?/ \ \. . \ VQS" , vt> C^ 14&* lf>Cx}~~< .. ' J>f } - -f ( w) >^ '^l ~ > u3 . ^ x f(-^) LIBRARY THE UNIVERSITY OF CALIFORNIA SANTA BARBARA PRESENTED BY RAYMOND WILDER Hmerican /IDatbematical Series E. J. TOWNSEND GENERAL EDITOR American Calculus By E. J. TOWNSEND, Professor of Mathematics, and G. A. GOODENOUGH, Professor of Thermodynamics, University of Illinois. $2.50. Essentials of Calculus By E. J. TOWNSEND and G. A. GOODENOUGH. #2.00. College Algebra By H. L. RIETZ, Professor, and A. R. CRATHORNE, Associate in Mathematics in the University of Illinois. $1.40. Plane Trigonometry, with Trigonometric and Logarithmic Tables By A. G. HALL, Professor of Mathematics in the University of Michigan, and F. H. FRINK, Professor of Railway Engineering in the University of Oregon. $1.25. Plane and Spherical Trigonometry. (Without Tables.) $1.00. Trigonometric and Logarithmic Tables. 75 cents. Analytic Geometry By L. W. BOWLING, Associate Professor of Mathematics, and F. E. TURNEAURE, Dean of the College of Engineering in the University of Wisconsin, xii + 266 pp. iamo. $1.60. Analytic Geometry of Space By VIRGIL SNYDER, Professor in Cornell University, and C. H. SISAM, Assistant Professor in the University of Illinois. xi+ 289 pp. I2mo. $2.50. Plane Geometry By J. W. YOUNG, Professor of Mathematics in Dartmouth College, and A. J. SCHWARTZ, William McKinley High School, St. Louis. 85 cents. School Algebra By H. L. RIETZ, Professor of Mathematical Statistics, A. R. CRATHORNE, Associate in Mathematics, University of Illinois, and E. H. TAYLOR, Pro- fessor of Mathematics in the Eastern Illinois State Normal School. First Course. 271 pp. i2mo. $1.00. Second Course. 235 pp. I2mo. 75 cents. Functions of a Complex Variable By E. J. TOWNSEND, Professor and Head of the Department of Mathe- matics in the University of Illinois, vii -f- 384 pp. 8vo. $4.00. HENRY HOLT AND COMPANY NEW YORK CHICAGO JUNCTIONS OF A COMPLEX VAEIABLE BY E. J. TOWNSEND, PH.D. n PROFESSOR OF MATHEMATICS, UNIVERSITY OF ILLINOIS NEW YORK HENRY HOLT AND COMPANY 1915 COPYRIGHT, 1915, BY HENRY HOLT AND COMPANY Stanbope jpress F. H. GILSON COMPANY BOSTON, U.S.A. PEEFACE THE present volume is based on a course of lectures given by the author for a number of years at the University of Illinois. It is in- tended as an introductory course suitable for first year graduate students and assumes a knowledge of only such fundamental prin- ciples of analysis as the student will have had upon completing the usual first course in calculus. Such additional information concern- ing functions of real variables as is needed in the development of the subject has been introduced as a regular part of the text. Thus a discussion of the general properties of line-integrals, a proof of Green's theorem, etc., have been included. The material chosen deals for the most part with the general properties of functions of a complex variable, and but little is said concerning the properties of some of the more special classes of functions, as for example elliptic functions, etc., since in a first course these subjects can hardly be treated in a satisfactory manner. The course presupposes no previous knowledge of complex numbers and the order of development is much as that commonly followed in the calculus of real variables. Integration is introduced early, in connection with differentiation. In fact the first statement of the necessary and sufficient condition that a function is holomorphic in a given region is made in terms of an integral. By this order of arrangement, it is possible to establish early in the course the fact that the continuity of the derivative follows from its existence, and consequently the Cauchy-Goursat and allied theorems can be dem- onstrated without any assumption as to such continuity. Likewise, it can thus be shown that Laplace's differential equation is satisfied without making the usual assumptions as to the existence of the derivatives of second order. The term holomorphic, often omitted, has been used as expressing an important property of single-valued functions, reserving the use of the term analytic for use in connection with functions derived from a given element by means of analytic continuation. While the Cauchy-Riemann viewpoint is that first introduced, attention is called to the Weirstrass development m the IV PREFACE chapter on series, and in subsequent discussions either definition of an analytic function is used as best suits the purpose in hand. In Chapter IV much use is made of mapping, thus enabling us to consider in connection with the definition of certain elementary functions some of their more important uses in physics. For the same reason in Chapter V the consideration of linear fractional transformation is especially emphasized and discussed as a kinematic problem. The discussion of series in Chapter VI lays the foundation for the consideration of the fundamental properties of single-valued functions discussed in the following chapter. In the final chapter, it is pointed out how these properties may be extended to the con- sideration of multiple-valued functions. The author wishes to express his appreciation of the helpful sug- gestions which have been given to him by Professor J. L. Markley of the University of Michigan, Professor A. Dresden of the University of Wisconsin, Professor W. A. Hurwitz of Cornell University, and to Dr. Otto Dunkel of the University of Missouri, who have read the proof sheets. He is also under obligations to his colleagues Dr. Denton and Dr. Kempner, who have read the manuscript. Finally, he wishes to express especially his obligations to Dr. George Rut- ledge, who has rendered him valuable assistance in the preparation of the manuscript. E. J. TOWNSEND. UNIVERSITY OP ILLINOIS July, 1915 ERRATA. Page 26, line 8, insert 5 after number. " 27, line 5 from bottom, for (4) read (8). " 38, replace z by t and z\ by t\. " , line 16 from bottom, for/00 read /(fl. " 55, Eq. (1), for J read J . " 71, Fig. 26, forC^'readC'.' " 88, line?, for f(x) read/00. " , line 6 from bottom, for Az read Aa. " 92, line 11 from bottom, for I read / . " 93, line 15, for (2) read (3). " 101, line 8 from bottom, for 2iy read 2ixy. " 104, line 4 from bottom, for Art. 21 read Art. 22. " 106, Fig. 36, interchange V and U. " 112, line 6 from bottom, for 4 u 4 read 4 u + 4. " 155, Ex. 20, for log ^ ^ read log ^ ~ J " 178, line 2 from bottom, for 2 4 read zt. " 195, lines 16-18, replace P by P lf and " at infinity " by P. " 240, line 15, for C read Ci. " 277, line 8 from bottom, for I read / . Jc Jc t " 286, line 14, transpose / and " when ". Jc " 299, line 7 from bottom, for z\ read z**. " 340, Eq. (6), for e ~^ T + ^ read e~^ T+ ^\ " " , line 8 from bottom, for (o, |) read (|, 0\ " 344, line 9, for r 2 read r 3 . 