The publication of that volume is now discontinued 
 and the chapters are issued in separate form. In these reissues 
 it will generally be found that the monographs are enlarged 
 by additional articles or appendices which either ampUfy the 
 former presentation or record recent advances. This plan of 
 publication has been arranged in order to meet the demand of 
 teachers and the convenience of classes, but it is also thought 
 that it may prove advantageous to readers in special lines of 
 mathematical hterature. 
 
 It is the intention of the publishers and editors to add other 
 monographs to the series from time to time, if the call for the 
 same seems to warrant it. Among the topics which are under 
 consideration are those of elliptic functions, the theory of num- 
 bers, the group theory, the calculus of variations, and non- 
 Euclidean geometry; possibly also monographs on branches of 
 astronomy, mechanics, and mathematical physics may be included. 
 It is the hope of the editors that this form of pubHcation may 
 tend to promote mathematical study and research over a wider 
 field than that which the former volume has occupied. 
 
 December, 1905. 
 
AUTHOR'S PREFACE. 
 
 In the following pages is contained a brief introductory 
 account of some of the more fundamental portions of the theory 
 of functions of a complex variable. The work was prepared 
 originally as a chapter for the volume called " Higher Mathe- 
 matics," pubUshed in 1896. It has been enlarged by the addition 
 of sections on power series, algebraic functions and their integrals, 
 functions of two or more independent variables, and differential 
 equations. Furthermore, the section on uniform convergence 
 has been extended, and the treatment of Weierstrass's theorem 
 and of Mittag-Leffler's theorem has been simplified. 
 
 It is hoped that the present work will give the uninitiatea 
 some idea of the nature of one of the most important branches 
 of modem mathematics, and will also be useful as an introduction 
 to larger works, such as those in English by Forsyth, Whittaker, 
 and Harkness and Morley; in French by Jordan, Picard, Goursat, 
 and Valine- Poussin; and in German by Burkhardt, Stolz and 
 Gmeiner, and Osgood. 
 
 New York, August, 1906. 
 
CONTENTS. 
 
 Art. I. Definition of Function • • . . Page i 
 
 2. Representation of Complex Variable . • 2 
 
 3. Absolute Convergence ,. 3 
 
 4. Elementary Functions 4 
 
 5. Continuity of Functions 5 
 
 6. Graphical Representation of Functions 7 
 
 7. Derivatives 8 
 
 8. conformal representation ii 
 
 9. Examples of Conformal Representation 13 
 
 10. Conformal Representation of a Sphere . 19 
 
 ■ II. Conjugate Functions io 
 
 12. Application to Fluid Motion 21 
 
 13. Singular Points 25 
 
 14. Point at Infinity 31 
 
 15. Integral of a Function 32 
 
 16. Reduction of Complex Integrals to Real 36 
 
 17. Cauchy's Theorem 37 
 
 18. Application of Cauchy's Theorem , . .' 39 
 
 19. Theorems on Curvilinear Integrals 42 
 
 20. Taylor's Series 44 
 
 21. Laurent's Series 46 
 
 22. Fourier's Series 48 
 
 2,3. Uniform Convergence 46 
 
 24. Power Series 54 
 
 25. Uniform Convergence of Pov^^er Series 56 
 
 26. Uniform Functions with Singular Points 57 
 
 27. Residues 61 
 
 28. Integral of a Uniform Function 63 
 
 29. Weierstrass's Theorem 66 
 
 30. Mittag-Leffler's Theorem 71 
 
 31. Singular Lines and Regions 78 
 
 32. Functions having n Values ,. 81 
 
 33. Algebraic Functions 83 
 
 34. Integrals of Algebraic Functions 85 
 
 35. Functions of Several Variables . 89 
 
 36. Differential Equations 91 
 
 IndSx 99 
 
 V 
 
FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Art. 1. Definition of Function. 
 
 If two or more quantities are such that no one of them suf- 
 fers any restriction in regard to the values which it can assume 
 when any values whatsoever are assigned to the others, the 
 quantities are said to be ♦* independent." 
 
 A quantity is said to be a function of another quantity or of 
 several independent quantities if the former is determined in 
 value whenever particular values are assigned to the latter. 
 The quantity or quantities upon the values of which the value 
 of the function depends, are said to be the '* independent vari- 
 ables " of the function. 
 
 A function is *' one-valued," or *' uniform," when to every 
 set of values assigned to the independent variables there cor- 
 responds but one value of the function. It is said to be 
 **w-valued" when to every set of values of the independent 
 variables n values of the function correspond. 
 
 The ''Theory of Functions" has among its objects the 
 study of the properties of functions, their classification accord- 
 ing to their properties, the derivation of formulas which exhibit 
 the relations of functions to one another or to their independ- 
 ent variables, and the determination whether or not functions 
 exist satisfying assigned conditions. 
 
• Ft/jfcf i(5n8 of a complex vakiablb. 
 
 Art. 2. Representation of Complex Variable. 
 
 A variable quantity is capable, in general, of assuming both 
 real and imaginary values. In fact, unless it be otherwise 
 specified, every quantity w is to be regarded as having the 
 " complex " form u-\-v V— i, u and v being real. It is cus- 
 
 tomary to denote r — i by t, and to write the preceding quan- 
 tity thus : M -}- ^^' I^ ^ is zero, w is real ; if u is zero, zv is a 
 ** pure imaginary." 
 
 A quantity 2 = x-\- ty is said to vary" continuously " when 
 between every pair of values which it may take, c^ = a^-\- ib^^ 
 ^2 = ^2 4" ^^2 » ^ and_;/ must pass through all real values inter- 
 mediate to a^ and a^ , b^ and b^ , respectively, either once or a 
 'inite number of times. 
 
 It is usual to give to a variable quantity ^ = ;r + ^^ a graphi- 
 cal representation by drawing in a plane a pair of rectangular 
 axes and constructing a point whose abscissa and ordinate are 
 respectively equal to x and y. To every value of z will corre- 
 spond a point ; and, conversely, to every point will correspond 
 a value of z. The terms '* point " and value, then, may be inter- 
 ohanged without confusion. When z varies continuously the 
 graphical representation of its varia- 
 tion, or its *' path," will be a continuous 
 line. This graphical representation is 
 of the highest importance. By means 
 of it some of the most complicated 
 propositions may be given an exceed- 
 ingly condensed and concrete expres- 
 sion. 
 
 By putting x = r cos 6, y = r sin 6, where r is a positive real 
 quantity, the point 
 
 z = r(cos 6 -{-t sin 6) 
 
 is referred to polar coordinates. The quantity r is called the 
 absolute value or " modulus " of z. It will often be written \js\. 
 is known as the ^argument " of 2. 
 
ABSOLUTE CONVERGENCE. 3 
 
 A function Is sometimes considered for only such values of 
 each independent variable as are represented graphically by the 
 points of a certain continuous line. In the study of functions 
 of real variables, for example, the path of each independent 
 variable is represented by a straight line, namely, the axis of 
 real quantities, or^;' = o. 
 
 *Art. 3. Absolute Convergence. 
 
 The representation of functions by means of infinite series 
 is one of the most important branches of the theory of func- 
 tions. In many problems, in fact, it is only by means of series 
 that it is possible to determine functions satisfying the condi- 
 tions assigned and to obtain the required numerical results. 
 Frequent use will be made of the following theorem. 
 
 Theorern. — If the moduU of the terms of a series form a 
 convergent series, the given series is convergent. 
 
 Let the given series be W = w^-\-w^-\- . . . + ^n + • • • 
 in which zv^ = r^ (cos b^ + e sin B^, w, = r, (cos 6^ + zsin 6^,) . . . 
 By hypothesis the series R= r^-\-r^-\- . . . + r„ + . . . is 
 convergent. Its terms being all positive, the sum of its first m 
 terms constantly increases with m, but in such a manner as to 
 approach a limit. The same will be true necessarily of any 
 series formed by selecting terms from R. The sum of the first 
 m terms of the series W is composed of two parts, 
 
 r, cos e^ + r,cose^ . . . + r^-. cos 6^,,, 
 i{r, sin 0^ + r, sin ^, + . . . + r^_, sin 0^.,), 
 
 and each of these in turn may be divided into parts which have 
 all their terms of the same sign. Every one of the four parts 
 thus obtained approaches a limit as m is increased ; for the 
 terms of each part have the same sign, and cannot exceed^ in 
 absolute value, the corresponding terms of R, Hence, as m is 
 increased, the sum of the first m terms of W approaches a 
 limit; which was to be proved. 
 
 A series, the moduli of whose terms form a convergent 
 series, is said to be " absolutely convergent." 
 
4 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Prob. I. Show that the series i -}- 2; -f- ^' + • • • + ^** + • • 'is 
 absolutely convergent, if | ^ | < i. 
 
 Art. 4. Elementary Functions. 
 
 In elementary mathematics the functions are usually con- 
 sidered for only real values of the independent variables. In 
 the case of the algebraic functions, however, there is no diffi- 
 culty in assuming that the independent variables are complex. 
 The theory of elimination shows that every algebraic equation 
 can be freed from radicals. Every algebraic function, there- 
 fore, is defined by an equation which may be put in a form 
 wherein the second rnember is zero and the first member is 
 rational and entire in the function and its independent variables. 
 
 Besides the algebraic functions, the functions most often 
 occurring in elementary mathematics are the trigonometric and 
 exponential functions and the functions inverse to them. The 
 definitions, by which these functions are generally first intro- 
 duced, have no significance in the case where the inde- 
 pendent variables are complex. However, the following 
 familiar series, 
 
 ^ = exp^= 1+^+1 + !^ + ^+..., 
 2 31 41 
 
 cos^=i--+^-^+..., 
 
 z* z^ z' . 
 smz=z-- + -^--^ + ... 
 
 which have been established for the case where the variables 
 are real, furnish most convenient general definitions for exp Zj 
 cos z, and sin z, these series being absolutely convergent for 
 every finite value of z. Defining the logarithmic function by 
 the equation__ 
 
 g\ogx — exp (log<3:) = Zy 
 it follows that 
 
 ^« _ ^ loga _ gxp {z log a). 
 
CONTINUITY OF FUNCTIONS, 
 
 The following equations also are to be regarded as equations 
 of definition : 
 
 sin^ 
 tan 2 — 
 
 COS-S: 
 cot 2 = , 
 
 sin^ 
 
 cos 2 
 
 I 
 sec 2 = . 
 
 I 
 cosec 2 = — , 
 
 cos-s sin ^ 
 
 It may be shown that the formulas which are usually obtained 
 on the supposition that the independent variables are real, and 
 which express in that case properties of and relations between 
 the preceding functions, still hold when the independent 
 variables are complex. 
 
 Prob. 2. Show that e'^e'* = e'*"^ ", m and n being complex. 
 
 Prob. 3. Deduce cos z = i{e'' + e-% sin z — --.(e'' — e'"). 
 
 Prob. 4. Deduce cos {z^ + z^ = cos z^ cos z^ — sin z^ sin z^ , 
 sin {z^ 4" ^3) = cos z^ sin z^ + sin 5, cos z^ , 
 
 Art. 5. Continuity of Functions. 
 
 Let a function of a single independent variable have a 
 determinate value for a given value c of the independent vari- 
 able. If, when the independent variable is made to approach 
 c, whatever supposition be made as to the method of approach, 
 the function approaches as a limit its determinate value at c, 
 the function is said to be " continuous " at c. 
 
 This definition may be otherwise expressed as follows : A 
 function of a single independent variable is continuous at the 
 point c, when, being given any positive quantity e, it is possible 
 to construct a circle, with center at c and radius equal to a 
 determinate quantity d, so small that the modulus of the 
 difference between the value of the function at the center and 
 that at every other point within the circle is less than e. 
 
 A function of several independent variables is said to be 
 continuous for a particular set of values assigned to those vari- 
 ables, when it takes for that set of values a determinate value 
 c, and for every new set of values, obtained by altering the 
 
FUNCTIONS OF A COMPLEX VARIABLE. 
 
 variables by quantities of moduli less than some determinate 
 positive quantity 6, the value of the function is altered by a 
 quantity of modulus less than any previously chosen arbitrarily 
 small positive quantity e. 
 
 A function of one independent variable is said to be con- 
 tinuous in a given region of the plane upon which its indepen- 
 dent variable is represented, if it is continuous at every point 
 in that region. 
 
 From the principles of limits, it follows that if two functions 
 are continuous at a given point, their sum, difference, and prod- 
 uct are continuous at that point. As an immediate conse- 
 quence, every rational entire function of ^ is continuous at 
 every finite point ; for every such function can be constructed 
 from 2 and constant quantities by a finite number of additions, 
 subtractions, and multiplications. 
 
 Let a function of a single independent variable be contin- 
 uous at c, and let it take at that point the value /, different 
 from zero. Suppose also that at any other point ^+^^ the 
 function takes the value / + ^^- Then 
 
 I I J/ 
 
 t + Jt t t{t-\- At) 
 
 If it be assumed that | J/ 1 < | / |, the modulus of the preceding 
 difference cannot exceed 
 
 \At\ 
 
 \t\{\t\-\At\)' 
 and will, therefore, be less than e if 
 
 eUl' 
 
 \At\< 
 
 \+e\t\ 
 
 Hence if a function is continuous and different from zero 
 at a point Cy its reciprocal is also continuous at c. It follows 
 at once that if two functions are both continuous at c, their 
 ratio is continuous at c, unless the denominator reduces to zero 
 
GRAPHICAL REPBESEKTATIOIS" OP PU2fCTI0IfS. 7 
 
 at that point. But every rational function of z may be expressed 
 as the ratio of two entire functions. It is therefore continuous 
 for all values of z except those for which its denominator 
 vanishes. 
 
 Consider the function exp^, 
 
 Hence if \Az\<\^ 
 
 but the limit of the third member is zero when \Az\ ap- 
 proaches zero. Hence exp z is continuous for all finite values 
 of z, 
 
 Prob. 5. Show that cos z and sin z are continuous for all finite 
 values of z. 
 
 Prob. 6. Show that tan z is continuous in any circle described 
 about the origin as a center with a radius less than ^n. 
 
 Art. 6. Graphical Re^presentation of Functions. 
 
 It was shown in Art. 2 that a plane suffices for the complete 
 graphical representation of the values of an independent vari- 
 able. In the same way it is convenient to use a second plane 
 to represent graphically the values of any one-valued function. 
 For example, if w ^=f{z) be such a function, to each point 
 X -\-iy o{ the independent variable will correspond a point 
 II + iv of the function. This point u + iv is called the " image " 
 of the point x -\-iy. If te^ is a continuous function of z, then 
 every continuous curve in the <^-plane will have an image in 
 the w-plane, and this image will be also a continuous curve. 
 
 Consider the expression u -^ iv = x"^ -\- y^ -\- lixy. Here 
 
8 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 u = X* -\- y and v = 2xy, Since to every value of 2 corre- 
 spond determinate values of x and j/, 
 and consequently determinate values 
 of u and v, this expression falls un- 
 der the general definition of a func- 
 tion of ^. It is evidently continuous. 
 Every straight line x = t parallel to 
 the axis of / is converted by means 
 of it into a parabola v^ = 4t\u — /'). 
 
 Prob. 7. Find the family of curves 
 into which the straight lines parallel to 
 the axis of y are converted by means of 
 the function u -}- w = x^ — y' -{- 2ixy. 
 of this family imersect. 
 
 Show that no two curves 
 
 Art. 7. Derivatives. 
 
 Let w = f{z) be a given function of 2. If k is an '' infini- 
 tesimal," that is, a variable having zero as its limit, and if the 
 expression 
 
 ' ■ f[z + h )-f{z) 
 
 h 
 
 has a finite determinate limit, remaining the same under all 
 possible suppositions as to the way in which /^ approaches zero, 
 this limit is said to be the " derivative" of the function y(^) at 
 the point z. In this case w = f{z) is said to be " monogenic " 
 at z. The derivative is written f'{z) or -r-. A function is said 
 to be monogenic in a region of the plane of the independent 
 variable if it is monogenic at every point of that region. 
 
 Consider now the circumstances under which a function 
 w :=. u -\- iv may have a derivative at the point z ^^ x -\- iy. 
 If 2: be given a real increment, x is changed into x ■\- Axy while 
 y is unaltered, so that Az = Ax\ and 
 
 Aw __ Au . Av 
 Az ~~ Ax "" ~Ax' 
 
DERIVATIVES. 9 
 
 If, on the other hand, z is given a purely imaginary incre- 
 ment, Az = iAyy and 
 
 Aw __ Au _, Av 
 
 Az ~~ iAy Ay ' 
 
 If the second members of these equations approach deter- 
 minate limits as Ax and Ay approach zero, and if these limits 
 are equal, 
 
 d^'^^d^~~"^dy^^y 
 
 Hence, equating real and imaginary parts, 
 
 du _ 9^ dv _ 'du 
 
 dx~'dy' dx~~dy* 
 
 which are necessary conditions for the existence of a derivative. 
 
 It can be shown that these conditions are also sufficient * 
 For let the increment of the independent variable be entirely 
 arbitrary, no supposition being made as to the relative magni- 
 tudes of its real and imaginary parts. Then the diffeiiential of 
 the function, that is, that part of the increment of the function 
 which remains after subtracting the terms of order higher than 
 the first, is 
 
 ^ \dJtr ^ dxl . ' \dy ' dy J "^ 
 
 Hence (^u x-'dA. l^u^ , .^\ dy^ 
 
 -; . du + idv _ Va-y a-y/ "^ W ^ dyi dx 
 - ' dx-\-idy~ J ,dy_ 
 
 •^ dx 
 
 which, by virtue of the conditions written above, is equal to 
 either member of the equation 
 
 dx '^ ^dx ~ ^dy dy 
 The value thus obtained is independent of -^, or, what is the 
 
 * For a complete discirssion see article by E. Goursat in the Transactions of 
 the Amer. Math. See, vol. i, p. 14. 
 
10 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 same thing, of the direction of approach to the point z. The 
 existence of a derivative of the function w depends, therefore, 
 
 only on the existence of partial derivatives ;— -, ;:— , — -, -- 
 
 ^x ox djy ^y 
 
 satisfying the specified equations of condition. 
 
 The same equations of condition express the tact that 
 w = « + iv^ supposed to be an analytical expression involving 
 X and 7, and having partial derivatives with respect to each, 
 involves ^ as a whole, that is, may be constructed from z by 
 some series of operations, not introducing x ox y except in the 
 combination x -\- iy. In other words, they indicate that x and 
 y may both be eliminated from w = (p(x, y) by means of the 
 equation z = x -\- iy. This property, however, is not sufficient 
 to define a function as monogenic, for not every function which 
 possesses it has a derivative with respect to z. 
 
 A monogenic function is necessarily continuous ; that is, 
 the existence of a derivative involves continuity. For, if 
 
 it follows that 
 
 where tf approaches zero with h. Hence f{z) is the limit of 
 f{z-\- h) when h approaches zero, or f(z) is continuous at the 
 point z. 
 
 The following pages relate almost exclusively to functions 
 which are monogenic except for special isolated values of z. 
 Functions which are discontinuous for every value of the inde- 
 pendent variable, and functions which are continuous but admit 
 no derivatives, have been little studied except in the case of 
 real variables * 
 
 * In this connection see G. Darboux, Sur les fonctions discontinues, Annales de 
 I'Ecole Normale, Series 2, vol. 4 (1875), pp. 51-112. For a systematic treatment 
 of functions of a real variable, see the German translation of Dini's treatise by 
 Liiroth and Schepp, Leipzig, 1892. For an illustration of a function constructed 
 from 2 by a series of arithmetical operations and discontinuous for a particular 
 value of z, see the expression given on page 53. 
 
CONFORMAL REPKESE^^TATIOJ?". 
 
 11 
 
 Art. 8. Conformal Representation. 
 
 Let z start from the point z^ and trace two different paths 
 forming a given angle at the point z^y and let z^ and z^ be arbi- 
 trary points on the first and second paths respectively. Then 
 
 z^— z^= r,(cos 6^, + ^ sin d^ = r/^\ 
 
 where r, denotes the length of the straight line joining z^ and 
 
 z^ , and 6*, denotes the inclination of this line to the axis of 
 reals. In the same way, for the point z^y there is an equation 
 
 z^ — z^= r, (cos 8^ + ^ sin 6*,) = r/^-^. 
 
 If now w is a one-valued monogenic function of z, in the 
 region of the <s:-plane considered, to the points z^, z^, z^ corre- 
 spond points w^yW^yW^'^ and for these points can be formed 
 the equations 
 
 ^1 — ^0 = Pi^'** » w^ — w^— p,^'*'. 
 
 From the supposition that w is monogenic, it follows at 
 once that, when z^ and z^ are assumed to approach z^^ 
 
 hmit ~ ° = limit — ^ ?. 
 
 z, — z, z,- z. 
 
 If the members of this equation are not equal to zero, it may 
 be put in the form 
 
 limit — ^ -" 
 
 limit 
 
 ^1- 
 
 w. 
 
12 FUNCTIONS OF A COMPLEX VABIABLE. 
 
 or 
 
 limit ^>^'<*»-^«> = limit !:v '<'''-••>. 
 
 Hence 
 
 limit (0,- 0,) = limit {6,- 0,) ; 
 
 and the images in the ze^-plane of the two paths traced by J3 
 form at w^ an angle equal to that at 2„ in the ^-plane. Accord- 
 ingly, if 2 be supposed to trace any configuration whatever 
 
 in a portion of the -s'-plane in which — — is determinate and not 
 
 equal to zero, every angle in the image traced by w will be 
 equal to the corresponding angle in the -a-plane. If, for exam- 
 ple, such a portion of the -sr-plane be divided into infinitesimal 
 triangles, the corresponding portion of the «;-plane will be 
 divided in the same manner, and the corresponding triangles 
 will be mutually equiangular. Such a copy upon a plane, or 
 upon any surface, of a configuration in another surface is called 
 a " conformal representation." 
 
 The modulus of the derivative — - := limit —— is the 
 
 dw ,. . 
 -r- = hmit 
 dz 
 
 Aw 
 
 Az 
 
 " magnification." Its value, which, in general, changes from 
 point to point, may be obtained from the relations 
 
 dw 
 d^ 
 
 ■={S)'+©'=(0+©' 
 
 '~' ^x -dy -dy dx 
 
 The theory of conformal representation has interesting ap- 
 plications to map drawing.* 
 
 *For the literature of the subject, see Forsyth, Theory oi Functions, 
 p. 500, and Holzmtiller, Einfiihring in die Theorieder isogonalen Verwandschat- 
 ten und der conforraen Abbildungen, verbunden mit Anwendungen auf mathe- 
 matische Physik. 
 
EXAMPLES OF CONFORMAL REPRESENTATION. 
 
 13 
 
 Art. 9. Examples of Conformal Representation. 
 
 Example I. — Let w = s -{- c. This function is formed 
 from the independent variable by the addition of a constant. 
 Putting for zi/, z^ and ^, respectively, u + iVy x + iy^ and a + ib, 
 one obtains 
 
 u-=.x-\-ay v=y-\-d. 
 
 Any configuration in the ^-plane appears, therefore, in the 
 w-plane unaltered in magnitude, and is situated with respect to 
 the axes as if it had been moved parallel to the axis of reals 
 through the distance a and parallel to the axis of imaginaries 
 through the distance d. The following diagrams represent the 
 transformation of a network of squares by means of the rela- 
 tion w = 2 -}- c. 
 
 y 
 
 X 
 
 Example II. — Let w =z cs. Writing w = pe^y zr=ire^^, 
 and c = r^e^^y the following equations result: 
 
 The origin transforms into the origin, all distances measured 
 from the origin are multiplied by a constant quantity, and 
 all straight lines passing through the origin are turned through 
 a constant angle. See the following diagrams. 
 
14 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Example III. — Let w = ^*. Writings ^ x-\-iy, the func- 
 tion becomes 
 
 w = e*^^ = ^*(cos^ 4- / sin^). 
 
 Every straight line x = t^ parallel to the axis of y is trans- 
 formed into a circle p =, ^* described about the origin as a 
 center, the axis of J becoming the unit circle. Points to the 
 right of the axis oi y fall without the unit circle, while points 
 to the left of this axis fall within. Every straight line y = t^ 
 parallel to the axis of x becomes a straight line v/u ■= tan /, 
 passing through the origin. The accompanying diagrams* 
 exhibit in a simple manner the periodicity expressed by the 
 
 equation 
 
 exp {2 -f 2n7ti) = exp (-3^), 
 
 where n is any positive or negative integer. 
 
 To every point in the w-plane, excluding the origin, corre- 
 spond an infinite number of points in the .^-plane. These 
 points are all situated on a straight line parallel to the axis of 
 
 *The figures of this and the following example are taken from Holzmilller's 
 treatise. 
 
EXAMPLES OF CONFORMAL REPRESENTATION. 
 
 15 
 
 y, and divide it into segments, each. of length 2;r. If z' be one 
 of these points, the general value of the inverse function is 
 
 log w z=z z' Ar 2ni7r, 
 
 where n is any positive or negative integer. 
 
 If any straight line beginning at the origin be drawn in the 
 ij£/-plane, there will correspond in the ^-plane an infinite number* 
 
 27t- 
 
 of straight lines parallel to the axis of x, dividing that plane 
 into strips of equal width. To any curve in the w-plane 
 which does not meet the line just drawn, will correspond in 
 the -s'-plane an infinite number of curves, of which there will be 
 one in each strip. 
 
 Example IV. — Let w = 'cos z. Writing w = u-{-w, z = 
 Xr\-iy, and employing as equations of definition cos (/r) = 
 cosh J, sin {iy) = / sinh y, the given function takes the form 
 
 Hence 
 
 u -f- iv == cos X cosh y — t sin x sinh y. 
 u = cos X cosh y, V ^= — sin ;r sinh^. 
 
16 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Any straight line, x = /,, parallel to. the axis of ^, is trans- 
 formed into one branch of a hyperbola, ^^ 
 
 = I, 
 
 cos' /, sin'' /, 
 
 having its foci at the points + i and — i. Any straight line, 
 ^ = /, , parallel to the axis of x, is transformed into an ellipse, 
 
 + 
 
 = I, 
 
 cosh' /, sinh' /, 
 
 having its foci at the same points, any segment of the straight 
 line equal in length to 27t corresponding to the entire curve 
 taken once. By means of these confocal conies, the w-plane 
 is divided into curvilinear rectangles, the conformal represen- 
 tation breaking down only at the foci, where the condition 
 dw 
 
 that ~ should be different from zero is not fulfilled. 
 az 
 
 periodicity of the function, expressed by the equation . 
 COS(<S' + 271) — cos-s, r 
 
 The 
 
 ) 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 15 
 
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 2 
 
 
 
 6 
 
 P 
 
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 K 
 
 L 
 
 M 
 
 N 
 
 
 
 P 
 
 A 
 
 B 
 
 X 
 
 
 B 
 
 A 
 
 F 
 
 
 
 N 
 
 M 
 
 L 
 
 K 
 
 J 
 
 I 
 
 H 
 
 G 
 
 F 
 
 E 
 
 D 
 
 C 
 
 B 
 
 A 
 
 P 
 
 
 
 
 
 2 
 
 1 
 
 16 
 
 15 
 
 14 
 
 13 
 
 13 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 i 
 
 3 
 
 2 
 
 1 
 
 16 
 
 15 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 is exhibited graphically 
 in the accompanying 
 diagrams. 
 
 It is interesting to 
 note in this example, 
 as also in the preceding 
 one, that the conformal 
 representation intro- 
 duces well-known sys- 
 tems of curvilinear 
 coordinates, the cartesian coordinates, x, f o( a point in the 
 
EXAMPLES OF COifFORMAL REPRESEN^TATION". 
 
 17 
 
 ^-plane serving to determine its image in the ze/-plane as an 
 intersection of orthogonal curves. 
 
 Example V. — Let w = z^. Writing w = « + «/, ^ = 
 X + iy^ the relations 
 
 « u = x^ ^ s^y^t 'v = s^y — y 
 
 follow at once. If one of the variables x,j/ be eliminated from 
 these two equations by means of the equation /x -\- my -\-n = Oy 
 representing a straight line in the ^--plane, equations are ob- 
 tained representing a unicursal cubic in the w-plane. 
 
 By putting w = p{cos (p -{- i sin 0), z = r(cos -{- t sin 6\ 
 the relations p = r\ = 36^, are obtained. Hence the 
 circle 
 
 r' — 2ar cos 6 -{• a* = c* 
 
 gives the curve 
 
 pi — 2ap^ cos — -f ^' = ^, 
 
 which enwraps three times the point corresponding to the 
 center. The accompanying figure represents this transfor- 
 mation, the straight Hne /<?^ giving the curve /e^. 
 
 div 
 
 To each point in the w-plane, excluding the origin, at which 
 = O and the conformal representation is not maintained, 
 
18 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 there correspond three points in the -s^-plane, having for their 
 -f- 2;r -f- 4?r 
 
 arguments -, 
 
 respectively. Any straight Hne 
 
 drawn from the origin in the w-plane will have, therefore, three 
 images in the -s^-plane, viz., three straight lines diverging from 
 the origin, and dividing the plane into three equal regions. 
 Any continuous curve in the w-plane not meeting the line just 
 drawn will be represented in the ^-plane by three curves, of 
 which one will be situated within each of these regions. In the 
 figure here given are exhibited the three conformal represen- 
 tations of a square formed in the 7£;-plane by lines u = t^, u = 
 t^,v=t^^v = t^y parallel to the axes. 
 
 If the relation between w and z be reversed, and z be 
 taken as a function of w, z will be a three-valued function, its 
 values giving rise to three branches which will remain distinct 
 and continuous except when w becomes equal to zero. 
 
 d c 
 a h 
 
 Prob. 8. U w = z -\ , show that circles in the ^-plane having 
 
 z 
 
 a common center at the origin transform into confocal ellipses. 
 
 Prob. 9. If IV 
 
 ;. show that the axis of reals in the s'-plane 
 
 transforms into the circle \7v\ = i, and the upper half of the ^-plane 
 into the interior of this circle. 
 
COKFORMAL REPRESENTATION^ OF A SPHERE. 
 
 19 
 
 Art. 10. CoNFORMAL Representation of a Sphere. 
 
 Let OPO' be a sphere having its diameter 00' equal in 
 
 length to unity. Con- 
 struct tangent planes at 
 at 6> and O' . Draw in 
 the tangent plane at 
 O rectangular axes Ox 
 and Oy ; and in the 
 other plane draw as 
 axes O'u, parallel to Ox 
 and measured in the 
 same direction, and O'v 
 parallel to Oy but meas- 
 ured in a contrary di- 
 rection. Join any point 
 z in the plane xOy to 
 C by a straight line, and let O'z meet the sphere in P. Draw 
 (9Pand produce it to meet the plane uO'v in w. 
 
 From the similar triangles O' Oz and OO'w 
 
 Oz 
 00' 
 
 00' 
 O'w 
 
 that is, 
 
 or Oz . O'w = 00' 
 
 w\ = rp= I, 
 
 To an observer standing on the sphere at O' rotation about 
 OO' from O'lc toward O'v is positive, while to an observer 
 standing on the sphere at O such a rotation is negative. 
 Hence 
 
 Z_xOz = — Z^O'w, or 6^ = — 0. 
 
 The following equation results : 
 
 wz 
 
 pre 
 
 t(<f> + 0) 
 
 The W' and ^-planes are therefore conformal representa- 
 tions of one another. Any configuration in one plane can be 
 formed from its image in the other by an ijiyersjon with respect 
 
20 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 to the origin as a center, combined with a reflection in the axis of 
 reals. Such a transformation was termed by Cayley a " quasi- 
 inversion." By it points at a great distance from the origin 
 in one plane are brought near together in the immediate neigli- 
 borhood of the origin in the other plane. 
 
 "Since the Hne O' Pz makes the same angle with the plane 
 tangent to the sphere at P as with the plane xOy^ any spherical 
 angle having its vertex at P is projected into an equal angle at 
 z. The sphere is thus seen to be related conformally to the 
 plane xOy, and it must be also so related to the plane uO'v, 
 
 The representation of the sphere upon a tangent plane in 
 the manner described above is termed a "stereographic pro- 
 jection." When to this representation is applied a logarithmic 
 transformation, that is, one inverse to the transformation 
 described in Example III of the preceding article, the so- 
 called * * Mercator's projection ' ' is obtained. 
 
 Art. 11. Conjugate Functions. 
 
 The real and imaginary parts of a monogenic function, 
 ze; = « 4" ^^> have been shown to satisfy the partial differential 
 equations 
 
 a« _ a^ 'd;v__ _'du 
 
 dx ~ dy 'dx ~ dy' 
 
 At any point, therefore, where u and v admit second partial 
 derivatives, one obtains 
 
 a^ , a^_ av , d'v _ 
 dx^'^dr ' dx'~^dy ' 
 
 that is, the functions u and v are solutions of Laplace's equa- 
 tion for two dimensions. Any two real solutions / and ^ of 
 this equation, such that p-\-iq is a monogenic function of 
 X + iy, are called " conjugate functions." * Thus the examples 
 of Art. 9 furnish the following pairs of conjugate functions: 
 
 * Maxwell, Electricity and Magnetism, 1873, vol. i, p. 227. 
 
applicatio:n' TO fluid motion". 21 
 
 x -\- a, y -\- d ] r^r cos {B^ + ^)» ^i^ sin (6^, ■\-B)\ ^* cos j, e^ s\x\y ; 
 cos X cosh J, — sin x sinhj ; x^ — 3^jk', Z^y^ — y^- Tlie second 
 pair is expressed in polar coordinates, but may be transformed 
 to cartesian coordinates by means of the relations 
 
 r = Vx' +/, cos e = - _ ^ sin 6 
 
 Vx'+f , Vx'+f 
 
 If one of two conjugate functions be given, the other is 
 thereby determined except for an additive constant. Let Uy 
 for example, be given. Then 
 
 , 'dv y . dVy 
 dv = — ax 4- —ay 
 
 'dx dy 
 
 — ax A ay. 
 
 dy ^ dx -"' 
 
 and therefore the value of v is 
 
 /( 
 
 dy ^ dx-"^ 
 
 The equations u =^ c^, v =^ c^, obtained by assigning con- 
 stant values to two conjugate functions, represent in the 
 7^-plane straight lines parallel to the coordinate axes. It 
 follows that the curves which these equations define in the 
 ^-plane intersect at right angles. Consequently, by varying 
 the quantities c^ and ^,, two orthogonal systems of curves are 
 obtained ; and c^ and c^ may be taken as orthogonal curvilinear 
 coordinates for the determination of position in the ^-plane. 
 
 Prob. lo. Show that if / and ^ are conjugate functions of u and 
 V, where u and v are conjugate functions of x andj, / and ^ will be 
 conjugate functions of x and y. 
 
 Prob. II. Show that if u and v are conjugate functions of x and 
 y, X and^ are conjugate functions of u and v. 
 
 Art. 12. Application to Fluid Motion. 
 
 Consider an incompressible fluid, in which it is assumed 
 that every element can move only parallel to the ^-plane, and 
 has a velocity of which the components parallel to the coordi- 
 
22 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 nate axes are functions of x and ^ alone. The whole motion 
 of the fluid is known as soon as the motion in the ^-plane is 
 ascertained. When any curve in the ^-plane is given, by the 
 "flux across the curve""* will be meant the volume of fluid 
 which in unit time crosses the right cyHndrical surface having 
 the curve as base and included between the ^-plane and a par- 
 allel plane at a unit distance. 
 
 The flux across^ any two curves joining the points js^ and 2 
 IS the same, provided the curves enclose a region covered with 
 the moving fluid. For, corresponding to the enclosed region, 
 there must be neither a gain nor a loss of matter. Let s^ be 
 fixed, and 2 be variable. Let ip denote the flux across any curve 
 s^2, reckoned from left to right for an observer stationed at 2^ 
 and looking along the curve toward 2. If /, m be the direction 
 cosines of the normal (drawn to the right) at any point of the 
 curve, and /, g be the components parallel to the axes of the 
 velocity of any moving element, the value of ip will be 
 
 
 where the path of integration is the curve joining z^ and z. 
 The function ^ is a one-valued function of z in any region 
 within which every two curves joining z^ to z enclose a region 
 covered with the moving fluid. 
 
 If z moves in such a manner that the value of rp does not 
 vary, it will trace a curve such that no fluid crosses it, i.e., a 
 " stream-line." The curves ^ = const, are all stream-lines, and 
 ^ is called the " stream-function." If p and q are continuous, 
 and if z be given infinitesimal increments parallel to x and y 
 respectively, one obtains 
 
 'dx~ ^' -dy ^* 
 If now the motion of the fluid be characterized, as is the 
 
 * Lamb's Hydrodynami s (1895). p. 6q. 
 
APPLICATION TO FLUID MOTION. 
 
 23 
 
 case in the so-called " irrotational" motion * by the existence 
 of a velocity-potential 0, so that 
 
 ^ 90 90 
 
 the following equations result : 
 
 a0_a^ 9^_ _a0 
 
 Hence + /^ is a monogenic function of ;tr + iy. The curves 
 = const., which are orthogonal to the stream-lines, are 
 called the " equipotential curves." 
 
 Consider, as an example, the motion corresponding to the 
 functionf w = z^. The equipotential curves are given by the 
 
 equations 
 
 « = ;tr* — 3;ry = CO n St. , 
 
 the stream-lines by 
 the equations 
 
 V = 'iyx'y —y^^= const. 
 