363, line 3 from bottom, for k read k + 1. /< /MT 364, line 3, for / read / . CONTENTS CHAPTER I REAL AND COMPLEX NUMBERS ARTICLE PAGE 1. Rational Numbers 1 2. Irrational Numbers 2 3. System of Real Numbers 4 4. Complex Numbers 5 5. Geometric Representation of Complex Numbers 6 6. Comparison of Complex Numbers 8 7. Addition and Subtraction of Complex Numbers 8 8. Multiplication of Complex Numbers 11 9. Division of Complex Numbers 16 CHAPTER II FUNDAMENTAL DEFINITIONS CONCERNING FUNCTIONS 10. Constants, Variables 20 11. Definition and Classification of Functions 21 12. Limits 23 13. Continuity 33 CHAPTER III DIFFERENTIATION AND INTEGRATION 14. Differentiation; Definition of an Analytic Function 43 15. Line-integrals 46 16. Green's Theorem 54 17. Integral of / (z) .'... 60 18. Change of Variable, Complex to Real 64 19. Cauchy-Goursat Theorem 66 20. Cauchy's Integral Formula 75 21. Cauchy-Riemann Differential Equations 82 22. Change of Complex Variable , . . 89 23. Indefinite Integrals 90 24. Laplace's Differential Equation 92 25. Applications to Physics 96 v VI CONTENTS CHAPTER IV MAPPING, WITH APPLICATIONS TO ELEMENTARY FUNCTIONS ARTICLE PAGE 26. Conjugate Functions 101 27. Conformal Mapping 104 28. The Function w = z n 114 29. Definition and Properties of e* 122 30. The Function IP = log z 133 31. Trigonometric Functions 144 32. Hyperbolic Functions 150 CHAPTER V LINEAR FRACTIONAL TRANSFORMATIONS 33. Definition of Linear Fractional Transformations 156 34. Point at Infinity 157 35. The Transformation u> = z + 159 36. The Transformation w = az 159 37. The Transformation w = az + 162 38. The Transformation w = - 166 z 39. General Properties of the Transformation w = - r-^ . 173 yz + 5 40. Stereographic Projection 184 41. Classification of Linear Fractional Transformations 190 CHAPTER VI INFINITE SERIES 42. Series with Complex Terms 198 43. Operations with Series 206 44. Double Series 213 46. Uniform Convergence 217 46. Integration and Differentiation of Series 222 47. Power Series 226 48. Expansion of a Function in a Power Series 238 CHAPTER VII GENERAL PROPERTIES OF SINGLE-VALUED FUNCTIONS 49. Analytic Continuation 245 60. Definition of Analytic Function 257 61. Singular Points, Zero Points 262 62. Laurent's Expansion 275 63. Residues 284 64. Rational Functions, Fundamental Theorem of Algebra 290 CONTENTS Vli ARTICLE PAGE 65. Transcendental Functions ....................................... 300 56. Mittag-Leffler's Theorem ........................................ 303 57. Expansion of Functions by Infinite Products ....................... 308 58. Periodic Functions .............................................. 317 CHAPTER VIII MULTIPLE-VALUED FUNCTIONS 59. Fundamental Definitions ........................................ 329 60. Riemann Surface for w 3 3 w 2 z = .......................... 338 61. Riemann Surface for w = Vz z + I/ - ..................... 343 T z z\ 62. Riemann Surface for w = log z ................................... 346 63. Branch-points, Branch-cuts ...................................... 347 64. Stereographic Projection of a Riemann Surface ..................... 354 65. General Properties of Riemann Surfaces ........................... 355 66. Singular Points of Multiple- valued Functions ...................... 358 67. Functions Defined on a Riemann Surface. Physical Applications .... 362 68. Function of a Function ......................................... 367 69. Algebraic Functions ............................................ 368 INDEX.. . ,\ . . 381 FUNCTIONS OF A COMPLEX VAEIABLE CHAPTER I REAL AND COMPLEX NUMBERS 1. Rational numbers. Some understanding of the nature of a number, the classes into which numbers may be divided, and the general laws governing the fundamental operations with them is essential to the study of the theory of functions. We obtain our first notion of numbers when we undertake to enumerate the indi- viduals composing a group of objects. The process of counting leads, however, only to the positive integers. We arrive at the same result when we assume the existence of unity and a certain mathematical process known as addition. Furthermore, the posi- tive integers obey the following law: Given any two positive integers a and 6(6 > a), there exists one and only one positive integer x such that a + x = 6. It becomes at once apparent that the positive integers do not completely serve the purpose of analysis when we attempt to solve the above equation for the case where a = 6. In order to give any interpretation at all to the solution in this case, it is necessary to introduce a new number called zero, denned by the identity a + = a. If a is allowed to be greater than 6, it is again necessary to ex- tend the domain of the number-system by the introduction of nega- tive numbers in order to give an interpretation to the solution of the above equation. Even with this extension of the number- system, it is impossible to solve all linear equations. Suppose, for example, it is required to find the value of x from the equation ax = b, a T*- 0. A number-system that includes only positive and negative integers is inadequate to interpret the result 6 2 REAL AND COMPLEX NUMBERS [CHAP. I. whenever b is not an integral multiple of a. A further extension of the number-system now becomes necessary and this extension is gained by the introduction of fractions. The numbers thus far discussed, that is integers including zero, and fractions, constitute a system of numbers called rational num- bers.