 In the following fig- 
 ure the stream-lines 
 are the heavy lines, 
 while the equipo- 
 tential curves are 
 dotted. 
 
 The fluid moves 
 i in toward the origin, 
 
 which is called a '• cross-point," from three directions, and 
 flows out again in three other directions. At the cross-point 
 the fluid is at a standstill, since at that point the velocity, for 
 which the general expression is 
 
 \/(g)'+(i)'. 
 
 * In irrotational motion each element is subject to translation and pure 
 strain, but not to rotation. 
 
 f F. Klein : Riemann's Theory of Algebraic Functions ; translated by 
 Frances Hardcastle (1893), p. 3. 
 
24 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 is equal to zero. The stream-lines in the figure represent the 
 motion of the fluid in each of six different angles, as if the fluid 
 were confined between walls perpendicular to the ^-plane. 
 
 It is of importance to note that if the function considered 
 be multiplied by i, the equipotential curves and stream-lines 
 are interchanged, since the function (p -\- tip then becomes 
 — ^ + i(t>. 
 
 An example of particular interest is 
 
 w 
 
 — //log 
 
 + ^ 
 
 Let z — a =^ ^/'^', 2 -\- a =^ r^e'^^ ; then 
 
 « = — // log -i, v= — ^{d, — e,). 
 
 The curves u = const., v = const, form two orthogonal sys- 
 tems of circles, either of which may be regarded as the stream- 
 lines, the other constituting the equipotential curves. 
 
 The velocities are everywhere, except at the points ± ^. 
 finite and determinate. If the circles r,/r, = const, be taken 
 as the stream-lines, each of the points ± ^ is a " vortex-point." 
 If the circles 6*, - ^, = const, be taken as the stream-lines, one 
 
1^ 
 
 SINGULAR POINTS. 25 
 
 of the points ± a is a " source," the other a "sink." In the 
 latter case, besides the hydrodynamical interpretation, a very 
 sinnple electrical illustration is afforded by attaching the poles 
 of a battery to a conducting plate of indefinite extent at two 
 fixed points of the plate. " 
 
 As another example may be taken the relation w = cos ^. 
 As has been shown, the curves x = const, form a system of 
 confocal hyperbolas, while the curves ^ = const, form an 
 orthogonal system of ellipses. Either system may be regarded 
 as stream-lines. In one case the motion of the fluid would be 
 such as would occur if a thin wall were constructed along the 
 axis of reals, except between the foci, and the fluid should be 
 impelled through the aperture thus formed. In the other case 
 the fluid would circulate around a barrier placed on the axis of 
 reals and included between the foci. 
 
 Besides their application to fluid-motion, conjugate func- 
 tions have important applications in the theory of electricity 
 and magnetism * and in elasticity .f 
 
 Art. 13. Singular Points. 
 
 Let w be any rational function of js. It can be written in 
 the form 
 
 «'-0(^)' 
 
 where /(2') and (p {£) are entire and without common factors. 
 This function is finite and admits an infinite number of suc- 
 cessive derivatives for every finite value of ^, except the roots 
 of the equation (-S") = o. Let a be such a root. Then the 
 reciprocal of the given function is finite and admits an infinite 
 number of successive derivatives at the point a. Such a point 
 
 * J. J. Thomson, Recent Researches in Electricity and Magnetism (1893), 
 p. 208. 
 
 \ Love, Theory of Elasticity (1892), vol. i, p. 331. 
 
26 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 is called a "pole." Any rational function having a pole at a 
 can be put by the method of partial fractions in the form 
 
 where A^y . . .^ A,, are constants, A^ being different from zero, 
 and tl){z) is finite at the point a. The integer k is said to be 
 the "order'* of the pole, and the function is said to have for 
 its value at a infinity of the >^th order. In accordance with 
 the definition of a derivative, w does not admit a derivative at 
 a. From the character of the derivative in the immediate 
 neighborhood of a, however, the derivative is sometimes said 
 to become infinite at a. 
 
 The trigonometric function cot^ has a pole of the first 
 order at every point z = nnty m being zero or any integer posi- 
 tive or negative. 
 
 The function w = log (2' — a) has for every finite value of 
 2, except z = a, a.n infinite number of values. U z — a is writ- 
 ten in the form Re*®, 
 
 w — log R -\- z{© + 2m7r)f 
 
 where log R is real, and m is zero or any positive or negative 
 integer. If 3 describes a straight line, beginning at a, S will 
 remain fixed, but R will vary. The images in the w-plane will 
 therefore be straight lines parallel to the axis of reals, dividing 
 the plane into horizontal strips of width 27t. If now the ^--plane 
 is supposed to be divided along the straight line just drawn, 
 and z varies along any continuous path, subject only to the 
 restriction that it cannot cross this line of division, there will 
 be a continuous curve as the image of the path of z in each 
 strip of the 2£/-plane. Each of these images is said to corre- 
 spond to a "branch" of the function, or, expressed otherwise, 
 the function is said to have a branch situated in each strip. 
 The line of division in the ^-plane, which serves to separate 
 the branches from one another is called a " cut." 
 
3IKGULAR POIN-TS. 27 
 
 At the point 2 = a no definite value is attached to the 
 function. As z approaches that point the modulus of the real 
 part of the function increases without limit, while the imagi- 
 nary part is entirely indeterminate. 
 
 Let 2^ be an arbitrary point, distinct from a, and let 
 
 log R^ + t&^ + 2m7rt 
 
 be any one of the corresponding values of the function. Sup- 
 pose that 2 starts from 2^ and describes a closed path around 
 the point a, the values of the function being taken so as to 
 give a continuous variation. Upon returning to the point 2^ 
 the value of the function will be 
 
 log R, + /©„ + 2{m + i)7[t, 
 or log R, + iQ^ -{-2(m— i)7ti, 
 
 according as the curve is described in a positive or negative 
 direction. By repeating the curve a sufficient number of times 
 it is evidently possible to pass from any value of the function 
 at z^ to any other. When a point is such that a ^-path en- 
 closing it may lead in this manner from one value of a function 
 to another value, it is called a " branch-point." In the case 
 of the function here considered, the point z =1 a is called 
 a "logarithmic branch-point," or a point of "logarithmic 
 discontinuity." 
 
 The function w = log ^^-)-{, where /{z) and (p{z) are entire, 
 
 (p{z) 
 
 has a point of logarithmic discontinuity at every point where 
 either /(^) or (p{z) is equal to zero. For, writing 
 
 /{z) = A{z - ay^{z - a.y^ . . . 
 0(^) = B{z — b,)^^{z — ^,)^» ... 
 
 the value of w may be written 
 
 A 
 w = log -B + ^">- log {z — a^) - ^q^ log {z — hi), 
 
 £> rn. n 
 
28 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Take now the function w = e'*. It has a single finite value 
 for every value of z except 2 = 0. If ^ is supposed to ap- 
 proach zero, the limit of the value of the function is indeter- 
 minate. 
 
 For let / + iq be perfectly arbitrary, and write 
 
 If now a + lb is the reciprocal of / + iq, so that 
 — / A _ —^ 
 
 the preceding equation may be written 
 
 e^TTb z= c -\- id. 
 
 But whatever the value of the integer m, q -\- imn may be 
 substituted for q without altering the value of ^ + id, and hence 
 both a and b may be made less than any assignable quantity. 
 
 The given function e^ therefore takes the value c -\- id at points 
 a + ib indefinitely near to the origin. A point such that, when 
 z approaches it, a fiinction elsewhere one-valued may be made 
 to approach an arbitrary value is called an • * essentiajl^ singu- 
 larity." ^ 
 
 Prob. 12. Show that for the function ^«^^ ^ = dt is an essei^ial 
 singularity. 
 
 Prob. 13. The function e ^* considered as a function of a real 
 variable is continuous for every finite value of z, and the same is 
 true of each of its successive derivatives. Show that when it is 
 regarded as a function of a complex variable, -? = o is an essential 
 singularity. 
 
 In order to illustrate still another class of special points 
 take the function 
 
 w=s/{z — a^{z — «,)... (^ — ^«). 
 
SINGULAR POINTS. 29 
 
 ni 
 
 This function has at every finite point, except <^j, «,,..., ^, 
 two distinct values differing in sign. At these points, however, 
 it takes but a single value, zero. From each of the points 
 a^, a^, . . . , a„\et a. straight line of indefinite extent be drawn in 
 such a manner that no one of them intersects any other, and 
 suppose the ^s-plane to be divided, or cut, along each of these 
 lines. Along any continuous path in the ^-plane thus divided 
 the values of the function form two distinct branches. 
 
 For, writing 
 
 js — a^ = r/'*i, z — a^z=z r,^''*9, . . , , ^ — a« = r,^**", 
 
 the function takes the form 
 
 A+h+ ... +t 
 
 w = Vr^r^ . . r„ e* 
 
 , No closed path in the divided plane will enclose any of the 
 points a^, a^, . . . , a„, and the quantities 6^, d,, . . . , 0„, after 
 continuous variation along such a path, must resume at the 
 initial point their original values. No such path, therefore, can 
 lead from one value of the function at any point to a new 
 
 value of the function at the 
 same point. If, however, the 
 cuts are disregarded and s 
 traces in a positive direction, 
 a closed curve including an odd 
 number of the points a^, ^,, 
 . . . y a„, and not intersecting 
 itself, then an odd number of 
 
 the quantities O^y 6^,, . . . , ^„ are each increased by 27r; and 
 the value of the function is altered by a factor ^(2*+i)t»^ and 
 so changed in sign. In the same way any closed path de- 
 scribed about one of these points, and enwrapping it an odd 
 number of times, leads from one value of the function to 
 the other. On the other hand, a simple closed path enclosing 
 an even number of these points, or- a closed path which en- 
 closes but one of the points, enwrapping it an even number of 
 times, leads back to the initial value of the function. It fol- 
 
80 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 lows that each of the points ^, , ^, , . . . , ^^ is a branch-point. 
 Any point in the ^-plane, closed paths about which lead from 
 one to another of k set of different values of a function, the 
 number of values in the set being finite, is called an " algebraic 
 branch-point." 
 
 As a further illustration, consider the function 
 w = 2^-{-(2 — a)\ 
 which is a root of the equation of the sixth degree, 
 
 w* —S^w* — 2(2 — ayw" + S-s'w' — 62(2 — a)w -\- (2— a)^— 2*^0. 
 
 The function has at every point, except 2 = and 2 ^a^ 
 six distinct values. Six branches are thereby formed which 
 can be completely separated from one another by making cuts 
 from the points 2 = and 2 = a to infinity. Putting a> for the 
 cube root of unity, these six branches can be written 
 
 1/2 , , vl/3 1/2 , , .1/3 
 
 w^ = 2 -{- {2 — a)' , w^= — 2 -^ {2 — a)^ , 
 w^ — z^^ -{- gd{2 — d)'\ w^-=^ — 2^ -\- gd{2 — df^^, 
 
 ^ W^ = 2'^^ + Q0^{2 — dj^^y Zf , = — 2^^ + ^\^ ~ ^)*^'» 
 
 The branches w, and w^, w^ and w^, w^ and w^ are interchanged 
 by a small closed circuit described about s = o, while a small 
 circuit described about 2-= a permutes cyclically the branches 
 w^, w^y w^y and also the branches w^, w^, w^. 
 
 All of the special points examined above, poles, points of 
 / logarithmic discontinuity, essential singularities, and branch- 
 points, are called singular points. In fact, a function, or a 
 branch of a function, is said to have a * * singular point ' * at each 
 point where it fails to have a continuous derivative,* or about 
 which as a center it is impossible to describe a circle of deter- 
 minate radius within which the function, or branch, is one- 
 Valued. 
 
 Any point not a singular point is called an " ordinary point." 
 
 * Continuity and, therefore, finiteness of the function are implied in the 
 existence of a derivative. 
 
POINT AT INFINITY. 31 
 
 An ordinary point at which a function reduces to zero is called 
 a **zero" of the function. 
 
 If in a certain region of the ^-plane a function is uniform 
 and has no singular points, the function is said to be ''synec- 
 tic " or *'holomorphic " in that region. If in a certain region 
 the only singular points of a uniform function are poles, the 
 function is said to be '* meromorphic " in that region. Under 
 similar conditions, a branch of a function is also described as 
 holomorphic or meromorphic. 
 
 Prob. 14. When w and z are connected by the relation w — g ^ 
 (z — /lY show that if z describes a circle about >^ as a center, w 
 describes a circle about g as a center, an angle in the ^-plane hav- 
 ing its vertex at /i is transformed into an angle in the w-plane t 
 times as great and having its vertex at g^ and that 2 = ^ is a branch- 
 point of w except when / is an integer. 
 
 Art. 14. Point at Infinity. 
 
 In determining the limiting value of a function when the 
 modulus of the independent variable z is increased indefinitely, 
 it is usual to introduce a new independent variable z' by the 
 relation z — \/z' , and consider the function at the point z' — o. 
 This is equivalent to passing from the ^-plane to another plane, 
 the^^-plane, related to the former by the geometrical construc- 
 tion described in Art. 10. It is often very convenient, however, 
 to go further and to suljgtitute for the ^-plane the surface of the 
 sphere of unit diameter touching the ^r-plane at the origin. No 
 difficulty is thus introduced since, as explained in the article 
 just cited, any configuration in the -a-plane obtains a conformal 
 representation upon the sphere; and the advantage is gained 
 that the entire surface upon which the variation of the inde- 
 pendent variable is studied is of finite extent. The point of 
 the sphere diametrically opposite to its point of contact with 
 the ^■-plane coincides with the point written above as z' = o. 
 It is called the point at infinity, 5' = 00 , since a point on the 
 sphere approaches it at the same time that its image in the 
 .s:-plane recedes indefinitely from the origin. 
 
32 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 The point at infinity may be either an ordinary or a singular 
 
 point. For the function ^, for example, it is an ordinary 
 
 I 
 point, since f» = e*' . For a rational entire function of the «th 
 
 degree it is a pole of order «. Consider it for the function 
 ^{z — a^){z — a^ . . .{z — a^), discussed in the preceding article. 
 Let a circle of great radius be described in the ^-plane inclosing 
 all the branch-points ^, , tf,, . . . , ^„. Its con formal representa 
 tion on the sphere will be a small closed curve surrounding the 
 point <s'= 00. This point must, therefore, be regarded as a 
 branch-point or not, according as the function changes value or 
 not when the curve surrounding it is described, that is accord- 
 ing as «, the number of finite branch-points, is odd or even. 
 When the point at infinity is taken into account, then, the 
 total number of branch-points of this function is always even. 
 The character of the point ^ = 00 for this function can be de- 
 termined directly, by changing z into i/z' and considering the 
 point 2' = o. 
 
 Prob. 15. Show that = 00 is an ordinary point for ~rr4j where 
 
 <p{z) and fp{z) are rational and entire if the degree of (p{z) does 
 not exceed that of tp(z)» 
 
 Art. 15. Integral of a Function. 
 
 Let w=/(z) be a continuous function of ^ in a given 
 region, and suppose z to describe a continuous path L from 
 the point z^ to the point Z. Let a series of points z,, z^, . . . ,-^„ 
 be taken on Z, and let Z^, /j, . . . , t„ be points arbitrarily chosen 
 on the arcs z^z^, z^z^, . . . , z^Z respectively. Form the sum 
 
 s={z,- z:)At:) + (^, - ^, wo + . . . + (^ - ^j/(/«). 
 
 If now the number of points z^y . . . , ^„ be increased indefi- 
 nitely in such a manner that the length* of each of the arcs 
 
 * It is assumed in regard to every path of integration that the idea of length 
 maybe associated with the portion of it included between any two of its points, 
 or, what is the same thing, that the path is rectifiable. This condition is evi- 
 dently satisfied if the current coordinates x and;)/ can be expressed in terms of 
 
IKTEGKAL OF A FUNCTIOK. 
 
 33 
 
 ^^3^fS^JS^f ,,,yZnZ approaches zero as a limit, the sum 5 ap- 
 proaches a finite Hmit which is inde- 
 pendent of the choice of the points z^^ 
 ^„ ...,^„and t,, t^,..., /«. 
 
 For take any other sum 
 
 5' = (v-^.)y(0+ 
 
 formed in a similar manner. Suppose 
 " for the sake of greater definiteness 
 and z/f . . . follow one another on the 
 
 that the points z^ , 
 line L in the order 
 
 Z^f Z^ , Z^ f Z^f Z^, Z, f , , , f 
 
 and form a third sum 
 
 in which b )th series of points occur. It may be shown that as 
 the number of points in each of the series -sr,, . . . and z/, ... is 
 increased, the differences S'^ — S and S^' — S' both approach 
 zero, from which it follows that the difference 5 — 5' has a 
 limit equal to zero. For example, the difference 5'' — 5 has 
 the value 
 
 (^. - ^.)[/(r.) -/('.)] + W - ^,)[/(r.) -/(/,)] 
 If M denotes the upper limit or bound of the quantities 
 
 |y(n)-AOI. \Ar,)-M)U \A-',)-M)\ 
 
 the modulus of 5" — 5 will be less than 
 
 dx dy . „ 
 
 any parameter / so that — and j- are continuous. For then the integral 
 
 / ^ dx'' + dy* is finite. See, in this connection, Jordan, Cours d'Analyse, 2d 
 Edition, Vol. I., p. lOO. 
 
34 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 But 1^, — -s-jI is equal to the chord of the arc s^o-S', , and must 
 therefore be less than or equal to this arc, and a similar result 
 holds for each of the quantities | z^ —z^ \A^\ ~^x I » • • • Hence 
 
 \S" -S\<Ml. 
 
 where / denotes the length of the path of integration. When 
 the number of points of division on the line L is increased, the 
 differences 
 
 /^^) - /('.). A^^ - At.)< /(n) - /(/,), . . . 
 
 approach zero, since f{z) is continuous.* M accordingly 
 decreases indefinitely and the difference S" — S approaches 
 zero. 
 
 The limit, the existence of which has just been demon- 
 strated, is called the integral of f\z) along the path L. It is 
 
 written I f{z)dz. The definition here given is similar to that 
 
 given for the integral of a function of a real variable. It is 
 unnecessary to specify the path of integration when the inde- 
 pendent variable is restricted to real values, since in that case 
 it must be the portion of the axis of reals included between 
 the limits of integration. 
 
 The following well-known principles, applicable to the case 
 of a real independent variable, may be readily extended to the 
 general case : 
 
 1. The modulus of the integral cannot exceed the length of 
 the path of integration multiplied by the upper bound of the 
 modulus of the function along that path. 
 
 2. The independent variable may be altered by any equa- 
 tion of transformation, but L' , the path of integration in the 
 transformed integral, must be such that it is described by the 
 new variable while z describes L. 
 
 3. If F{z) is any one-valued function having everywhere 
 a continuous function f{z) for its derivative, the equation 
 
 must be true. 
 
 * For a complete discussion it should be shown that the continuity oif{z) is 
 necessarily '•uniform." See Jordan, Cours d'Analyse, 2d Edition, vol. i, p. 183. 
 
INTEGRAL OF A FUNCTION. 35 
 
 To prove the third principle, write F{Z) — F{z^ in the 
 form 
 
 Since the derivative of F{z) \sf{z)y 
 
 F{2,„+;) - F{2„,) = [/(^,„) + ^J(^«.+, - ^m), 
 
 where //,„ has zero for its limit * when z^j^^ is made to approach 
 z^. Hence 
 
 F{Z) - F{z,) = limit 2f{z,,){^^^. - ^n) + limit :2'//,«(^„+, - z^) ; 
 
 or, since the second term of the right-hand member is equal 
 to zero, 
 
 F{Z)-F{.,) = J[Az)dz. 
 
 If no function F{z) fulfilling the preceding conditions is 
 known, the value of the integral requires further investi- 
 
 Consider as an example the integral j — ^ taken from the 
 
 point ^ = — I to the point z = i, the path of integration being 
 the upper half of the circumference of a unit circle described 
 about the origin as a center. Writing z = exp {t6), z will 
 describe the required path while 6 varies from tt to o. 
 
 The equations -, = e'""'^, dz = ie^^dd, 
 z 
 
 dz 
 
 — = ie-^^dB = i cos d dd -\- sin 6 dS ^ id (sin B) — d (cos B\ 
 
 Z" 
 
 follow at once. Hence for the path specified 
 
 J —^-=1 Cd (sin B) — Cd (cos <9) = — 2. 
 
 ~^ It n 
 
 The application of the direct and more famihar method 
 i?'ves the same result: 
 
 + ^ J r- -I r- -1 
 
 dz III III 
 
 2. 
 
 J z' L zj,=-i L zj^^^i 
 
 1 
 
 *The "uniformity" of continuity is involved here. See Jordan, Cours 
 
 d' Analyse, 2d Edition, vol i, p. 184. 
 
36 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 For a path along the axis of reals between the limits of 
 integration this result is unintelligible. The discontinuity of 
 
 the differential, —i, at the point 2 = 0, prevents the considera- 
 
 JS 
 
 tion of such a path ; and that the result should be negative 
 when the differential is at every point of the path positive 
 has no significance. The introduction of the complex variable 
 furnishes a perfectly satisfactory explanation of the result. 
 
 Prob. 16, Show that the integral of — along any senii-circum- 
 
 z 
 
 ference described about the origin as a center is equal to tti. 
 
 Art. 16. Reduction of Complex Integrals to Real. 
 The integral 
 
 may be written in the form 
 
 or, separating the real and imaginary terms, 
 
 / {udx — vdjy) -\- i I {^dx + udy). 
 
 Hence the calculation of the integral may be reduced to 
 the calculation of two real curvilinear integrals. 
 
 The equations 
 
 9« _ ?)V 'du _ _'dv 
 'dx ~ d/ 9j dx 
 
 which express the condition that 2/ + ^^ should be monogenic, 
 express also that 
 
 udx — vdy, vdx -\- udy 
 
 are the exact differentials of two real functions of the variables 
 X, y. Consider the case where these functions are one-valued. 
 
cauchy's theorem. 37 
 
 Denoting them by P(x, y) and Q{x^ y) respectively, the inte- 
 gral may be written 
 
 \_P(,X, Y) - I\x„y:i\ + i\_Q{X, Y) - (2(^., jy.)], 
 
 (•^o»Jo) 3^rid (X, F) being the initial and terminal points re- 
 spectively of the path of integration. 
 
 Art. 17. Cauchy's Theorem. 
 
 Cauchy's Theorem furnishes the necessary and sufficient 
 conditions that a uniform function /(z), having continuous 
 partial derivatives with respect to x and j', should yield within 
 a region bounded by a continuous closed curve a one-valued 
 integral, that is, an integral the value of which, when the lower 
 limit is fixed, depends simply on the upper limit, and not on the 
 path of integration. It will be more convenient, before consider- 
 ing Cauchy's Theorem, to demonstrate the following lemma: 
 
 Lemma. — Let ^ be a portion of the ^-plane, having a bound- 
 ary 5 which consists of a closed curve not intersecting itself, 
 or of several closed curves not intersecting themselves or one 
 another. If at every point of the region A, including its 
 boundary 5, a function W oi the real variables x and j is one- 
 valued and continuous and has continuous partial derivatives 
 
 — — , — — , the relations 
 
 fm. = -Jl^,.,y (a) 
 
 •exist, the integrals in the first members being taken along the 
 boundary in the positive direction, and those in the second 
 members being taken over the enclosed area. 
 
 Denote by A the inclination to the axis of x of the exterior 
 normal at any point of the boundary,* that is, the normal drawn 
 
 * It is assumed that the boundary has a determinate tangent at every point. 
 If the boundary of a given region is not of this sort, the theorem holds for any 
 interior curve ot which this assumption is true. 
 
36 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 to the right as the boundary is described in a positive direction. 
 If any straight line parallel to the axis of x be traced in 
 the direction of increasing values of x, at each point where 
 it passes into the area A^ 
 cos A is negative, and there- 
 fore in the first member of 
 {\) dy=. cos \ (is is negative. 
 At each point where this 
 straight line passes out ol 
 the area A, cos A, and there- 
 fore dy^ in the first member 
 of equation (i), is positive. 
 Hence in the first member 
 of equation (i) the differ- 
 entials Wdy corresponding 
 to a given value of y^ and taken in the order of increasing 
 values of ;r, have signs which, compared with the signs of the 
 corresponding values of W, first differ, then agree, and so 
 on alternately. In order now to compare the integral in the 
 first member of equation (i) with the integral in the second 
 member, it is necessary to take dy as essentially positive. 
 The sum of the differentials in the first member, correspond- 
 ing to a fixed value of jj/, must therefore be written in the 
 form 
 
 where W^, PF, , . . . are the corresponding values of W taken in 
 the order of increasing values of x. But performing now in 
 the second member of equation (i) an integration with respect 
 to X, the same result is obtained, so that the two members of 
 equation (i) become identical, and the equation is verified. 
 
 To obtain equation (2) the same method is used. It is 
 necessary in this case to observe that if a line parallel to the 
 axis of jK is traced in the direction of increasing values of j^, at 
 each point where it enters A, dx in the integral of the first 
 
APPLICATION OF CAUCHY'S THEOEEM. 39 
 
 member must be taken as positive; and at each point where 
 this line passes out o( A, dx in that integral must be taken as 
 negative. 
 
 By means of the preceding lemma, Cauchy's Theorem is 
 easily proved. This theorem may be stated as follows ; 
 
 Theorem. — If, on the boundary of and within a given region 
 Aj a one-valued function w = f{z) is monogenic, and its deriv- 
 ative f{z) is continuous,* the integral ff{z)dz taken along 
 the boundary 5 is equal to zero. 
 
 For writing the integral in the form 
 
 J^wdz = J^{udx — vdy) + i J{udy + vdx\ 
 the preceding lemma gives 
 
 but since at every point of A 
 
 the given integral reduces to zero. 
 
 Art. 18. Application of Cauchy's Theorem. 
 
 From Cauchy's Theorem it follows that, if two different 
 paths Z, and Z, lead from the point z^ to the point Z, and if 
 along these paths and in the region inclosed between them a 
 given function f{z) has no critical points, the integrals of the 
 function taken along these two paths are equal. For two such 
 paths taken together, one described directly, the other re- 
 versed, constitute a closed curve, and the integral taken along 
 
 * Otherwise expressed, the one-valued function /"(sr) has no singular points on 
 the boundary of or within A, (xf.z) is holomorphic in A. It has been shown by 
 Goursat that this theorem can be proved without assuming the continuity of the 
 derivative. , See Transactions Amer. Math, Soc, vol. I. p. 14. 
 
40 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 it is equal to zero. But, since reversing the direction of the 
 path of integration is equivalent to changing the sign of the 
 integral, the equation 
 
 J^A^yz - f^Mdz = o 
 
 is obtained. 
 
 The result just established may be stated in the following 
 theorem : 
 
 Theorem I. — If a function is holomorphic in any simply 
 connected region bounded by a continuous closed curve, the 
 integral of the function, from a fixed lower limit in that region 
 to any point contained therein, is independent of the path of 
 integration, and is a one-valued function of its upper limit. 
 
 A region whose boundary is composed of disconnected 
 curves is not necessarily characterized by the property stated 
 in the theorem. Take, for example, the function 
 
 and suppose that o < |^, | < |^, | < . . . < | ^„ ]. With the ori- 
 gin as a center, construct a system of concentric circles (7,, 
 ^2, • . •, Cn, C^ passing through ^,, C^ through ^„ and so on. 
 Denote by 5o the region inclosed within the first circle 6',, by 
 S^ that inclosed between C^ and C,, and so on, the portion 
 of the plane exterior to the last circle C^ being denoted by 5«. 
 At an initial point z^ interior to one of these regions, assign to 
 w one of the two values possible, and consider the branch of 
 w resulting from a continuous variation. Then however z may 
 vary within any such region, this branch of w will be a mono- 
 genic function, and its derivative will be continuous. Having 
 regard to the branch-points ^,, ^,, . . ., «„, it is. evident that in 
 the regions 5o, 5, . . . it will be one-valued, and in the regions 
 5j, 5,, . . . , it will be two- valued. Thus in the regions 5,, 5^, 
 . . . , the branch fulfils the required conditions, but the boundary 
 does not. The theorem is applicable only to S^. It may be 
 observed that in every other region two paths may be drawn 
 joining the same two points such that the branch is not one- 
 valued in the enclosed portion of the ^-plane. 
 
APPLICATION" OF CAUCHY^S THEOREM. 41 
 
 Theorem II. — If/(-s') is holomorphic in any simply connected 
 region ^ bounded by a continuous closed curve, the integral 
 
 i f{z)dz, taken from a fixed lower limit z^ in that region to any 
 
 point Z contained therein, is a holomorphic function of its 
 upper limit. 
 
 Let L be any path from z^ to Z. When the upper limit is 
 at the point Z -\- dZ, L followed by a straight line from Z to 
 Z -\- dZ c^n be taken as the path of integration. Hence 
 
 *Z+dZ PZ _ . . pZ+dZ 
 
 *Z+dZ nz+dz 
 
 pz+dz nz pz-vdz 
 
 pZ+dZ pZ+dZ 
 
 The first term is equal to f(Z)dZ. The modulus of second 
 term is equal to or less than M\dZ\j where M is the upper 
 bound of |/(z) — f{Z)\ along the line joining Z to Z-\-dZ, 
 But since f{z) is continuous, the Hmit of J/ when Z -\- dZ 
 approaches Z is zero. Hence 
 
 £'^'Az)dB - £j{z)dz = lAZ) + ^¥Z- 
 
 where rf approaches zero with dZ. The integral therefore has 
 y(Z)fora derivative, and is holomorphic in S. 
 
 In the case of a region bounded by several disconnected 
 closed curves, of which one is exterior to all the others, 
 Cauchy's Theorem may be stated in the following form : 
 
 Theorem III. — Let a function f\z) be holomorphic in a 
 region A bounded by a closed curve ^ and one or more closed 
 curves C^, C^, . . . interior to C. The integral of f{z) taken 
 along C will be equal to the sum of its 
 integrals taken in the same direction 
 along the curves C^, C^, . . . 
 
 For the integral of f{z) taken in a 
 positive direction completely around the 
 boundary of A is equal to zero. But 
 tne curves (.7,, C^, . . . are then described in the direction oppo- 
 
42 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 site to that in which C is described. Hence if all the curves 
 are described in the same direction, the result may be written 
 
 If there is but one interior curve, so that the region A is 
 included between two curves C and C^, the integral taken along 
 every closed curve containing C, but interior to C has the 
 same value, viz., the common value corresponding to the paths 
 C and 6*,. 
 
 Art. 19. Theorems on Curvilinear Integrals. 
 
 Theorem I. — li fiz) be continuous in a given region except 
 
 at the point a, the integral j f{z)dzy taken around a small circle 
 
 t, having its center at a, will approach zero as a limit simulta- 
 neously with the radius r of the circle c, provided only 
 
 lim {z — d)/{z) = o when z = a. 
 
 For let the upper bound of the modulus of {z — a)f(z) on 
 the circle c be denoted by M. Then at every point of c^ 
 
 . ^ X - M _M 
 mod i\z) - '. r - — , 
 
 and consequently 
 
 mod J^f{z)dz %—J ds %.27tM. 
 
 closed curve C containing the point a, is equal to zero, except 
 when « = I. When « = i, this integral is equal to ini. 
 
 For the value of the integral will be the same if any 
 circle described about <? as a center be taken as the path of 
 integration. Let then z — a =: r^'^ where r is a constant and 
 varies from o to 2n. The integral becomes 
 
 -• /»2»r — {n-\)ie 
 
THEOREMS Oi^ CURVILINEAR IKTEGRALS. 43 
 
 which reduces to zero excepi when « = I. If « = i, its value 
 is 2niy whence 
 
 / 
 
 27tl, 
 
 Theorem III. — If f{z) is a function holomorphic in a given 
 region 5, C a closed curve the interior of which is wholly 
 within Sf and a a point situated within (7, then 
 
 f I^dz = 2niAcf)' 
 
 For describing about <3; as a center a small circle c of radius 
 r, the equation 
 
 *J^ z — a ^' 2— a 
 
 is obtained. But at every point of c, 
 
 where, by choosing r sufficiently small, the modulus of tf may 
 be made less than any fixed positive quantity. Hence 
 
 ^cz—a ^c2 — a *^c z — a 
 
 ,"jut by the preceding theorems the first term of the right-hand 
 member is equal to 2nif{a), and the second term is equal to 
 zero. 
 
 If the equation of the theorem just established be differ- 
 entiated with respect to a, the following important formulas, 
 expressing the successive derivatives of a holomorphic function 
 at a given point, are obtained: 
 
 
44 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 The integrals in the first members of these equations are all 
 finite and determinate for every position of a within the curve 
 C. Therefore any function holomorphic in a given region ad- 
 mits an infinite number of successive derivatives at every 
 interior point. Each of these derivatives being monogenic 
 must be continuous. Hence the following: 
 
 Theorem IV. — If/C-s^) is holomorphic within a given region, 
 there exists an infinite number of successive derivatives of 
 f{z)y which are all holomorphic within the same region. 
 
 Denote by r the shortest distance from the point a to thd 
 
 curve C, Then at every point of this curve |^ — ^| > r. Let 
 M be the upper bound of the modulus /(^) on Cy and / the 
 length of C, Then 
 
 n fU\ ^ r M . ^ Ml 
 
 and consequently mod/^"^ {a) ^ 
 
 I .2 ... n Ml 
 
 27r r"-^* 
 
 In particular, if C is a circle having a for its center, 
 
 mod /<«) {a) < . 
 
 Art. 20. Taylor's Series. 
 
 Theorem. — Let f{z) be holomorphic in a region 5, and let 
 C be any circle situated in the interior of 5. 
 If a be the center and a-\- 1 any other point 
 interior to Cy 
 
 fia + =Aci) + tf'{d) + /-/"(«) + . . . 
 
 1.2 
 
Taylor's series. 45 
 
 From the preceding article, denoting a variable point on C 
 byC, 
 
 _ I fAQdQr t I r , ^ +' ^ 1 
 
 = A") + tna) + /-/"(«) + . . . + . r /'"('^) + ^, 
 
 where 
 
 , '^- 27tiJ^(Z-ar^\Z-a-t) ^- 
 
 By taking « sufficiently great the modulus of R may be 
 made less than any given positive quantity. Let M be the 
 upper bound of the modulus of f{s) on the circle Cy p the 
 modulus of /, and r the modulus of C — <^ or radius of C. Then 
 
 ^^Jc r''^\r-p) ^r — p\rl ' 
 
 which, since p < r, has zero for its limit when ^ = 00. 
 Writing now z for a-\-t, Taylor's Series becomes 
 
 The series is convergent and the equality is maintained for 
 every point ^ included within a circle described about <3: as a 
 center with a radius less than the distance from a to the nearest 
 critical point oi f{z). 
 
 When a is equal to zero, Taylor's Series takes the form 
 /W = /(o) + Bf'(0) + f-f"(o) + . . . + -^f'\0) + . . . , 
 
 1.2 1.2.. .72 
 
 expressing /(^r) in terms of powers of -s". This form is known 
 as Maclaurin's Series. 
 
46 functions of a complex variable. 
 
 Art. 21. Laurent's Series. 
 
 Theorem. — Let S, a portion of the ^--plane bounded by two 
 concentric circles C, and C^, be situated in the interior of the 
 region £f in which a given function /{js) is holomorphic. If a 
 be the common center of the two circles, and a -\- t a. point 
 interior to 5, /{a + t) can be expressed in a 
 convergent double series of the form 
 
 tn = oo 
 m = — 00 
 
 With a -{-t diS di center construct a circle 
 c sufficiently small to be contained within 
 the region 5. If then 6', be the greater of 
 the two given circles, it follows from Article i8 that 
 
 27ti*^Cx Q — a — t 27ti*^c^ C, — a — t 2ni ^cZ — ^ — ^ 
 But from Article 19, 
 
 whence 
 
 •^^ ^^ 2ni^c^C, - a - t 2ni^c,c-a - t 
 The two integrals of the right-hand member may be written : 
 
 where 
 
 I r t-^'AOdz 
 ^' ~ 27ii^^^{Z - ^)^+xc -a-ty 
 
 _i_ /- (C - ^)-^7(CKC 
 
 But |/| <|C — ^1 at every point of C,, and |/|>|C — «| at 
 every point of (7,, so that i?, and R^ both have zero for a limit 
 
LAURENT'S SERIES. 47 
 
 when « = 00 . The value oif{a -\- 1) can therefore be expressed 
 in the form 
 
 Since in the region 5 the function f{z)/{z — <3:)'"+* is holomor- 
 phic for both positive and negative values of m^ A^. maybe 
 written 
 
 where C is any circle concentric with C^ and C^ and included 
 between them. 
 
 The series thus obtained is convergent at every point a-\-t 
 contained within the region S. It is important to notice, how- 
 ever, that when the positive and negative powers of / are con- 
 sidered separately, the two resulting series have different 
 regions of convergence. The series containing the positive 
 powers of / converges over the whole interior of the circle C^ ; 
 while the series of negative powers of / converges at every 
 point exterior to the circle C^. The region 5" can be regarded, 
 therefore, as resulting from an overlapping of two other 
 regions in which different parts of Laurent's Series converge. 
 