* A characteristic property of such numbers is that they may always be expressed in the form - , where a and 6 are integers prime CL to each other and a ^ 0. By the aid of the symbols for the funda- mental operations of arithmetic rational numbers can always be expressed by a finite number of digits. It is possible and often con- venient to express such numbers by means of an infinite sequence of digits, but it is not necessary to do so. Thus f is a rational number, but when expressed in the form of a decimal fraction we have i = 0.3333 .... 2. Irrational numbers. If we undertake to solve equations of a higher degree than the first, the system of rational numbers often proves insufficient. For example, if we have given the equation z 2 - 2 = to find the value of x, we have x = "V/2, a result that has no interpretation in the domain of rational numbers. To show that no such interpretation is possible, assume j- = V2, a and 6 being integers prime to each other. We have then = 2, a* = 26*. The number 2 is then a factor of a 2 and as all prime factors appear an even number of times in a perfect square, 2 must appear an even number of times in a 2 . Consequently, 2 must also appear as a factor of 2 6 2 an even number of times. This, however, is impos- sible, as it must then appear as a factor of 6 2 itself and indeed an even number of times. As 2 cannot be a factor of one member of the identity an even number of times and of the other an odd num- ber of tunes, the assumption that \/2 is a rational number is not valid. * For a more complete discussion of rational numbers the reader is referred to Pierpont, Theory of Functions of Real Variables, Vol. I, Chap. I. ART. 2.] IRRATIONAL NUMBERS 3 We shall see later that it is characteristic of a new class of num- bers, called irrational numbers to distinguish them from the num- bers discussed in the preceding article, that they do not admit of expression in the form =- To see more clearly the nature of irrational numbers, let us con- sider the totality of rational numbers. Suppose we separate these numbers into two sets such that each number of the first set is greater than every number of the second set. Such a separation of the rational system of numbers is called a partition.* We have, for example, a partition if we select any rational number a and put into one set A\ all those rational numbers that are equal to or greater than a and into a second set A 2 all rational numbers that are less than a. In this case the number a is itself an element of the set A\. We may likewise establish a partition by putting into the set A\ ah 1 of those rational numbers greater than a and into A 2 ah 1 those equal to or less than a. In this case the number a belongs to set A 2 . It will be noticed that by the first partition there is a smallest number in AI and by the second partition there is a largest number in A 2 . In each case this number is the rational number a itself. It is possible, however, to establish a partition of the entire sys- tem of rational numbers in such a manner that in the pne set AI there shall be no smallest number and at the same time in the sec- ond set A 2 , there shall be no largest number. For example, let us consider again \/2. As we have seen, this number is not a rational number. Put into set AI all of those rational numbers whose squares are greater than 2 and into A 2 all rational numbers whose squares are less than 2. The two sets AI and A 2 then fulfill the conditions that each number of AI is greater than any number of A 2 and there is no smallest number in AI and no largest number in A 2 ; for, no matter how near to 2 the square of a particular rational number may be, there are always other rational numbers whose squares lie between the square of the one selected and 2. The notion of the partition of the system of rational numbers affords a convenient means of defining irrational numbers. For this purpose suppose the totality of rational numbers to be divided in any manner whatever into two groups AI, A 2 having the following properties : * Introduced by Dedekind, SietigkeU und irrationale Zahlen, Braunschweig, 1872. 4 REAL AND COMPLEX NUMBERS [CHAP. I. (1) Each number of the set A\ shall be greater than any number of the set A* (2) There shall be no smallest number in AI and no largest number in .1 : . In the case where a was the smallest rational number in A i or the largest one in At, it could be said that the partition defined uniquely the rational number o. In the present case, it can no longer be said that the partition defines a rational number; for, every rational number belongs either to set AI or set A 2 , and since by (2) there can be no smallest number in AI and no largest one in AZ, the partition can not define a number in either set. Conse- quently, the partition may be said to define a new number; we call such a number an irrational number. The fundamental operations of arithmetic may be defined for irrational numbers in a manner consistent with the corresponding definitions for rational numbers.* 3. System of real numbers. The rational numbers and the irra- tional numbers taken together constitute a system of numbers known as real numbers. It is this system of numbers that lies at the basis of the calculus of real variables. This system constitutes a closed group with respect to the fundamental operations of arithmetic and obeys certain laws already familiar to the student from his study of algebra. For any numbers a, b, c of this system, we have from the definitions of those fundamental operations I. For addition: (1) The commutative law: a + 6 = 6 + a. (2) The associative law: a + (6 + c) = (a + 6) -}- c. II. For multiplication: (1) The commutative law: ab = ba. (2) The associative law : a(6c) = (ab) c. (3) The distributive law : (a + 6) c ac + be. (4) Factor law : If ab = 0, then a = or 6 = 0. It is customary to introduce subtraction and division as the in- verse operations of addition and multiplication. From the defini- tion of these inverse operations and the foregoing fundamental laws follow, as purely formal consequences, all of the rules of operation for real numbers, f * See Fine, The Number-System of Algebra, Art. 29. t Ibid., Arts. 10, 18. ART. 4.] REAL NUMBERS, COMPLEX NUMBERS 5 We assume the existence of a one-to-one correspondence between the totality of real numbers and the points on a straight line; that is to say, we assume that to each real number can be assigned a definite point on the line and conversely to every such point there may be assigned one and only one real number.* This assumption makes possible a geometric interpretation of the results of our dis- cussion and the applications of analysis to geometry. 4. Complex numbers. It will be observed that all real numbers arise from the assumption of a single unit, namely 1. By assuming the additional fundamental unit v 1, which we shall represent by i, a very important extension of the number-system thus far discussed can be made. By the use of these two units, 1 and i, we can construct the numbers of the type a -f- ib, where a and 6 are real numbers. It becomes necessary to extend the number-system so as to include numbers of this type if the solution of the equation ax- + bx + c = 0, where b 2 4 ac < 0, is to have any meaning. Such numbers are called complex numbers and form the basis of that special branch of the theory of functions to be considered in this volume. It will be seen that since a and 6 may take all real values, therefore including zero, real numbers are a special case of complex numbers, that is, complex numbers where 6 = 0. In considering the arithmetic of complex numbers, the question arises as to what is to be under- stood by such terms as "equal to," "greater than," etc., and by the fundamental operations of addition, subtraction, etc. More- over, it cannot be assumed in advance that the laws of operation with real numbers may be extended without qualification to this broader field. Since real numbers appear as a special case of com- plex numbers, it is necessary to define these expressions and the fundamental operations in such a manner that the corresponding relations between real numbers shall appear as special cases. These definitions will be considered in the following articles. Complex numbers involving more than two units have been used by mathematicians. For example, Hamilton, a distinguished Eng- lish mathematician, introduced higher complex numbers known as * For references to the mathematical literature where this subject is discussed see: Encyclopedic des Sciences Mathematiques, Tome I, Vol. I, pp. 146-147, or Staude's Analytische Geometrie des Punktes, der geraden Linie, und der Ebene, p. 422 (10). 6 REAL AND COMPLEX NUMBERS [CHAP. I. quaternions. For this purpose, he made use of the unit 1 and the additional units i, j, k, connected by the following relations: = j* = Jt 2 = ijk = -I. No use will be made of quaternions or of other higher complex numbers in this volume, and the subject is mentioned merely to illustrate the possibility of further extensions of the number concept. 5. Geometric representation of complex numbers. The as- sumption which we have made as to the one-to-one correspondence between points on a straight line and the totality of real numbers, makes it possible to give a geometric representation to complex numbers. For this purpose, we introduce a system of rectangular coordinates similar to those used in Cartesian geometry. To represent the number * a + i&, lay off on OX, called the axis of reals, the distance a and on OY, called the axis of imaginaries, the distance 6. Draw through A a line parallel to OF and through B a line parallel to OX. The intersection P of these lines represents the complex number a + ib. The numbers a and 6 may be any real numbers, positive or negative. From these considerations, it follows that there exists a one-to-one correspondence between the points of the plane and the totality of complex numbers. We shall refer to the plane, used in this way, as the complex plane. From the relation between the points of the complex plane and the totality of complex numbers, it follows that the complex numbers constitute a continuous system. By making use of the trigonometric functions, it is possible and frequently convenient to represent complex numbers hi another form. From Fig. 1, we have a = p cos 8, b = p sin. 6. We may therefore write a + ib = p(cos 6 + i sin B). The distance OP = p is called the modulus of the complex num- ber, and the angle B is called the amplitude of the complex number. * The first mathematician to propose a geometric interpretation of the imagi- nary number V-l was Kiihn of Danzig hi 1750-1751. The idea was extended by Argand hi 1806 to include a representation of complex numbers of the form a + b V 1, a representation that was later used by Gauss. The complex plane is frequently referred to as the Argand plane or the Gauss plane. ART. 5.] - ,1 fl GEOMETRIC REPRESENTATION FIG. 1. It will be observed that for any given number a + ib the modulus p is a single-valued function of the real numbers a and b, while the amplitude 6 is a multiple-valued function of these numbers. The number a 2 + b 2 = p 2 is frequently referred to as the norm of the com- plex number a + ib. The value of 6 lying in the interval IT < d = TT is called the chief amplitude. The amplitude is measured positively in a counter-clockwise direction. The modulus is always to be con- sidered as positive, and hence is often referred to as the absolute value of the complex number. We frequently indicate the modu- lus or absolute value of any com- plex number a by placing a vertical line before and after the number, thus | a \, read "the absolute value of ." Other geometric interpretations of complex numbers are possible. We shall have occasion later to point out, for example, how complex numbers may be represented by points on a sphere by showing that there exists a one-to-one correspondence between the points of the complex plane and those on the surface of a sphere. From what has already been said, it will be seen that complex numbers are directed numbers, that is, numbers that have both magnitude and direction. Consequently, we may when convenient think of the complex number a + ib as represented by the plane vector joining the corresponding point of the complex plane with origin. Such physical magnitudes as force, velocity, acceleration, electric intensity, etc., have direction as well as numerical value and may be represented therefore by complex numbers, provided their directions are confined to a plane. The factor i rotates the given number through an angle ^ . Thus ia, as we have seen, indicates that a distance a is to be laid off on a line perpendicular to the axis of reals. In the complex number a = p(cos B + i sin 6), the magnitude of the number is p, while the direction in which this 8 REAL AND COMPLEX NUMBERS [CHAP. I. magnitude is measured is determined by the factor in the paren- thesis. 