 Writing z ior a -\- t, Laurent's Series takes the form 
 
 f{z) = A, + Aiz - a)+A,{z - ay + . . , 
 
 + A ^, (z- a)-' + A_,(z - a)-' + . . . 
 Consider as a special numerical example the fraction 
 
 ^ = L_ L_ . __J__ 
 
 (<3: - I) (^ - 2) (^- 3) 2(Z-I) Z — 2^2{Z~2,) 
 
 If 1^1 < I, all three terms of the second member, when 
 developed in powers of z, give only positive powers. If 
 I < 1^1 < 2, the first term of the second member gives a series 
 of negative descending powers^ but the others give the same 
 series as before. If 2 <|^|< 3, the first and second terms 
 both give negative powers. If \z\ > 3, all three terms give 
 
48 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 negative powers, and the development of the given fraction 
 can contain no positive powers. Thus a system of concentric 
 annular regions is obtained in each of which the given frac- 
 tion is expressed by a convergent power-series. Laurent's 
 Series gives analogous results for every function which is holo- 
 morphic except at isolated points of the ^-plane. 
 
 Art. 22. Fourier's Series. 
 
 Let w =■ /{z) be holomorphic in a region S^y and let it be 
 periodic, having a period equal to oo, so that^<8:+ noo) = f{2)r 
 where n is any positive or negative integer. Denote by 5„ the 
 region obtained from S^ by the addition of noo to 2 ; and sup- 
 pose that the regions . . . , 5_„ , . . . , 5. 1 , 5^ , 5, , . . . , 5„ , ... 
 meet or overlap in such a manner as to form a continuous strip 
 Sy in which, of course, the function w will be holomorphic. 
 Draw two parallel straight lines, inclined to the axis of reals at 
 an angle equal to the argument of oo, and contained within the 
 strip 5. The band T included between these parallels will be 
 wholly interior to S. 
 
 2itiz 
 
 By means of the transformation z' = e '^ the band T in 
 the ^-plane becomes in the ^'-plane a ring T' bounded by two 
 concentric circles described about the origin as a center, z and 
 z + noo falling at the same point z\ Since w is holomorphic 
 in a region including T, and 
 
 dw dw dz 00 _ vtx» dw 
 dz' ~ dz dz' ~~ 2ni • '^ dz' 
 
 w regarded as a function of z' will be holomorphic in T', 
 Hence, by Laurent's Theorem, 
 
 w = ^2 A^z'^, 
 
 »«= -00 
 
 the quantity a in the general formula of the preceding article 
 being in this case equal to zero. Substituting for z' its value, 
 the preceding equation becomes 
 
UNIFORM CONVERGEJS'CB. 41) 
 
 where 
 
 In the latter integral the path is rectilinear. Denoting its 
 independent variable by C for the purpose of avoiding confu- 
 sion, the value of w becomes 
 
 
 
 »«= -00 
 
 »t = l * 
 
 Art. 23. Uniform Convergence. 
 
 Let the series W := ^o + ^i + ^a +• • • + ^n + . • . , each 
 term of which is a function of z^ be convergent at every point 
 of a given region S. Denote by Wn the sum of the first n 
 terms of W. If it is possible, whatever the value of the posi- 
 tive quantity e, to determine an integer p, such that whenever 
 n> p 
 
 \W- ^n|<e 
 
 at every point of S, the series W\?, said to be uniformly con- 
 vergent in the region 5. 
 
 For convergent series in general the determination of p 
 will depend on the value of z. In the case of uniformly 
 convergent series / can be determined simultaneously for all 
 points in the region 5. 
 
 Uniformly convergent series can in many respects be treated 
 in exactly the same manner as sums containing a finite num- 
 ber of terms. 
 
60 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Theorem I. — If in a region S a series of continuous functions 
 
 is uniformly convergent, the sum of the series is a continuous 
 function of z. 
 
 For at any point z, W may be written in the form W = W^-{-R\ 
 and at a neighboring point z\ W = W ^ -\-R!, Hence 
 
 n n 
 
 and 
 
 \w-w\<\w^-wj\+\R\-\-m. 
 
 But by choosing n sufficiently great, \R\ and \R'\ may both be made 
 less than any given positive quantity e/3 for all values of z and z' 
 in S. Having chosen n thus, W ^ becomes the sum of a finite 
 number of continuous functions. It is then continuous, and, by 
 making \z—z'\ less than a suitable quantity d, "^ — W'\ may be 
 made less than e/3. But under these suppositions 
 
 \W-W'\<e, 
 
 W is therefore continuous at the point 0. 
 
 Theorem II. — If in a region S a series of continuous functions 
 
 is uniformly convergent, the integral of the series, for any finite 
 path L in the region, is the sum of the integrals of its terms : 
 
 J Wdz = J w^dz^J Widz+, . . +J 'w^dz+. . . 
 
 For, writing W = W^+R, it is possible to choose n so that, 
 however small e may be, ]R\<£ at every point of Z. If n be so 
 chosen. 
 
 fwdz ^ f^VJz + fRdz. 
 
UNIFORM CONVERGENCE. 61 
 
 But, by Article 15, denoting by / the length of the path L, 
 
 mod / Rdz<elj 
 which, when w= 00, has zero for its limit. Hence 
 fwdz= lim Jwjz. 
 
 L n = oo L 
 
 From the preceding demonstration we have at once the following 
 result : 
 
 Theorem III. — If in a simply connected finite region S a uni- 
 formly convergent series of holomorphic functions is integrated 
 term by term, the resulting series is uniformly convergent in the 
 same region. 
 
 For in a simply connected region the integral of a holomorphic 
 function is independent of the form of the path of integration. 
 Only paths whose lengths have a finite upper bound need, there- 
 fore, be considered. 
 
 Theorem IV. — If, in a region 5, the series of uniform functions 
 
 is convergent, and the series 
 
 dz az az 
 
 is uniformly convergent, and if further the terms of PT' are con- 
 tinuous in the same region, W will be the derivative of W. 
 
 For, integrating W^ from a to z along a path L contained in 
 S, we have, by Theorem II, 
 
 J W'dz='w^(z)-w,(a)+, . .+w^(z)-w^(a)+... 
 = W{z)-W{a). 
 
 But since W is continuous, it is the derivative of the first mem- 
 ber, and therefore of the second member, and of the function W, 
 
52 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Theorem V. — If in a finite region S the terms of a uniformly 
 convergent series 
 
 are holomorphic, the sum of the series is holomorphic, its deriva- 
 tive being the sum of the derivatives of its terms. 
 
 For let C be the boundary of 5, and let C be a closed curve 
 interior to C. Let ^ be a positive number such that the distance 
 between C and C is everywhere greater than d. Then if z is any 
 point interior to C, we will have, when (^ varies along C, 
 
 \t:-z\>3. 
 
 The given series being uniformly convergent, we can write 
 
 W=-W„-\-R, 
 
 where \R\<s when n is taken sufficiently great. Accordingly if 
 L be the length of C, we will have in the equation 
 
 the modulus of the last term less than 
 
 
 It follows that the series 
 
 converges uniformly. But this gives at once, if we divide by 2^i, 
 From the preceding demonstration we have at once; 
 
UNIFORM CONVERGENCE. 53 
 
 Theorem VI. — If a series of holomorphic functions is uni- 
 formly convergent in a given region S, the series formed by the 
 derivatives of its terms will be uniformly convergent in the same 
 region. 
 
 To illustrate by an example that uniformity of convergence 
 is essential to the preceding theorems, take the series 
 
 W = ^-+I 
 
 I+Z ,(l+Z^)(l+2^+0* 
 
 At the point z=i each term is continuous, and the series 
 is convergent, having the value 1/2. The series is, however, 
 discontinuous at z=i. For, writing it in the form 
 
 the sum of the first n terms is seen to be 
 
 W =^~. 
 
 / 
 
 But W is the limit of TF^ when w=oo, and is therefore 
 unity at every point z for which lzl<i, and zero at every point 
 for which l^l >i. 
 
 If now this series be considered for the points within and 
 
 upon a circle described about the origin as a center with an 
 
 assigned radius less than unity, the remainder after n terms, or 
 
 z" 
 I— pr„=-— — - can, by a suitable choice of n, be made less in 
 
 absolute value than any given quantity. In such a region, then, 
 the series converges uniformly, and, by Theorem I, can have no 
 point of discontinuity. A similar result holds for the region 
 exterior to any circle described about the origin as a center with 
 an assigned radius greater than unity. 
 
54 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Art. 24. Power Series. 
 
 The nost elementary and at the same time the most impor- 
 tant series of functions which enters into the theory of functions 
 is of the form 
 
 where a^^ a^, a^, . . . , a^, . . . are constants. 
 
 If this series is convergent for a certain value Z of the varia- 
 ble 2, it will be convergent for every value of z for which \z\ < \Z\, 
 For if the modulus of z is less than that of Z, the series 
 
 z z^ 2** 
 
 is an absolutely convergent geometrical progression. Since, now, 
 the series 
 
 a^-^a,Z-^a^Z''+. . .+a^Z" + . . . 
 
 is convergent, the moduli of its terms must have a finite upper 
 bound A. We can accordingly use its terms as multipliers foi 
 the corresponding terms of the geometrical progression, and we 
 will obtain an absolutely convergent series. But this series will be 
 the given series 
 
 a^^a^z-\-a^z^^-. . .+fl^z" + . . . 
 
 subject only to the condition that |z|<|Z|. 
 
 It is obvious that every power series of the form here given 
 converges for z = o. When we consider other values of z three 
 cases arise: 
 
 (i) The series may converge for every finite value of z, as, 
 for example, 
 
 z^ z** 
 
 I+Z + - + ... + +... 
 
 2 I . 2 . . . W 
 
POWER SERIES. 55 
 
 (2) The series may diverge for every value of z, except z=o, 
 as, for example, 
 
 1+Z+42H. . .+w**2**+. .. 
 
 (3) The series may converge for some values of z different 
 from zero and diverge for others. For example, the series 
 
 z z^ z^ 
 
 12 n 
 
 converges for 3= — i and diverges for z=i. 
 
 In the third case the modulus of the values of z for which the 
 series converges must have a finite upper bound. Call this R. 
 The circle of radius R described about the origin as a center is 
 known as the circle of convergence. For this circle we have the 
 following theorem: 
 
 Theorem. — A power series is convergent at every point inte- 
 rior to its circle of convergence, and is divergent at every point 
 exterior to its circle of Convergence. 
 
 No general statement can be made as to the convergence or 
 divergence of the series upon the circumference of the circle of 
 convergence. The series may converge at all points of the cir- 
 cumference, as, for example. 
 
 2^ 2** 
 
 I+2+-2 + ... + -2 + . 
 
 or it may diverge at all such points, as, for example, 
 
 1+Z+22H. . .+wz«+. . . , 
 
 or finally, as already illustrated, it may converge at some points 
 and diverge at other points of this circumference. 
 
56 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Art. 25. Uniform Convergence of Power Series. 
 
 Theorem I. — A power series is uniformly convergent in every 
 circle described about the origin as a center with a radius less 
 than R. For, if R' < R, the series 
 
 K|+K|i?'+...+K|i?'-+... 
 
 is convergent; and, consequently, whatever the value of the posi* 
 tive quantity £, we can find an integer p such that \in>p 
 
 For all values of z within the circle of radius R', the sum of 
 the series will then differ from the sum of its first n terms by a 
 quantity less than £ in absolute value. Hence the series is uni- 
 formly convergent within the circle of radius R'. 
 
 Theorem II. — If a power series is uniformly convergent in a 
 given circle, the series obtained by integrating its terms or by 
 differentiating its terms is uniformly convergent in the same 
 circle. 
 
 This theorem follows at once from Theorems III and VI of 
 Article 23. Since R is the upper bound of R\ the series of primi- 
 tives and the series of derivatives have exactly the same circle of 
 convergence as the given po\ver series. We have also as an im- 
 mediate consequence of Theorems II and V of 'Article 23: 
 
 Theorem III. — The primitive of a power series is the sum of 
 the primitives of its terms; and the derivative of a power series 
 is the sum of the derivatives of its terms. 
 
 As a result of these theorems, we have that, so far as continuity, 
 differcntiabihty, and integrabilily are concerned, a power series 
 has within its circle of convergence the same properties as the 
 simi of a finite number of powers. 
 
UKIFORM FUNCTIONS WITH SINGULAR POINTS. 57 
 
 Art. 26. Uniform Functions with Singular Points. 
 
 Theorem I. — A function holpmorphic in a region 6" and 
 not equal to a constant, can take the same value only at iso- 
 lated points of 5. 
 
 For in the neighborhood of any point a interior to 5, by 
 Taylor's theorem, 
 
 Az) -M = {z- aY'{a) + (l=|)! /"(«) + . . . 
 
 Unless y(^) is constant over the entire circle of convergence of 
 this series, the derivatives /'{a)^ f"{d), . . . cannot all be 
 equal to zero. Let/"^''^(^) be the first which is not equal to 
 zero. Then 
 
 f{z) - f{a) = (z-ay[-J^-M 1 f^^'^'Y) J ^-a)+ . . .1 
 
 Since the series within the brackets represents a contin- 
 uous function, if \z — a\ be given a finite value sufificiently 
 small, the modulus of the first term of the series will ex- 
 ceed the sum of the moduli of all the other terms, and the 
 same result will hold for every still smaller value of \z—a\. 
 For values of z, then, distant from a by less than a certain 
 finite amount, /(^r) — fia) is different from zero. 
 
 If, on the other hand, the function is constant over the en- 
 tire circle, described about ^ as a center, within which Taylor's 
 series converges, it will be possible, by giving in succession 
 new positions to the point a, to show that the value of the 
 function is constant over the whole region 5. 
 
 Theorem II. — Two functions which are both holomorphic 
 in a given region 5 and are equal to each other for a system of 
 points which are not isolated from one another, are equal to 
 each other at every point of S. 
 
 For let f{z) and <i){z) be two such functions. By the pre- 
 ceding theorem, the difference /(<s:) — 0(^) must be equal to 
 zero at every point of 5. 
 
58 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Theorem III. — A function which is holomorphic in every 
 part of the <s:-plane, even at infinity, is constant. 
 
 For, a being any given point, whatever the value of z^ 
 
 Az) = Aa) + (^- aV'{a) + ... + Y^fr^„f"V) + •■■ 
 
 But by Article 19, r being the radius of any arbitrary circle 
 having its center at ^, and M being the upper bound of the 
 modulus oi f{z) on the circumference of this circle, 
 
 J ^(n)r \ =1.2.. . tlM 
 
 But M is always finite, and r may be made indefinitely great, 
 Hencey^'*^(.3:) = o for all values of «, and 
 
 A^) =/{")■ 
 
 Theorem IV. — If a,function/(^), holomorphic in a region 5, 
 is equal to zero at the point a situated within S, the function 
 can be expressed in the form 
 
 /(^) = {z- ar,p{B), 
 where m is a positive integer, and (p{z) is holomorphic in 5 and 
 different from zero at a. 
 
 For in the neighborhood of the point a, by Taylor's Theorem, 
 /(z)=/{a) + {z-ay{a)+.., 
 LetyM (^) be ^he first of the successive derivatives at ^ which 
 is not equal to zero. Then 
 
 ■^^ ^ ^ Li '2. . ,m ^ 1.2 . . . {m+ ly ^ ' J* 
 
 which is the required form. The point a is a, zero oi f{z), and 
 m is its order. 
 
 Theorem V. — If the point ^ is a singular point of a given 
 function /(-s"), but is interior to a region 5, in which the recip- 
 rocal oi f{z) is holomorphic, the function can be expressed in 
 the form 
 
 {z - aY ' 
 
 where m is a positive integer, and x{^) is holomorphic in the 
 neighborhood of a. 
 
UNIFOKM FUNCTIONS WITH SINGULAR POINTS. 59 
 
 For by the preceding theorem 
 
 y^ = (^ - ^)"0(^)» 
 
 where (J){z) is holomorphic and not equal to zero at z-=. a. 
 Hence 
 
 f{z\ = ^ ^ - ^(^^ 
 
 •/^^ {2-a)^'(p{2) {z-aY 
 
 Further, since in a region of finite extent including the 
 
 point a 
 
 X(z) = A,-\A,{z - ^) + ..., 
 
 -^ ' {z—aY~ ^z — a^^^*' 
 
 a being an ordinary point for ^(-s"). 
 
 The point « is a pole oi f{z) and m is its order. 
 
 Theorem VI. — A function, not constant in value, and hav- 
 ing no finite singular points except poles, must take values 
 arbitrarily near to every assignable value. 
 
 For suppose that f{z) is such a function, but that it takes 
 no value for which the modulus o{ f{z) — A\.s less than a given 
 positive quantity e. Then the function 
 
 f(?) - A 
 will be holomorphic in every part of the ^-plane, which, by 
 Theorem III, is impossible unless /(^) is a constant. 
 
 Theorem VII. — A function /(z), having no singular point 
 except a pole at infinity, is a rational entire function of z. 
 
 For the only singular point of /( — J is a pole at the origin. 
 
 Hence 
 
 /©= 
 
 ^+...+^+*(.), 
 
 where (f){z) is holomorphic over the entire plane, including the 
 point at infinity. (p{z) is consequently equal to a constant A^, 
 The given function therefore can be written in the form 
 
60 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Theorem VIII. — A function /(z) whose only singular points 
 are poles is a rational function of z. 
 
 The poles must be at determinate distances from one an« 
 other ; otherwise the reciprocal of f{2) would be equal to zero 
 for points not isolated from one another. The number of poles 
 cannot increase indefinitely as \z\ is increased; for then the 
 
 reciprocal of / ( - ) would have an infinite number of zeros indefi- 
 nitely near to the origin. The total number of poles is there- 
 fore finite. Let a^ b, , , . denote them. In the neighborhood 
 of a the function can be expressed in the form 
 
 a being an ordinary point for 0(^). In the neighborhood of by 
 (p{2) can be expressed in the form 
 
 a and b being both ordinary points for tp{2). Proceeding in 
 
 this way the given function will be expressed as the sum of a 
 
 finite number of rational fractions and a term which can have 
 
 no singular point except a pole at infinity. This term is a 
 
 rational entire function. 
 
 Theorem IX. — If the function /(z) has no zeros and no 
 
 singular points for finite values of z, it can be expressed in 
 
 the form /(z) = e^^^\ where £^{z) is holomorphic in every finite 
 
 region of the z-plane. 
 
 f(z) ■ 
 For'^-Ap can have no singular points except at infinity, since 
 
 /(2) 
 
 in every finite region of the ^-plane /{z) and /'(z) are holomor- 
 phic and /(z) is different from zero. Hence, choosing an arbi- 
 trary lower limit z^y the integral 
 
 is holomorphic in every finite region. The function f{z) con- 
 sequently must take the form 
 
RESIDUES. 61 
 
 where g{z) = h{z) + log/CzJ. 
 