6. Comparison of complex numbers. The question very natu- rally arises as to how two complex numbers may be compared with each other. Given the two numbers a = a + ib, /3 = c + id. We say that a = /3, when we have the relations a = c, b = d. Expressed in terms of polar coordinates, equality involves the con- dition that the two numbers shall have equal moduli and shall have amplitudes that are either equal or differ by some multiple of 2 IT. It will be observed that the two equal numbers a and /3 are represented by the same point in the complex plane. Since the moduli of complex numbers are real, their magnitudes may be compared one with another in the same manner as any other real numbers. Thus of two complex numbers a and /3, it is possible to say that the modulus of a is greater than or less than the modulus of /3; that is, we may write iiVm 7. Addition and subtraction of complex numbers. We define the sum of two complex numbers a + ib and c + id as the complex number (a + c) + i (b + d), obtained by adding the real parts and the imaginary parts separately. It is not to be assumed without proof that the laws of addition enumerated for real numbers in Art. 3 hold for complex numbers. It may be easily shown, however, that such is the case. For this purpose, suppose we have given any three complex numbers a = a + ib, /3 = c -f- id, y = e + if. That the commutative law holds is shown as follows : We have a + j8 = (a + c) + i (b + d) = (c + a) + i (d + 6) (Art. 3) = + a. To show that the associative law likewise holds we proceed as follows: + 03 + 7) = [a + (c + e}\ + i [b + (d +/)] = [(a + c) + e] + i [(b + d) + /] (Art. 3) = (a + 0) + y. ART. 7.] ADDITION AND SUBTRACTION 9 The addition of complex numbers can be easily performed geo- metrically. In Fig. 2, let a = a + ib and ft = c + id be represented by the points R and S, respectively. Complete the parallelogram having OR and OS as two sides. The point P then represents the sum a + ft; for, drawing through R a parallel to OX and dropping from P the perpendicular PM, we have from the equality of the two triangles. RMP and ONS RM = c, MP = d, and consequently the coordinates of P are (a + c, 6 + d). Hence, the point P represents the complex number (a + c) + i (b + d} = a + ft. ' FIG. 2. The result of geometric addition may be conveniently obtained also as follows: To add ft to a, draw from R (Fig. 2) a line parallel to OS, extending in the same direction and equal to it in length. The terminal point P of the line thus drawn represents the number a + ft. To add several numbers in succession, all that is necessary- is to draw from the point P representing the sum of the first two numbers a line parallel to the line on which the modulus of the third point is measured, and upon the line thus drawn to lay off from P a segment equal to OP and extending in the same direction. The terminal point of this line represents the sum of the first three num- bers. To this result a fourth number may be added in the same way, etc. An important relation between the absolute values of two complex numbers is suggested by the geometric considerations already in- troduced. This relation may be stated as follows: THEOREM I. Given two complex numbers a and ft; we have the following relation between their moduli, namely, From elementary geometry, it is known that any side of a triangle is greater than the difference of the other two sides and less than the sum of those sides. Referring now to Fig. 2, we have OR-RP = OP = OR + RP. (1) 10 REAL AND COMPLEX NUMBERS [CHAP. I. But we have OR = | a |, RP = OS=\ft OP = | a + |. (2) Hence, substituting these values in (1), we have the result given in the theorem. An equality sign is to be used only when the points representing a and ft lie on a straight line passing through the origin. We may state also the following theorem. THEOREM II. For a given complex number a = a + ib, we have + ib\. (1) The proof of this theorem, like that for Theorem I, follows at once from geometrical considerations. As the point a moves around the circle (Fig. 3) | a | and | 6 | change values. Let us find the maxi- mum value of | a | + | 6 |, subject to the condition that +|6|2= |a + i6|2 = r2, (2) where r is the radius of the given circle. By the ordinary methods for computing a maximum, we find that the function in question has a maximum if FIG. 3. Consequently, the maximum value of | a | + | 6 | is "N/2 | a + ib |, from which the relation (1) must hold for the various values of a upon the circle. We define the difference a ft of two complex numbers, a = a + ib, ft =. c + id, as the complex number (a c) + i (b d). Here, as in addition, the laws of operation with real numbers hold for com- plex numbers and follow as a consequence of the laws of addition and the definition of subtraction. The reader can easily verify this statement. Let a and ft be represented by R and S, respectively (Fig. 4). To subtract ft from a geometrically, construct the parallelogram having OR as a diagonal and OS as one side. The point P repre- sents a ft, since the sum of fee and the complex number repre- sented by this point is ft. The same result is obtained by drawing the line RP parallel to SO and equal to it in length. ART. 8.] MULTIPLICATION 11 8. Multiplication of complex numbers. We may define the prod- uct of two complex numbers a, P by the relation a/3 = (a + ib) (a' + ib') = (aa' - 66') + i(ab' + 6a'). If the given numbers are written in the form a = pi(cos 81 + i sin 0i), P = P2(cos 6 2 + i sin 2 ), then the product a/3 becomes a/3 = PI(COS 0i + * sin 0i) p2(cos 2 + i