 Theorem X. — If two functions /(z) and 0(z) have no singu- 
 lar points in the finite portion of the z-plane except poles, and 
 if these poles are identical in position and in order for the two 
 functions, and their zeros are also identical in position and 
 order, there must exist a relation of the form 
 
 f{z) = 0(^)^(^), 
 
 where g{j3) is holomorphic in every finite region of the ^-plane. 
 
 For the ratio of the two functions has no zeros and no 
 singular points in the finite portion of the 2-plane. 
 
 Art. 27. Residues. 
 
 If a uniform function has an isolated singular point a, it 
 is expressible by Laurent's series in the region comprised be- 
 tween any two concentric circles described about a with radii 
 less than the distance from a to the nearest singular point. 
 Hence in the neighborhood of a 
 
 The coefficient of {z — d)'"^ in this expansion is called the 
 "residue" of/(^) at the point a. 
 
 If any closed curve C including the point a be drawn in the 
 region of convergence of this series, and f{z) be integrated 
 along C in a positive direction, the result will be 
 
 L 
 
 f{z)dz = 27tiB^. 
 c 
 
 The following may be regarded as an extension of Cauchy's 
 theorem : 
 
 Theorem I. — If in a region 5 the only singular points of the 
 one-valued function f{z) are the interior points ^, «',... , the 
 
i 
 
 62 FUNCTIONS OF- A COMPLEX VARIABLE. 
 
 integral J f(z)dz taken around its boundary C in a positive 
 direction is equal to 
 
 where B, B' , , , . are the residues of f{z) at the singular points. 
 For the integral taken along C is equal to the sum of the 
 integrals whose paths are mutually exterior small circles de- 
 scribed about the points a^ a\ , , , 
 
 The following theorems are immediate consequences of the 
 preceding : 
 
 Theorem II. — If in a region having a given boundary Cthe 
 only singular points of the one-valued function f(z) are poles 
 interior to C, an equation 
 
 exists, M denoting the number of zeros and N the number of 
 poles within C, each such point being taken a number of times 
 equal to its order. 
 
 For in the neighborhood of the point a 
 
 f{z) = {z- ar<p{z) 
 
 where (p{z) is finite and different from zero at a^ and m is a, 
 
 positive integer if <a: is a zero, a negative integer if ^ is a pole. 
 
 Hence 
 
 /\z) __ m (l>\z) 
 
 f(z) z-a^ <t>(zy 
 
 The integrand, therefore, has a pole at every zero and pole of 
 /(^), and its residue is the order, taken positively for a zero, 
 and negatively for a pole. 
 
 Theorem III. — Every algebraic equation of degree n has n 
 roots. 
 
 For let f{z) represent the first member of the equation 
 ^« _|_ ^j^~-' -|_ . . . _|- ^^ = o. Since f{z) has no poles in the 
 
IXTEGRxVL OF A UNIFORM FUXCTIOIN". 63 
 
 finite part of the ^-plane, the number of roots contained within 
 any closed curve C will be given by the integral 
 
 
 But taking for C a circle described about the origin as a 
 center with a very great radius, this integral is 
 
 j_ r n.-' + (n-,)a,^-' + ... ^^ ^ j_ fndz 
 
 where e has zero for a limit when |^|= oo . Hence the limit 
 of the preceding integral, as \z\ is increased, is n. 
 
 Prob. 17. Show that if 2 = 00 is an ordinary point of/(s^), that 
 is, if /(^) is expressible for very great value of 2 by a series contain- 
 ing only negative powers of z, the integral of/(2;) around an infinitely 
 
 great circle is equal to 27ti into the coefficient of — . This coeffi- 
 
 z 
 
 cient with its sign changed is called the residue for 2 = 00 . 
 
 Prob. 18. Show that the sum of all the residues oi f{z), of the 
 preceding problem, including the residue at infinity, is equal to 
 zero. 
 
 Prob. 10. If -77-^ is a rational function of which the numerator 
 tp{z) 
 
 is of degree lower by 2 than the denominator, and if the zeros 
 
 ^j!, , ^, , . . . , ^„ of the denominator are of the first order, show that 
 
 Art. 28. Integral of a Uniform Function. 
 
 It was shown in Article 18 that, if a function /"(-s-) is holo- 
 morphic in a simply connected region S, its integral taken 
 from a fixed lower limit contained in 5 to a variable upper 
 limit ^ is a uniform function of z within 5". If F{z) is a function 
 which takes a determinate value F{zq) at 2- = ^y and is uniform 
 while 2 remain^ within S, having at every point f{z) for its 
 derivative, the integral of f{z) from z^ to z is equal to 
 F{z) — F(2^). If F^(^z) is another function fulfilling these con- 
 
64 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 editions, so that the integral of f{z) can be written also in the 
 form FJ^z) — F^(z^), the functions F(z) and FXz) differ only by 
 a constant term ; for 
 
 Suppose now thaty"(^) is still uniform in 5, but that it 
 has isolated critical points ^, , ^,, . . . interior to 5. Any two 
 paths from z^ to Zy which inclose between them a region con- 
 taining none of the points ^,, <3:,, . . ., will give integrals identi- 
 cal in value. Let the two paths Z, , L include between them 
 a single critical point a^\ and consider the integrals along 
 these two paths. The integral along Z, will be equal to the 
 integral along the composite path Z,Z~'Z, where the exponent 
 — I indicates that the corresponding path is reversed ; for the 
 integral along L'^L is equal to zero. But L^L~^ is a closed 
 curve, or" loop," including the critical point a^y and, assuming 
 that it is described in a positive direction about a^, the inte- 
 gral along it is equal to 27iiB^y where B^ is the residue oi f{z) 
 at a^. Hence 
 
 rf{z)dz = 27iiB^ + ff{z)dz. 
 
 If now the two paths Zj, L from z^ to z include between 
 them several critical points a^, a^, «^, . . ., draw intermediate 
 paths Z-„ . . ., L^, so that the region between any two consec- 
 utive paths contains only one critical point. The integral 
 along Z, will be equal to the integral along the composite path 
 L^L'^L^ . . . L^'^L^L'^L, since the integrals corresponding to 
 L'^L^, . . ., L^'^L^y L~^L are all equal to zero. But L^L'^y 
 L^L^'\ . . ., L^L'^ are all closed paths or loops, each including 
 a single critical point, so that, assuming that each is described 
 in a positive direction and that B^, B^, ^^, . . . denote the resi- 
 dues of f{z) at the critical points, 
 
 £^A^)dz = 2ni[B^ + ^, + ^^ + . . .) + fj{z)dz. 
 
 It has been assumed in the preceding that neither of the 
 paths Zj, L intersects itself. In the case where a path, for 
 
IKTEGRAL OF A UNIFORM FUNCTION^. 65 
 
 example Zj, intersects itself in several points c^, <;,,..., it is 
 possible to consider Z, as made up of a path Z/ not intersect- 
 ing itself, together with a series of loops attached to Z/ at 
 the points c^y c^, . . . Each of these loops encloses a single 
 critical point Uk and, if described in a positive direction, adds 
 to the integral a term 27tiB^ Each such loop described in a 
 negative direction adds a term of the form — 2niB^, It is evi- 
 dent that the form of each loop and the point at which it is 
 attached to Z/ may be altered arbitrarily without altering the 
 value of the integral, provided no critical point be introduced 
 into or removed from the loop. In fact all the loops may be 
 regarded as attached to Z/ at z^. 
 
 It can be proved by similar reasoning that the most gen- 
 eral path that can be drawn from z^ to z will be equivalent, so 
 far as the value of the integral is concerned, to any given path 
 Z preceded by a series of loops, each of which includes a sin- 
 gle critical point and is described in either a positive or nega- 
 tive direction. The value of the integral is therefore of the 
 form 
 
 f{z)dz + 27ii{m,B, + m^B^ + •••)» 
 
 I 
 
 where m^^ m^^ . . . are any integers positive or negative. 
 
 /*' dz 
 
 As an example consider the integral / . The only 
 
 critical point is ^ = ^. Any path whatsoever from z^ to z is 
 equivalent to a determinate path, for example, a rectilinear 
 path, preceded by a loop containing a and described a certain 
 number of times in a positive or negative direction. If w de- 
 note the integral for a selected path, the general value of the 
 integral will be w + '2'nni. If now a straight line be drawn 
 joining z^ to a, and if along its prolongation from a to infinity 
 the -sr-plane be cut or divided, the integral in the ^-plane thus 
 divided is one-valued. But, with the variation of z thus re- 
 stricted, any branch of the function log {z — a) is one-valued. 
 Select that branch, for example, which reduces to zero when 
 z=^ a-\- \. It takes a determinate value for z = z^, and its 
 
66 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 derivative for every value of z is . Hence, denoting it 
 
 by Log {z - a), 
 
 /* dz z — a 
 
 — — = Log {z-a)- Log {z, -a) = Log ^— -. 
 
 For a path not restricted in any way, the value of the inte- 
 gral is 
 
 z — a 
 
 /dz z — a 
 = Log h 2nni = log 
 ,,z -a ^ z,-a^ ^ 
 
 ?o — ^ 
 
 Prob. 20. If -rT-\ is a rational function of z of which the numer- 
 i\z) 
 
 ator is of degree lower by 2 than the denominator, and if the zeros 
 
 /Zj, «,,..., d!„ of the denominator be of the first order, show that 
 
 t/«o i\){z) X ip\av) ^ z^ — Uy 
 
 where 2<p{av)/tp'(ay) = o. (See Prob. 19, Art. 27.) 
 
 1 
 
 Art. 29. Weierstrasss Theorem. 
 
 Any rational entire function of z, having its zeros at the 
 points ^,, ^,, . . ., ^^, can be put in the form 
 
 A(z- a,p{z - a,y^ . . . (^ - ^^)''m, 
 
 where ^ is a constant and «,, «,, . , ,, n„ are positive integers. 
 More generally, any function which has no singular point in the 
 finite portion of the ^-plane and has the points a^, . . ., a„ as 
 its zeros, is of the form 
 
 e^'\z - a,p ..,{z- a^Ym, 
 where ^{z) is holomorphic in every finite region. 
 
 The extension of this result to the case where a function 
 without finite singular points has an infinite number of zeros is 
 due to Weierstrass. It is effected by means of the following 
 theorem : 
 
 Theorem. — Given an infinite number of isolated points a^^ 
 
WEIERSTEASS'S THEORBM. 67 
 
 a^j . . ., a„, . . ., a function can be constructed holomorphic ex- 
 cept at_ infinity and equal to zero at each of the given points 
 only. 
 
 For the given points can be taken so that 
 
 k, I <kJ <•••!««'<•• •> 
 
 I an I increasing indefinitely with n. Consider the infinite product 
 
 (P{s) = nil _^)^^«w, 
 1 ^ ii„' 
 
 where Pn{^) denotes the rational entire function 
 Any factor may be written in the form 
 
 
 But since 
 
 the path of integration being arbitrary except that it avoids 
 the points a^, a,, . . ., the product may be expressed as 
 
 00 /»2 
 
 ne'f'n^'\ in which tp„{z) = —J 
 
 z^dz 
 
 In any finite region of the z-plane it will be possible to 
 
 assume that | z | ^ P < |^^|, if P and m be suitably chosen, since 
 \ay\ increases indefinitely with n. Divide the product into 
 two parts 
 
 n (I- -)- 
 
 and 
 
68 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Since when n>m, \aj >pj the integrand of the exponent 
 
 ^'^ ' Jo a.»(a„-«) 
 is holomorphic in the circle |2;| <|0. Accordingly, (l>„(z) is in the 
 given region a holomorphic function of its upper limit. 
 But we may write 
 
 m 
 
 00 
 
 Consider now the series 1(1) Jz). For the modulus of each term 
 we have 
 
 '^"^'^'" KHfcp7)' 
 
 where / denotes the length of the path of integration. But, if the 
 path of integration be taken as rectilinear, we will have l<p. 
 Hence each term o! the series is less in absolute value than the 
 corresponding term of a convergent geometrical progression in- 
 dependent of z. The series is, accordingly, uniformly conver- 
 gent and, by Theorem V of Article 23, represents a function holo- 
 morphic in the given region. The exponential 
 
 also must be holomorphic. The other part of the product 
 
 m 
 
 ^■(-f.) 
 
 i7 I-- Uf«« 
 
 containing only a finite number of factors is everywhere holo- 
 morphic, vanishing at all of the points a^, a^, . . . , which are 
 situated within the given finite region. But this region may be 
 extended arbitrarily. The product therefore fulfils the required 
 conditions. 
 
 In the preceding demonstration it was tacitly assumed that 
 none of the given points a^, ^2, . . . was situated at the origin. 
 To introduce a zero at the origin it is necessary merely to mul- 
 tiply the result by a power of z. 
 
 The most general function without finite singular points 
 
WEIERSTKASS'S THEOREK. 69 
 
 having its only zeros at the given points a^, a^, . . ., ^« . . ., can 
 
 be expressed in the form 
 
 where ^^) is holomorphic except at infinity ; for the ratio of 
 any two functions satisfying the required conditions is neither 
 infinite nor zero at any finite point. 
 
 By means of Weierstrass's theorem it is possible to express 
 any function, F^s), whose only finite singular points are poles as 
 the ratio of two functions holomorphic except at infinity. For, 
 construct a function ^-(-s-) having the poles of F{2) as its zeros. 
 The product F{2). ip{s) = (p(z) will have no finite singular point. 
 The given function can, therefore, be written 
 
 which is the required form. 
 
 In applying Weierstrass's theorem to particular examples, 
 it will rarely be found necessary to include in the polynomial? 
 Fn{^) SO many terms as were employed in the demonstration 
 given above. It is quite sufficient, of course, to choose these 
 polynomials in any way which will make the product converge 
 for finite values of <sr to a holomorphic function. Factors of the 
 form / „ K 
 
 \ aj 
 
 where /*„(^) is chosen in such a manner, are called " primary 
 factors." 
 
 As an application of Weierstrass's Theorem take the reso- 
 lution of sin 2 into primary factors. The zeros of sin z are o, 
 ±;r, ±2;r, . . ., ±«;r, .... Consider factors of the form 
 
 \ fZTt' 
 
 SO that Pn{^) contains only one term — , and 
 
 nn 
 
 r' zdz 
 
 nninn — z) 
 ^ ' 
 
70 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 00 
 
 The series I(p^{z) will converge uniformly in any given finite 
 
 m 
 
 region. For if p and m be suitably chosen we will have 
 
 |2|<^<w;r. 
 Hence 
 
 \ mn/ 
 
 where / is the length of the path of integration from the origin 
 to the point z. If this path be taken as rectilinear, we will have 
 / < iO and (lf^(z) will be less in absolute value than the correspond- 
 ing term of the convergent numerical series 
 
 
 
 A similar result holds for the series I ^^{z). These series ac- 
 
 — m 
 
 cordingly represent holomorphic functions in any region for 
 which |z| < /O. Hence the expression sought is 
 
 sin z=ze^^'^n ( i — — ]en^. 
 - 00 \ nnj 
 
 the value w=o being excluded from the product. It will be 
 shown in the next article that e^^^^ = i. 
 
 Prob. 21. If (jJx and co^ be two quantities not having a real ratio, 
 the doubly infinite series of which the general term is — 
 
I 
 
 mittag-leffleb's theorem. 71 
 
 is absolutely convergent ii p'>2. Hence show that the product 
 
 .(.)=.iz(i-i) 
 
 
 where co=mo)i + no)2, defines a holomorphic function in any finite 
 region of the 2-plane. This function is Weierstrass's sigma func- 
 tion, and is the basis of his system of elliptic functions. 
 Prob. 22. Show that the product 
 
 'K-i)' 
 
 I 
 
 2(1 + 2 
 
 defines a function holomorphic in every finite region of the 2-plane. 
 This function is the reciprocal of the gamma function r(z) or, in the 
 notation employed by Gauss, 77(2— i). It may also be defined as 
 the limit when w= 00 of the product 
 
 z(z+i)(z-\-2) . . . (z-^-n) -- 
 
 n • 
 
 i-2'3 . . . n 
 
 Prob. 23. Assuming the relation that 
 
 r{i-\-z)=zr(z\ 
 
 show that 
 
 I I sinrrz 
 
 r(z)*r(i-2)~~S~' 
 
 Art. 30. Mittag-Leffler's Theorem. 
 
 Any uniform function j(z) with isolated singular points 
 ai, a2, . . . can be represented in the neighborhood of one of 
 these points by Laurent's series; viz., 
 
 }{z)=Ao+Ai{z-aJ+A2(z-aj2^... 
 +Bi{z-aJ-^+B2{z-aJ-2+... 
 
 Hence /(,)=^(,)+G„(j-^), 
 
72 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 where <j>(z) is holomorphic in a region containing the point <z„, 
 and GJ-— — j is holomorphic over the whole plane excluding 
 
 the point a„. If fl„ is a pole of /(z), GJ —^ — ) consists of a finite 
 number of terms; otherwise, it is an infinite series. If the num- 
 ber of singular points is finite, and the function gJ ) is 
 
 formed at each such point, we can obtain by subtracting the 
 sum of these functions from j(z) a remainder which has no 
 singular point in the finite part of the plane. This remainder 
 can therefore be expressed as a series of ascending powers G(z) 
 converging for every finite value of z. The function /(z) can 
 accordingly be written in the following form: 
 
 m=G(z)+IG„{^), 
 
 which is analogous to the expression of a rational function by 
 means of partial fractions. 
 
 The extension of this result to the case where the number 
 of singular points is infinite is due to Mittag-Leffler. Let ai, 
 ^2, . . . , cin, ... be the singular points of the one-valued func- 
 tion /(z), and suppose that 
 
 kil<la2l< . . . U„|<. . . , 
 
 \a„\ increasing without limit when n is increased indefinitely. 
 Let, further, G„(—2 — ) be the series of negative powers of 
 
 z — a^ contained in the expansion of /(z) according to Laurent's 
 series in the neighborhood of a„. 
 
 The function gJ ), having no singular pomt except at 
 
 a„, may be developed by Maclaurin's series in the form 
 
mittag-leffler's theorem. ■ 73 
 
 \z a^/ 
 
 and the series will converge uniformly within a circle described 
 about the origin as a center with any determinate radius |0„< \aj^. 
 Hence, for any point within the circle [2| = (0„, 
 
 Fn(z) representing the first p terms of the development of 
 
 GJ ) by Maclaurin's theorem, and R the remainder, which 
 
 by a suitable choice of p may be made less in absolute value 
 than any given quantity. 
 
 Choose the positive quantities £i, ^2, • - - , £„,... so that 
 the series £i + £2 + - • • + ^« + - • • is convergent. Choose also in 
 connection with each of the points <ii, fi^2j • • • j ^«> • • • , a suitable 
 integer p such that 
 
 mod[Gi(^^J -Fi(zij < £i, ifl2l^^i< |ai|; 
 
 mod|^G2^j^j-i^2(2)J<^2, if \z\<P2<la2\; 
 and, in general, 
 
 mod[G„^^3^j -i^n(2) J < c,:, if \z\<Pn<\an\. 
 Consider now the series 
 
 in any finite region of the plane, the points ai, ^2, . . . , a^, . , . 
 