sin 2 ) = Pip2[(cos0icos0 2 sin0isin0 2 ) +z(sin0icos0 2 +cos0isin0 2 )] = Pip2[cos (0i + 2 ) + i sin (0j + W This relation gives us the rule for multiplication, which may be stated in words as follows: The product of two complex numbers is a complex number whose modulus is the product of the moduli of the two given numbers and whose amplitude is the sum of the given amplitudes. The associative, commutative, distributive, and factor laws for multiplication hold for complex numbers as for real numbers. The commutative law for example can be established as follows: Given as before a, /3, we have a/3 = pip2[cos (0! + 2 ) + i sin(0i+ 2 )] - P2 pi[cos (0 2 + 00 + i sin(0 2 + 0j)] = 0a. (Art. 3) In a similar manner the associa- tive, distributive, and factor laws may be established. From the definition of a product, we are able easily to give a geo- metric interpretation of multiplication. Represent the two complex numbers a and /3 by the points P and Q, respectively. Thus far we have been able to carry out the geometric operations introduced without reference to the magnitude of the unit. We must now FIG. 12 REAL AND COMPLEX NUMBERS [CHAP. I. make use of a unit length. Lay off on OX the distance OA, which we take as the unit of length. Connect the point P with A. On OQ as a base construct the triangle OQR similar to the triangle OAP. We have then from the figure OR-.OP: :OQ:1; that is, 0& :*::*:!, or P1P2 = OR. Furthermore, we have by construction /LQOR = ZAOP; hence Z.AOR = 01 + 0* Consequently, from the definition of multiplication it follows that the point R represents geometrically the required product; for, it has the modulus pipz and the amplitude 0i + 2 . The rule of multiplication may be extended to the product of any finite number of complex numbers. Suppose, for example, we have i = PI (cos 0i + i sin 0i), "2 = P2(cos 2 -\- 1 sin 2 ), a n = p n (cos 6 n + i sin B ). We obtain as the continued product of these numbers . . . p n [cos(0i+ + n )+i sin (0i + + n )] When pi = pa = = p n = 1, we have an important theorem, known as De Moivre's theorem, namely: (cos 0i + i sin 0i) (cos 2 + i sin 2 ) . . . (cos n + i sin n ) = cos (0i + 2 + + n ) + i sin (0i + 2 + 0). If we also put 0i = 2 = = n = 0, we have the form of the theorem most frequently used, namely: (cos + t sin 0)" = cos nd + i sin nd. This theorem gives us a method of raising any complex number to an integral power; for, we have [p(cos + i sin 0)] n = p n (cos nd + i sin n0). ART. 8.] MULTIPLICATION 13 The power of a complex number may be found geometrically as follows: Let PI represent the number a = p(cos + i sin 6}. On OPi construct the triangle OPiP 2 similar to OAPi; then P 2 rep- resents the second power of the given complex number. On OP 2 as a base construct another tri- angle OP 2 Ps similar to OAPi or to OPiP 2 . The point P 3 of this triangle represents the cube of the given number. Continuing in this way, we may raise the given number to any required integral power. It may be shown that De Moivre's theorem holds likewise for negative and fractional pow- ers.* As an iHustration, let us consider the n th roots of a complex number a. We have then Pi. - /_*fs \ ~J e , e \ a n = p"{ cos - + i sm - ), \ n nj where p" is always to be considered positive and real. We obtain the remaining n th roots of a by replacing 9 by 6 + 2 kir, k = 1, 2, 3, . . ., n 1. Thus the n roots are I/ /T nl e , 0\ p" cos + i sin- > V w / p" COS t sin \ - , / 3" p"| cos + 2 (n - 1) TT sin 2 (n - 1) TT> That each of these numbers is an n ih root of a is seen at once from the fact that its n th power gives the number a = p(cos 9 + i sin 6) . * See Hobson, Plane Trigonometry, 2<* Ed. Arts. 181 and 186. 14 REAL AND COMPLEX NUMBERS [CHAP. I. If 8 be the chief amplitude of a, that is if IT < = TT, then -/ e e\ p"\ cos f- ism - \ n nj is called the chief or principal value of the root. For example, consider the positive real number a = a. The two square roots of a are Va(cos + i sin 0), Va(cos TT + i sin TT) . The principal value of a? is Va(cos + i sin 0) = Va; for, in this case 6 = n amp a" = 2-0 = falls in the interval TT < = TT. If we consider the number a = a, we have as the two square roots /-/ TT . . . ir\ /-/ 3r . . . 37r\ Va( cos^ + isin-J, Va( cos-^- + zsm )> and the pruicipal value of a? is Va( cos ^ + i sin ^ ) = * Va. \ Z z/ As a further illustration of the use of De Moivre's theorem, let us consider the n roots of unity. Here 6 = 0, and we have as the roots cos + i sin 0, 2r . , 2ir cos -- \- i sin > n w . cos -- h * sm n n 2 (n - 1) TT . . . 2 (n - 1) TT cos -+ism n n If we denote the second of these roots by w, then the n roots may be .written 1, CO, CO 2 , . . . , CO"" 1 . ART. 8.] Since we have MULTIPLICATION 15 cos 6+2 for 0+2 far / . . . 0W J = cos- + i sin- cos- \ n nj\ 2kir . . 2far\ sin I i sin we may write the n roots of any complex number a in the form a", wa", a> 2 a", . . . , w^a", (2) l^ where a" denotes one of the n roots of a, for example the principal value of the root; that is to say, the n roots of any complex number are the products of some one value of the root into the n roots of unity. In certain cases the roots of a complex number may be deter- mined graphically. It is not pos- sible, however, to do this in all cases; for, it is not always possi- ble to make the construction by means of the ruler and compasses. Let us consider the fourth roots of the number a, represented by the point P (Fig. 7) . Denote the mod- ulus of a by p and its chief amplitude by 0. The principal value of the root is then given by FIG. 7. i i/ . . . 6\ a* = pM cos- + i sm jl- To determine this root, it is necessary to construct an angle j and to lay off a distance equal to p*. We can construct the angle j by dividing twice in succession the angle by the methods of elementary geometry. We can find the line segment that represents p* by con- structing the mean proportional between OP and 1 and then con- structing the mean proportional between that result and 1. In this manner the point Q is determined representing the principal value of the fourth root of a. That Q does represent a fourth root of a, may be shown by constructing, as indicated in Fig. 6, the fourth 16 REAL AND COMPLEX NUMBERS [CHAP. I. power of the number represented by Q. There are three other fourth roots of a. From (2) they are seen to be a*o>, a*a> 2 , a*a> 3 , where u denotes a fourth root of unity. To multiply by co, or, or to 3 is to rotate the line OQ in a counter-clockwise direction through an angle of 90, 180, 270, respectively, about as a center. To find geometrically the four points representing the fourth roots of a is equivalent to constructing a regular inscribed polygon of four sides in a circle having OQ as a radius and Q as one of the vertices. Each vertex of this polygon is a fourth root of a, as may be verified by constructing its fourth power. To determine graphically all of the n th roots of any complex num- ber involves the division of the chief amplitude into n equal parts, the laying off of a distance equal to the n th root of the given modu- lus, and the inscribing of a regular polygon in a circle. For the special case of the n th roots of unity, the problem reduces to the construction of a regular polygon inscribed in a circle of unit radius; for, the modulus is 1 and in this case the chief amplitude is zero. As has already been pointed out this construction is not always possible by means of a ruler and compasses. The construction is however possible if and only if we have * n = 2 l p\pz . . . , where I is an in- teger and pi, pz . . . are distinct prime numbers of the form 2 2 ' + 1. For example, it is possible if n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ~. . and impossible if n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, .... 9. Division of complex numbers. Given two complex numbers a = a + ib, = a' + ib f . Since division is the inverse operation of multiplication, we must so define the quotient of a by that the result multiplied by /3 gives a. In accordance with this relation, we define the quotient of a divided by /3 by means of the identity a _ a + ib aa' + bb' . a'b - ab' ft ~ a' + ib' ~ a' 2 + b' 2 ~* * a' 2 + b' 2 ' * See Monographs on Modern Mathematics, edited by J. W. A. Young, p. 379. ABT. 9.] DIVISION 17 Writing a, j3 in the form a = pi(cos 0i + i sin t ), /3 = p^cos 2 + i sin 2 ), the quotient of a by /3 may be written as follows: 3 = - [cos (0i - 2 ) + i sin (0i - 2 )1. P P2 This form of the definition may be expressed in words as follows: The quotient of one complex number by another is a complex number whose amplitude is the amplitude of the dividend minus that of the divisor, and whose modulus is the modulus of the dividend divided by that of the divisor. We have already pointed out in another connection that the amplitude of a complex number is multiple-valued. This, however, does not affect the quotient; for, an increase of the amplitude of the dividend or the divisor by a multiple of 2 IT increases or de- creases likewise the amplitude of the quotient by the same multiple of 2 TT and hence the result remains unchanged. From the definition of division, we have for the reciprocal of a complex number a 1 _ (cos + i sin 0) a p(cos + i sin 0) = -[cos( 0) + isin( 0)] P = - (cos i sin 0) . p We may perform geometrically the operation of division as follows: Let P, Q, represent the two complex numbers a = PI (cos 0i + t sin 0i) and p 1 = P2(cos 2 + i sin 2 ), respectively. Draw the line OM (Fig. 8) making the angle 2 with OP. Construct on OP the triangle ORP similar to OQA , when A = 1 . The point R represents the quotient; for. it has the amplitude 0i 2 and its modulus is > since we have from the two similar triangles, P2 ORP and OAQ, OR : 1 : : pi : P!J, and, hence, = OR. Pz 18 REAL AND COMPLEX NUMBERS [CHAP. I. It will be observed that - has no significance when pa = 0; for, P division by zero is meaningless. The general laws of division for real numbers hold for complex numbers and are a consequence of the definition of division and the laws of operation governing multiplication. We have now defined fhe funda- mental operations of arithmetic with reference to complex num- bers. Moreover, we have seen' that the general laws of opera- tion in the arithmetic of real num- bers may be extended without modification to complex numbers. We are now in a position to in- troduce the complex variable and functions of it. Certain funda- mental notions concerning the functions to be considered will be discussed in the next chapter. EXERCISES 1. Express \- y z in the form A + iB, where a, 0, y are given complex numbers. "7 2. Perform graphically the operations indicated by (o0 y) + y, (0 -=- y) 1, where a=2 + i3, = 1 + i 2, y = 3 - i2. 3. Represent graphically the square roots of 3 + i 2, 2 + i 3, J + * i- 4. Represent geometrically the four values of (1 + v I)*. 5. Locate the points representing the sixth roots of 1 and of 1. 6. Give an illustration of two complex numbers the sum of whose moduli is equal to the modulus of their sum; also two complex numbers the sum of whose moduli is greater than the modulus of their sum. 1. Under what conditions do the relations FIG. 8. hold when a, b are real numbers? Under what conditions do they hold when a, b are replaced by the complex numbers a, 0? "7 8. Prove the associative and distributive laws for multiplication of complex numbers. 9. If a and are complex numbers, where a ^ 0, ^ 0, prove that o0 5^ 0. ART. 9.] EXERCISES 19 10. Interpret geometrically (a0) m = a" 1 ^ 1 ; a m a n = a m + n . 11. A boat is being rowed from the west to the east bank of a stream at the rate of three miles per hour. At the same time it is being carried north by the current at the rate of two miles per hour. Represent the velocity by a point in the complex plane. What represents the speed? 