74 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 being excluded. Since \a^\ increases indefinitely with «, it is 
 possible, in any finite region of the z-plane, to assume that 
 kl<Pm<kml- Separate from the series its first w-i terms. 
 These terms will have a finite sum. The remaining terms of the 
 series taken in order will be less in absolute value than £^> 
 £„+i, . . . respectively, \z\ being less than the least of the quanti- 
 ties p„j Ptn+ii ' • • Accordingly, the series 
 
 |[°-(;^J-''-«] 
 
 is absolutely convergent for every value of z except a^ a^, . . . , 
 a^, . . . It is evident, further, that in any given finite region, 
 from which the points a^, aj, . . . , a„, . . . are removed by means 
 of small circles described about them as centers, the series 
 converges uniformly. In such a region any term of the series 
 is holomorphic; and, therefore, by Theorem V of Article 23, the 
 series defines a holomorphic function. 
 
 The point a^ is an ordinary point for the difference 
 
 since in its neighborhood this difference may be developed as a 
 convergent series containing only positive powers of z — a^. In 
 the same way each of the points a^, aj, . . . , o„, . . . is an ordinary 
 point for the function 
 
 ^(^)-?^"fe)-'^«4 
 
 This function, therefore, can have no singular point except at 
 infinity, and must be expressible as a series G{z) containing 
 only positive powers of z and converging uniformly in any 
 finite region of the z-plane. Hence the function f{z) may be 
 put in the form 
 
mittag-leffler's theorem. 
 
 75 
 
 in which the character of each singular point is exhibited. 
 
 As an appHcation of Mittag-Leffler's theorem consider cot 2. 
 Its singular points are ^ = o, ± tt, ± 2;r, . . . . In the neigh- 
 borhood of ^ — o, cot ^ is holomorphic ; and in the neigh- 
 
 borhood of -^ = njt, n being any positive or negative integer, 
 
 I 
 
 cot z — 
 
 z — nn 
 
 is holomorphic. The series 
 
 + 00 
 
 z— nn 
 
 in which m \s an arbitrary positive integer, is not convergent 
 for finite values of z, even when |^| < mn. The series. 
 
 .z — nn nn_ 
 
 nn(z — nn) 
 
 \ nnl 
 
 is, however, absolutely convergent at every point for which 
 1^1 < mn. For the modulus of any term is equal to 
 
 nn \ rnti 
 
 nitt 
 and, therefore, less than the corresponding term in the series 
 
 z\ 
 
 \ mn] 
 A similar result holds for the series 
 
 I 
 
 -^ + 
 
 — -T 
 
 nn nnj 
 
 It is easy to see now that the reasoning employed in the 
 demonstration of Mittag-Leffler's theorem may be applied to 
 show that the series 
 
76 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 where the summation does not include n =o, defines a func- 
 tion holomorphic in any finite region of the ^-plane, the points 
 O, ± ;r, ± 2;r, . . . being excluded. The difference 
 
 I 
 
 cot 2 
 
 Z 
 
 r— ^+-1 
 
 can have no singular point except at infinity. It must, there- 
 fore, be expressible as a series G{z) of positive powers of <sr, 
 having an infinite circle of convergence. Hence 
 
 I jti^ r I I "1 
 
 — 00 l— — ' 
 
 The next step is to determine G{z). It is to be observed 
 that, if G{z) is a constant, its value must be zero, since 
 cot (— 2:) = — cot z. If G{z) is not a constant, differentiation 
 of the preceding expression for cot z gives 
 
 I I ^"^ I 
 
 ~ s"h?^ ^ ^'(^^ ~? ~ ^ {z - nnf 
 
 It follows, by changing z into z -{- n, that 
 
 G\z + 7t) = G'{z), 
 
 Hence G'{z) is periodic, having a period equal to n ; and as the 
 point z traces a line parallel to the axis of reals, G'{z) passes 
 again and again through the same range of values. But G\z)j 
 being the derivative of G{z)y is holomorphic for every finite 
 value of z. It can, therefore, become infinite, if at all, only 
 when the imaginary part of z is infinite. If z be written in 
 the form x-\-iy, the value of G'{z) may be expressed as 
 
 Q'U\ = ' , ^ I / 2/4cos;tr4-/sin;tr) y 
 
 ^ ^ (x -\- iyf ' ^ {x-^iy—nn)^ \(cos 2;tr+^sin 2;r)— ^>/ 
 
 When J = ± 00 the first and last terms of the second 
 member vanish. In regard to the series it can be proved that. 
 
mittag-leffler's theorem. 77 
 
 for any given region is which y is finite and different from 
 zero, an integer r can be found such that the sum of the moduli 
 of those terms for which \n\> -k is less in absolute value than 
 any previously assigned quantity e. As \y\ is increased the 
 modulus of each of these terms is diminished. The modulus 
 of their sum, therefore, cannot exceed e when 7 =±00. But 
 whenj^= ±00 the sum of any finite number of terms of the series 
 is zero. Hence the limit of the whole series is zero. G'(z)y 
 therefore, never becomes infinite. Hence, by Theorem HI, 
 Article 26, it is constant, and is equal to zero. It follows that 
 G{z) is equal to zero. 
 
 The expression for cot z is accordingly 
 
 cot ^ = ^ + ^[^3^ + ^]. 
 
 The logarithmic derivative of the product expression for 
 sin Zy given in the preceding article as an example of Weier- 
 strass's theorem, is 
 
 cot z=g'{z)^- + ^ r — ^ — h— T 
 
 Hence ^(^) in that expression is a constant. Making z = o, 
 its value is seen to be unity. 
 
 Prob. 24. From the expression for cot z deduce the equation 
 
 where the summation does not exclude n =^ o. 
 Prob. 25. Show that the doubly infinite series 
 
 where go=z mco^ + nao^ , defines a function whose only finite singular 
 points are z = go. This function is Weierstrass's ^function. (Com- 
 pare Problem 21.) 
 
 Prob. 26. -Prove that 
 
78 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 Prob. 27. Prove that ^\z) = — 2^7—^: — y* where the summa- 
 tion does not exclude 00=0. 
 
 Art. 31. Singular Lines and Regions. 
 
 The functions whose properties have been considered in the 
 preceding articles have been assumed to have only isolated sin- 
 gular points. That an infinite number of singular points may- 
 be grouped together in the neighborhood of a single finite 
 point is evident, however, from the consideration of such ex- 
 amples as 
 
 w = cot-, w = e^°^^^ ^. 
 z 
 
 In the former an infinite number of poles are grouped in the 
 neighborhood of the origin. In the latter an infinite num- 
 ber of essential singularities are situated in the vicinity of the 
 point z =. a. 
 
 It is easy to illustrate by an example the occurrence of lines 
 and regions of discontinuity. Take the series * 
 
 The sum of its first n terms is 
 
 I 
 
 z — I 
 
 which converges to unity if |^|< i, and to zero if |-s^!> I. 
 Hence the circle \z\=. i is a line of discontinuity for this 
 series. 
 
 Consider now any two regions 5, and 5,, the former situated 
 within, the latter without, the unit circle. Let (p{2) and ip{z) 
 be two arbitrary functions both completely defined in these 
 regions. The expression 
 
 ^(z)ii{£)+<p{z)[_i-e{z)\ 
 
 * This series is due to J. Tannery. See Weierstrass, Abhandlungen aus der 
 Functionenlehre (1886), p. 102. 
 
SINGULAR LINES AND REGIONS. ^ 79 
 
 will be equal to 0(z) in S^ and ^(z) in S^. In regions com- 
 pletely separated from one another by a singular line, the same 
 literal expression may thus represent entirely independent 
 functions. 
 
 For a single continuous region, however, in the interior of 
 which exist only isolated critical points, the character of the 
 function in one part determines its character in every other 
 part. Let 5 be such a region, and assume that its boundary is a 
 singular line. In the neighborhood of any interior point a, not 
 a critical point, the given function is expressible as a power 
 series, viz. : 
 
 M = /(«) + (^ - ^yx") +■■■+ -r^^/'"' w + • • • 
 
 This series will converge uniformly over a circle described 
 about « as a center with any determinate radius less than the 
 distance from a to the nearest singular point. It serves for the 
 calculation of/(^) and all its successive derivatives at any point 
 ^ interior to this circle. From the preceding power series, ac- 
 cordingly, can be obtained another 
 
 A^) = Ab) + (^ - i>V'{.b) + ... + If^yLmb) +..., 
 
 representing the f{z) within a circle described about /^ as a 
 center. In general, the point b can be so chosen that a portion 
 of this new circle will lie without the circle of convergence of 
 the former power series. At any new point c within the circle 
 whose center is b, the value of the function and all its succes- 
 sive derivatives can be calculated ; and so, as before, a power 
 series can be obtained convergent in a circle described about c 
 as a center and, in general, including points wot contained in 
 either of the preceding circles. By continuing in this manner 
 it will be possible, starting from a given point a with the ex- 
 pression oi f{z) in ascending powers, to obtain an expression of 
 the same character at any other point k which can be connected 
 with <^ by a continuous line everywhere at a finite distance 
 from the nearest singular point. It follows that the character of 
 
80 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 the function everywhere within 5 can be determined completely 
 from its expression in ascending power series in the neighbor- 
 hood of a single interior point. 
 
 The process here described, whereby from a single ascending 
 power series representing a function in the neighborhood of a 
 given point of the z-plane one can derive a succession of similar 
 series, the totality of which determines the function throughout a 
 connected region limited only by the singularities of the function, 
 is known as the process of " analytical continuation." Each of 
 the series obtained is called an ** element " of the function. Ac- 
 cording to the theory of functions of a complex variable as pre- 
 sented by Weierstrass, the infinite number of elements connected 
 together by the process of analytical continuation are said to 
 constitute the definition of an " analytical function." 
 
 It will be impossible by the process just explained to derive 
 any information in regard to a function at points exterior to the 
 connected region S covered by the circles of convergence of its 
 elements. Moreover, as has been shown by an example, an 
 expression which gives a complete definition of /(z) within S 
 may carry with it the definition of an entirely independent func- 
 tion outside of 5. 
 
 As an example of a function having a singular region con- 
 sider the function defined by the scries 
 
 I -f- 2^ -f- 2^* + 2<S'' 4" . . . , 
 
 which represents a function without singular points in the 
 interior of the circle j^s"! = I. For points on or without this 
 circle the series is divergent ; and, further, it is impossible to 
 obtain from it an expression converging when \s\ = i. The 
 function thus defined, consequently, exists only in the region 
 interior to the unit circle. By changing j3 into 1/2 a series 
 I 2 , 2 , 2, 
 
 is obtained, representing a function which has no existence in 
 the interior of the unity circle. Functions in connection with 
 which such regions arise are called "lacunary functions."* 
 
 * Poincar6, American Journal of Mathematics, Vol. XIV; Harkness and 
 Morley, Theory of Functions (1893), p. 119 
 
FUNCTIONS HAYING n. VALUES. 81 
 
 Art. 32. Functions Having n Values. 
 
 Let the function w =/{^) take at the point ^^ of a given re- 
 gion 5 a value w^^K Suppose that along any continuous path, 
 beginning at :s^, and subject only to the conditions that it shall 
 remain in the interior of 5 and shall not. pass through certain 
 isolated points a^ , a^ , . . . , w is continuous and has a contin- 
 uous derivative. If it is impossible, when 2 traces such a path, 
 to return to the point -s-^ so as to obtain there a value of w dif- 
 ferent from w^^\ w is uniform in the region 5. On the other 
 hand, certain paths may lead back to z^ with new values of w. 
 
 Suppose that at each point of S, except «, , ^, , . . . , zf has 
 n different values, and that starting from such a point z^ and 
 tracing any continuous curve not passing through <?;, , ^j, . . . , 
 the several values of w give rise to n branches ze/j , w, , . . . , w^, 
 each of which is characterized by a continuous derivative. In 
 the neighborhood of a^ any one of the points a^^ a^^ , . . 
 these branches are said to be distinct or not, according as small 
 closed curves described about this point lead from each value of 
 w back to the same value again, or cause some of the branches 
 to interchange values. In the latter case the point is a branch 
 point. 
 
 About any branch point Uk as a center describe a small cir- 
 cle ; and suppose that, starting from any point of it with the 
 value Wa. corresponding to a certain branch, the values 
 w^ ^Wy ... are obtained by successive revolutions about ak , 
 the original value being reproduced after p revolutions. In- 
 troduce now a new independent variable 2' such that 
 
 z' = {z-a,fK 
 
 It can be shown that when z makes one revolution about 
 ak , z' makes only one /th part of a revolution about the ori- 
 gin of the .a'-plane, and that to a complete revolution of z^ 
 
82 FUNCTIONS OF A COMPLEX VARIABLE 
 
 about the origin of the ^'-plane correspond / revolutions of z 
 about a^. Considering then the branch w^ as a function of z' , 
 the origin cannot be a branch point, for whenever z' describes 
 a small circle about it, the value w^. is reproduced. The 
 branch Wa^ must accordingly be expressible by Laurent's 
 series in the form +«, 
 
 — 00 
 
 or, substituting for z' its value, 
 
 1 3 
 
 w^ = A,-\- A,{z - a^V + AJ,z - a^* + . . . 
 
 + ^_.(-sr -^,r^ + yi_,(^ - ^,)"^+ . . . 
 
 This expression makes plain the relation between the different 
 branches of a function in the neighborhood of a branch point. 
 When the development of a branch in the neighborhood of one 
 of its branch points gives rise to only a finite number of terms 
 containing negative powers, the branch point is called a " polar 
 branch point." 
 
 Consider the functions 
 
 P^ = W^W^ + W^W^ •\- . . .-\- Wn-^Wny 
 
 P^ = W^W^ . . .Wn. 
 
 Each of these functions is unchanged in value when several or all 
 of the quantities w^, ee/,, . , * y w^ are interchanged, and is con- 
 sequently a one-valued function of z within 5. Hence w must 
 satisfy an equation of the «th degree, 
 
 w^ + P^w''-' + P^ tv^-' + . . . + p^ = o, 
 the coefficients of which are one-valued functions of z having 
 only isolated critical points within S. When the entire ^-plane 
 can be taken as the region S, and those branch points at which 
 the branches do not all remain finite are polar branch points, 
 the only other critical points being poles for one or more 
 branches, the functions P^, P^, . . . , P„ are rational functions 
 of z. In this case w is an algebraic function of z. 
 
ALGEBRAIC FUifCTIONS. 83 
 
 Art. 33. Algebraic Functions. 
 
 Any algebraic function satisfies an equation of the form 
 F(z, w) = o, where i^(z, w) is a rational entire function of z and w. 
 It this equation is of the n\h degree in w, to any value of z will 
 correspond, in general, n dififerent values of w, but for special 
 values of z, two or more values of w may be equal. 
 
 The principle of continuity appHed to the values of an alge- 
 braic function would lead us to expect that, when F{a, 'w) = o 
 has q roots equal to h, it should be possible, whatever the value 
 of the positive number £, to determine a positive quantity d such 
 that, whenever \z — a\<d, the equation i^(z, 'w) = o would give q 
 and only q values of w satisfying the condition \w — h\<£. 
 
 It is necessary in the demonstration of this fundamental prop- 
 erty of algebraic functions to consider only the case where a and 
 h are both zero; for every other case can be reduced to this one 
 by means of the substitution z=a+2', w = h-\-'u/. Write the 
 function Fiz, w) in the form 
 
 F{z, 'w) = P^ + P^w-\-, . .-VPqW^-\-. . .+P„7t;«, 
 
 in which, when z=o, P^==P^ = . . .=Pq.^=^o, but Pq takes a 
 value different from zero. This expression can be put in the 
 form 
 
 F{z,w)=P^w%i + U^-V\ 
 where 
 
 W^Pq^'"^W Pq' 
 
 Describe about the points z=o and 7£;=o as centers, in the 
 is-plane and le^-plane respectively, circles C and T, of radii r and 
 p. It is possible to choose r and p sufficiently small to satisfy 
 the following conditions: (i) whenever z and w are interior to C 
 and r, 
 
 \U\<h\ 
 
84 
 
 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 (2) whenever w is on the circumference T, and z is interior to C, 
 
 \V\<i. 
 
 It is evidently possible to satisfy the first condition. The ine- 
 quality 
 
 l^'V, 
 
 + ... + - 
 P 
 
 shows, further, since P^, . . . , Pg-^ all approach zero with r, 
 that for any value of |0, r can be chosen sufficiently small to sat- 
 isfy the second condition. 
 
 But for any assignable position of z within C, the number of 
 roots of the equation F{z, w) = o contained within F is, by 
 Theorem II, Article 27, equal to 
 
 2mJ ■ 
 
 dw^ 
 
 [PgW%i + U+V)] 
 
 •dw, 
 
 T PgW%i-\-U-{-V) 
 
 or the total variation of any branch of 
 
 hglPgW^ii + U+V)], 
 
 when w describes the circumference T, divided by 27Ti. But 
 
 log [PgW^{i + U+ F)]=log Pg + q log w+log (i + U+ V). 
 
 The first term is constant; the total variation of the second term 
 is 2mq; and, since 1C7+ F| < i when w is on the circumference jT, 
 the argument of i-\-U-\-V must return to its original value, and 
 the total variation of log(i + C/+F) is zero. The number of 
 values of w within F is, therefore, equal to q. 
 
 Those values of z for which two or more values of w are equal 
 must satisfy the equation obtained by eliminating w between 
 
 F{z,w)=o, 
 
 dw 
 
 F(z,w)=o. 
 
INTEGRALS OF ALGEBEAIC FUNCTIONS. 85 
 
 For every other finite value of z, the equation 
 
 dF(z, w) 
 dw dz 
 
 dz '^F{z, w) 
 dw 
 
 gives at once a single determinate value for the derivative of w. 
 
 It follows from the preceding Article that any branch Wa of 
 w must be expressible in the neighborhood of any singular point 
 a A; by a series of the form * 
 
 Wa = A^+A,(z-ak)^ +A^(z-ak^ +. . . 
 
 I 2_ 
 
 -{■B,iz-ak)~^ + B^(z-ak) ^+... 
 
 uniformly convergent in a small circular band surrounding the 
 point dk. If dk is not a branch point, p = i. 
 
 Art. 34. Integrals of Algebraic Functions. 
 
 In determining the value of the integral of an algebraic func- 
 tion w=/{z) along any path joining z^ to z, it is possible by virtue 
 of Cauchy's Theorem to alter the path of integration arbitrarily, 
 provided that no singular point is contained in the region enclosed 
 between its original and final positions. By employing the same 
 reasoning as in Article 28, any path joining z^ to z may be reduced 
 to a determinate path, preceded by a system of loops, of which 
 each encloses a single singular point. The value of the integral 
 corresponding to a loop surrounding a branch point requires 
 special consideration. If z describes such a loop, w returns to 
 Zq with an altered value. When, however, the initial point is 
 fixed, the value of the integral is not altered by varying arbi- 
 
 * For examples see Briot and Bouquet, Fonctions elliptiques (1875), PP- 40> 
 57; Chrystal, Algebra, vol. 11 (1889), pp. 356, 370. 
 
86 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 trarily the form of the loop, provided that no singular point is 
 introduced into or removed from the loop. 
 
 To show that a given loop, containing a branch point and 
 attached to the path of integration at a point c^, different from 
 Zq, may be transformed into one whose initial point is z^, it is 
 necessary to observe that the variable passes first from Zq to c^ 
 and then around the loop to c^ again. If now, before continuing 
 along the remaining part of the path, z be required to retrace its 
 way to Zq and then return to Ci, the value of the integral will not 
 be altered thereby; for the integral resulting from the path c^z^c^ 
 is equal to zero. The loop, however, has been made to begin and 
 end at z^i and it is followed by a path which begins at z^. 
 
 For any algebraic function, therefore, just as for a function 
 without branch points, the most general path of integration can 
 be reduced to a determinate path, having the same limits, pre- 
 ceded by a system of loops of which each encloses a single sin- 
 gular point. 
 
 The integral around such a loop enclosing a a-, a singular 
 point but not a branch point for the branch of /(z) considered, is 
 equal to ±27:iBk, where Bk is the residue of this branch of /(z) 
 at a A, and the plus or minus sign is taken according as the loop 
 is described in a positive or negative direction. 
 
 Consider now a loop enclosing a branch point a^. It can be 
 reduced to a special form, consisting of a small circle described 
 
 about a^^ as a center and a line, 
 straight or curved, joining this 
 circle to z^. The term Q^ to be 
 added to the integral on account 
 of this loop will be obtained by 
 integrating w=/(z) from z^ along 
 the line joming z^ to tnc circle, around the circle, and back along 
 the same line again to z^. The parts resulting from tracing the 
 line joining z^ to the circle in opposite directions do not cancel; 
 since on account of the nature of the branch point w does not 
 take its former system of values when z retraces its path to z^. 
 
 If now the integral of /(z) along any determinate path from 
 
INTEGKALS OF ALGEBRAIC FUl^CTIONS. 87 
 
 Zq to z be denoted by I{z), the general value of the integral, J{z), 
 resulting from an arbitrary path between the same limits, is 
 
 J(z)=I(z) + 2Q„, 
 
 where Q^ is the value of the integral along the wth loop in the 
 reduced form of the path of J(z). 
 
 If the upper limit z of the integral J(z) is situated in the neigh- 
 borhood of a critical point, w is expressible in a region containing 
 z by the uniformly convergent series 
 
 I _£ 
 
 w=AQ+A^(z-aky ^A^iz-aky +. . . 
 
 ' I 2 
 
 +B,{z-ak) ^+R,iz-ak) ^+... 
 
 The integral, therefore, except for a constant term, which includes 
 IQ^i is equal to 
 
 J{z)=A^(z-ak) + j^A,(z-ak) ^ +j^A^(z-ak) ^ +. . . 
 
 +~-^B,(z-ak)~^~-^--^B^(z-ak)~P~ + . , .+pBp.,(z-ak)'^ 
 
 pi p 2 
 
 _- 1 p _£_ 
 
 +Bp\og{z-ak)-pBp+z{z-ak) p --Bp+^{z-ak) ^-... 
 As an example consider the integral 
 
 where the initial value of the radical vi— z^ is +i. If under 
 the integral sign z be replaced by z/, where / is a real quantity vary- 
 ing from zero to unity, the resulting integral 
 
 /(z) = z/^-=4== 
 
 will correspond to a rectilinear path joining the origin to z. 
 
B8 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 In J{z) the only singular points of the integrand are z= ±1. 
 The integral for the circumference of a small circle described 
 about either of these points as a center, by Theorem I of Article 
 19, approaches zero as a limit simultaneously with the radius of 
 the circle. A loop enclosing the point +1, therefore, gives a 
 term equal to 
 
 r' dz n^ dz _ />* dz _ 
 
 the radical taking a negative sign on the way back to the origin 
 by virtue of the fact that z has turned around the branch point 
 2 = 1. In the same way, a loop enclosing the point 2= — i will 
 give, if the initial value of the radical is positive, 
 
 
 dz 
 
 When z describes a loop about either of the points ±1, the radical 
 returns to the origin with its sign changed. Hence, if z describe 
 in succession two loops about the same branch point, the total 
 effect on the value of the integral is zero. If the path of the in- 
 tegral J^{z) is equal to that of the integral J{z) preceded by a 
 single loop enclosing the point +1 or the point — i, the value 
 of J^{z) will be 
 
 Tz—J{z) or —Tz—J{z) 
 
 respectively. If the path of J^{z) consist of two loops, the first 
 about z = I, the second about z= — i, followed by the path of /(2),* 
 
 /i(z) = 27r+/(z). 
 
 An arbitrary path from z^ to z gives an integral of the form 
 
 2mz-\-I{z) or (2W + i)7r— /(z), 
 
rUKCTIONS OF SEVERAL VARIABLES. 89 
 
 where n is an integer positive or negative and I{z) is the integral 
 for a rectilinear path. 
 
 Prob. 28. If i?=V(2— <Zi) . . . (2— a„), and the rectilinear inte- 
 eral value of / ^ is 
 
 i^ 
 
 2m^Ai+ . . . +2w^^^+Z or 2miAi+ . . . +2m„A^+AK—Z, 
 where mi, . . . , /^„ are any integers, positive or negative. 
 
 Art. 35. Functions or Several Variables. 
 
 Let f(Zi, Z2) be a function of two independent variables holo- 
 morphic with respect to each when z^ and z^ are interior to the 
 regions A^ and A^ respectively. Let Q and Cg be two closed 
 curves drawn in these regions, and let a^ and aj be points con- 
 tained within these curves. Then 
 
 rAz^z^), ... . ' 
 
 J - — —dz, = 2mf(a,,Z2) 
 
 *^Cx ^1 — "1 
 
 X 1 — —dz^=2mf{a^, a^), 
 
 so that 
 
 /fe, ^2) 
 
 Differentiating this integral with respect to the parameters a^ ^j* 
 gives the general result 
 
 f* r /(Zi, z^dz^dz^ 
 
 (2;ri) 
 
 'ba^'ba^ 
 
90 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 It follows that /(2i, Zj) has an infinite number of successive par- 
 tial derivatives holomorphic under the same conditions as4tself. 
 
 Let M be the upper bound of the modulus of /(Zj, Zj) when 
 Zj and Z2 vary along the curves Q and Cj respectively; r^ and rj* 
 the shortest distances from a^ and a^ to these curves; l^ and /j, 
 the lengths of these curves: then 
 
 ^ 1-2 . . . p'i-2 . . . q MIJ2 
 
 < (27:)' r/+^r2«+^* 
 
 If Cj and C2 are circles described about a^ and aj as centers, 
 /i = 27rri, l^ = 2nr2, and 
 
 It is easy now to extend Taylor's Series to the case of a function 
 of two variables. Let/(Zi, Zj) be holomorphic as long as z^ and 
 Z2 remain within circles Cj and C^ described about a^ and a^ as 
 centers. Let flj + Zj, ^2 + ^2 be points chosen arbitrarily within 
 these circles. Then 
 
 I / 3 3 \^, 
 + 7:rV'3^+''3a;M'^''"''^) 
 
 (•-v Ci \ 3 
 
DIFFERENTIAL EQUATIONS. 91 
 
 The proof that the remainder approaches zero as a limit is anal- 
 ogous to that given in the case of a single variable. 
 
 Corresponding results can be obtained for functions having 
 any nimiber of independent variables. 
 
 Art. 36. Differential Equations.* 
 Consider the differential equation 
 dw 
 
 where /(z, 7£;)is holomorphic when z and w are near the points 
 
 Zq and Wq respectively. By the transformation 'W='WQ-h'u/j z= 
 
 dii/ 
 z^\z\ the equation becomes -jy = ^{z', v/), where 9^(2', w) is 
 
 holomorphic when 2' and w' are both near zero. Without loss 
 of generahty, therefore, the discussion can be restricted to the 
 special case where /(z, w) is holomorphic with respect to z and 
 w^ when z and w are confined to small regions containing z = o 
 and w = o respectively. 
 
 If the given differential equation admits an integral, holomor- 
 phic in the neighborhood of z = o, and vanishing at that point, 
 this integral will be unique; for all its successive differential co- 
 efficients at the point z = o can be obtained from the given differ- 
 ential equation. It is sufficient to differentiate that equation 
 once, and make z = o, w=o, in order to find the second differen- 
 tial coefficient; to differentiate again and make the same substi- 
 tution to find the third differential coefficient, and so on. In this 
 way is obtained the development. 
 
 'dw^ 
 dz 
 
 «'=(:^)/+r^(S)„^'+- • •=«.^+''.^'+- • ■ 
 
 If this development can be proved to converge when \z\ is suffi- 
 ciently small, w thus defined satisfies the differential equation. 
 
 div 
 For -7- and/(z, w) have the same value for z=o; and their suc- 
 
 * Briot and Bouquet, Fonctions EUiptiques, p. 325 ; Picard, Traite d' Analyse, 
 vol. n, p. 291. 
 
92 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 cessive differential coefficients with respect to z of any order what- 
 
 dw 
 soever are also equal for 2=0. Hence -7- and/(z, w) are equal. 
 
 Describe small circles C and C about the points z = o, w=o 
 as centers with radii r, /. Let M be the upper bound of the 
 modulus of /(z, w) within or upon these circles. If now the 
 function 
 
 F(z,w) = 
 
 Hi'-?) 
 
 be constructed, it will be holomorphic within the circles C and 
 C. Its development in a convergent series of ascending powers 
 of z and w, is found by multiplying together the series for 
 
 and 
 
 z w 
 
 r r 
 
 and introducing into each term the constant factor M, 
 
 The successive partial derivatives of F{z, w) are all positive 
 and such that 
 
 ?>zpdw^ \izl<\ 'dzp'dw<i y^rg- 
 
 Consider now the differential equation 
 
 dW ^, „,, 
 
 If it has an integral W^ holomorphic in the neighborhood of 
 2=0, the integral will be expressible in the form 
 
 The coefficients in this series are all positive, and for every value 
 of w 
 
DIFFERKXTIAL EQUATIONS. 93 
 
 The series given above for w, therefore, is convergent at every 
 point where the series for W converges. But it is easy to demon- 
 strate the existence of the function W. For the equation 
 
 dW M 
 
 dz 
 
 (-7)(-f) 
 
 may be written in the form 
 
 / _W\dW M 
 V r'l dz~ z 
 
 I — 
 r 
 
 The two members are the derivatives respectively of 
 
 Mr log [i-^. 
 
 W-~ and 
 2r 
 
 If the logarithm be chosen so that it vanishes when z = o, it will 
 be holomorphic within the circle |z| = r. Since W is to vanish 
 when z=o, the relation between W and z should be 
 
 or 
 
 
 where the radical is equal to + 1 for z = o. 
 
 The function W thus defined satisfies the equation 
 
 dW 
 
 ~-j~=F{z, W); it vanishes when z = o; and it is holomorphic 
 
 in the interior of a circle having for its center the origin, and for 
 its radius p the root of the equation 
 
 2Mr I p\ 
 i + -^log(^i--j=o, 
 
94 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 that is, p=r(i-e''^Mr\ 
 
 The series for W, consequently, converges in the interior of the 
 circle of radius p. The series for w must converge in the same 
 circle. Hence the given differential equation admits an integral 
 vanishing for z = o, and holomorphic within the circle of radius 
 p and center at the origin. 
 
 The preceding discussion can be extended without modifica- 
 tion to the case of n equations : 
 
 
 -3r=/„(2,^i,^2,...,wJ, 
 
 The functions in the second members are supposed to be holo- 
 morphic with respect to z, Wj, . . . , w^ within a circle of radius 
 r described about z=o, and circles of radius / described about 
 w;i=o, ...,w„ = o. 
 
 If, further, M denotes the upper bound of the moduli of/, 
 fii'-'ifn i^ ^^^ regions considered, the associated differential 
 equations are 
 
 dz dz '" dz ^^^'^i^^^y-'y^nh 
 
 where 
 
 M 
 
 F(z,W,,W,,..,,W) = 
 
 (-;-)(-^^)-(.-^-)- 
 
DIFFERENTIAL EQUATIONS. 95 
 
 The functions W^, PTj, . . . , TF„ all vanish for z=o and are 
 identical, so that only one equation 
 
 dW M 
 
 dz 
 
 (-r)(-7) 
 
 need be considered. The radius p of the circles, within which 
 all the developments converge, is 
 
 ,=r\i~e («+^)^'-). 
 
 P- 
 
 As an example take the differential equation 
 dw 
 
 assuming as initial conditions 2 = 0, 110 = 0. This equation defines 
 'Z£; as a holomorphic function of z in any region in which w re- 
 mains finite. Suppose that w becomes infinite for some finite 
 value a of the variable z. To determine the nature of the point 
 2 = a, make the substitution 
 
 vf 
 The given differential equation is transformed to 
 
 dw' 
 
 the initial conditions being 2' = o, ii/ = o. This equation defines 
 i£/ as a holomorphic function of z' in the neighborhood of 2' = o, 
 and, consequently, of z in the neighborhood of z = a. The given 
 differential equation is satisfied, therefore, by a function w whose 
 only finite critical points are poles. 
 
96 FUNCTIONS OF A COMPLEX VARIABLE. 
 
 The values of 2 for which w takes an assigned value may be 
 found bv means of the integral 
 
 /"" dw I /'"' dw j_ n"" dw 
 „ i+w^ 21 Jq w — i 2i^o w-^-i 
 
 If w describes two paths symmetrical with respect to the origin, 
 z acquires values numerically equal but of opposite signs. It 
 follows that w is an odd function of z. A loop enclosing the point 
 'W=ij described in the positive direction n times, adds to the inte- 
 gral a term equal to nn. A loop described about w^ —i in a 
 positive direction n times similarly gives —mz. The function w 
 is thus periodic, having a period equal to n. 
 
 It is possible to express w as the ratio of two functions having 
 no finite critical points. Assume 'W = 'wjw2. The given differ- 
 ential equation takes the form 
 
 (dw, \ Idw^ \ 
 
 This equation can be satisfied by making 
 
 dw. dw^ 
 
 and 2=0, 'Z£'i=o, ^2 = 1 ^^^7 ^^ taken as initial conditions. From 
 these equations can be obtained 
 
 
 
 w^- 
 
 dw^ 
 dz 
 
 d'w, 
 dz' 
 
 d'w, 
 dz' 
 
 dHv, 
 dz' 
 
 
 
 w^- 
 
 dw, 
 dz 
 
 d'w^ 
 dz'~ 
 
 d'w, 
 dz'' 
 
 d'w^ 
 ~ dz* 
 
 Hence, 
 
 when 
 
 z = 
 
 ■Oj 
 
 
 
 
 , , /dw,\ /d'w,\ /d?w\ (d*w,\ 
 
DIFFERENTIAL EQUATIONS. 97 
 
 and 
 
 The series for w^ and w^ are, therefore, 
 
 Wi = -- 1 . . . =sin0, 
 
 ' I I -2 -3 I -2 -3 •4-5 
 
 ^2=1- -\ : — ...=cos2: 
 
 1-2 I -2 -3 -4 
 
 sin z 
 
 whence w = = tan z. 
 
 cos z 
 
 Prob. 29. Show that the integral of — = w, with the initial condi- 
 tions z=o, 10=1, is the series 7£'=i4-zH h. . . = exp. z. 
 
 Prob. 30. Show that the equation ;7-^+«^=o is equivalent to the 
 
 du dv , , .,,..., 
 
 system, ■j-='^, -i-=—u\ and that with the initial conditions z=o, 
 
 u=a, v=b, the solution by series gives u=a cos z+b sin 0. 
 
 Prob. 31. Show that the equation -ty=(i — w2)(i — ifeW^ with the 
 
 az 
 
 initial conditions z=o, w=o, Vi — w^= + i, Vi — )^2'Z£;2= + i, is equiv- 
 alent to the system ~=uv, j-=—v'w, —=—k^wu, with th^ initial 
 
 conditions z=o, w=o, u=i, v=i, and that the functions w, u, v have 
 no finite critical points except poles. The functions are Jacobi's 
 ■elliptic functions sn z, en z, dn z, respectively. 
 
INDEX. 
 
 Absolute convergence, 3 
 Algebraic functions, 83 
 Analytical continuation, 80 
 Argtiment, 2 
 
 Bound, 33 
 Branch point, 27 
 
 Cauchy's theorem, 37 
 Complex integrals, 36 
 Complex variable, 2 
 Conformal representation, 11 
 Conjugate functions, 20 
 Continuation, Analytical, 80 
 Continuity of function, 5 
 Continuity of variable, 2 
 Convergence, Absolute, 3 
 Convergence, Uniform, 49 
 Curvilinear integrals, 42 
 Cut, 26 
 
 Derivative, 8 
 Differential equations, 91 
 
 Elementary functions, 4 
 Element of function, 80 
 Equipotential curves, 23 
 Essential singular point, 28 
 
 Fluid motion, 21 
 Fourier's series, 48 
 Function, Definition of, i 
 Functions having n values, 81 
 Functions of several variables, 89 
 
 Oamma function, 71 
 Goursat's demonstration, 10 
 Graphical representation, 7 
 
 Holomorphic function, 31 
 
 Independent variable, i 
 Infinity, Point at, 31 
 
 Integral, Definition of, 32 
 
 In egral of algebraic function, 85 
 
 Integral of uniform function, 63 
 
 Lacunary functions, 80 
 Laurent's series, 46 
 
 Magnification, 12 
 Mercator's projection, 20 
 Meromorphic function, 31 
 Mittag-LeflSer's theorem, 71 
 Modulus, 2 
 Monogenic function, 10 
 
 Non-monogenic function, 10 
 
 Ordinary point, 30 
 Orthogonal systems, 21 
 
 Path, 2 
 
 Pole, 26 
 
 Power series, 54 
 
 Primary factors, 69 
 
 Rectifiable curve, 32 
 Residues, 61 
 
 Series of functions, 49 
 Singular lines and regions, 79 
 Singular points, 25 
 Sink, 25 
 
 Spherical representation, 19 
 Stereographic projection, 20 
 Stream lines, 22 
 Synectic function, 31 
 
 Taylor's series, 44 
 
 Uniform convergence, 49 
 
 Vortex point, 24 
 
 Weierstrassian elliptic functions, 71 
 Weierstrass's theorem, 66 
 
 99 
 
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