12. A fly-wheel two feet in diameter is revolving counter-clockwise at the rate of 180 revolutions per minute. Express as a complex number of the form a + ib the velocity of a point on the rim of the wheel where the radius vector through that point makes an angle with a fixed initial position. CHAPTER II FUNDAMENTAL DEFINITIONS CONCERNING FUNCTIONS 10. Constants, variables. We shall make the same distinctions between constants and variables as in the realm of real variables. If a complex number assumes but a single value in any discussion, it is called a constant. The numbers thus far considered serve as illustrations. If, on the other hand, a number is allowed in any discussion to assume various complex values, it is called a variable. A complex variable z may be written in the form x + iy, where x and y are real variables. We shall speak of a connected portion of the complex plane as a region or domain. Any point ZQ is said to be an inner point of a region if it can be made the center of a circle of radius different from zero such that all points within this circle are points of the region. If the circle can be taken so small that it contains no points of the region, then ZQ lies exterior to the region. If the circle contains both points of the region and points exterior to it, however small the radius of the circle be taken, then z is called a boundary point of the region. If the boundary is included in the region, then it is spoken of as a closed region, otherwise it is called an open region. Unless the contrary is stated, the term region will be used to desig- nate an open region, that is, a connected portion of the complex plane consisting only of inner points. A given region may be finite or it may be infinite. The inner points of an infinite region may all lie exterior to a given curve in which case the curve is the boundary of the region if its points are boundary points. By a neighborhood of a given point ZQ, we shall understand a small region about ZQ having ZQ as an inner point. For most purposes it will be found convenient to choose a neighborhood for which I z - Zo | < P, that is a circle of radius p about the point ZQ. In referring to the points of a neighborhood of ZQ exclusive of the point ZQ itself, we shall speak of the region as a deleted neighborhood of ZQ. 20 ART. 11.] CLASSIFICATION OF FUNCTIONS 21 11. Definition and classification of functions. The definition of a function has undergone radical changes since it was first intro- duced in connection with real variables. Leibnitz, for example, associated the term with any expression standing for certain lengths connected with curves, such as tangents, radii of curvature, normals, etc. Euler, who wrote the first treatise on the theory of functions, defined a function as an analytic expression in which one or more variables appear. We must set aside such a definition because there are numerous illustrations of related and interdependent vari- ables, both in pure mathematics and in theoretical physics, where as yet no analytic expression has been found. These early defini- tions of a function also assumed that a continuous function can always be represented geometrically by a continuous curve having a definite tangent at each point. This condition involves the require- ment that every continuous function shall have a definite derivative for each value of the variable. Subsequent researches show that this is an erroneous assumption, and that there exist functions defined by analytic expressions that are continuous throughout an interval and that do not possess a derivative at any point of that interval. The study of Fourier's theory of heat led Dirichlet in 1837 to formulate the following definition of a function of a real variable, which is the one commonly accepted by mathematicians at the present time: * // for each value of a variable x, there is determined a definite value or set of values of another variable y, then y is called a function of x for those values of x. This definition does not necessitate the existence of any analytic relation between y and x. It is to be observed that it is not neces- sary that x should have every value in an interval; it may take, for example, only a set of values. What is essential in the definition is that for every value that x does take, there is thereby determined a definite value or definite values of y. An important step in the evo-r lution of the idea of a function was made when Cauchy gave to the. variable complex values, and extended the notion of a definite inte- gral by letting the variable pass from the one limit of integration to the other through a succession of complex values along arbitrary paths. The work of Cauchy and the subsequent work of Riemann and Weierstrass laid the foundation for the development of the gen- eral theory of functions. * See Encyclopedic des Sciences Math., Tome II, Vol. I, Fasc. 1, p. 13. 22 DEFINITIONS CONCERNING FUNCTIONS [CHAP. II. We shall understand the complex variable w to be a function of the complex variable z in a given open or closed region S if for each value of z in this region w has a definite value or set of values. Here, as in the case of functions of a real variable, the function may be denned also with respect to a set of values rather than for all values of z of a given region. Unless otherwise stated, however, it will be under- stood that z takes all values of a given region. Then, if w is a func- tion of z, we may write w = u(x, y) + i v(x, y) = /(z), where u, v are real functions of the two real variables x and y. If w has but one value for each value of z, w is said to be single-valued; if it takes two or more values for some or all of the values of z, then w is called a multiple-valued function of z. A possible criticism of Dirichlet's definition is that it is not suffi- ciently restrictive. Without introducing some additional proper- ties, such as continuity, differentiability, etc., it is impossible upon the basis of this definition of a function to build a theory of analysis permitting the operations and developments similar to those of the calculus of real variables. In later articles we shall discuss certain characteristic properties that the functions to be considered in this volume must possess. Before doing so we shall define certain general classes of functions and discuss some of the fundamental conceptions that are of importance in the consideration of their properties. One of the important classifications of functions is their division into rational functions and irrational functions. By a rational in- tegral function or polynomial is understood a function of the form a n z n + a n -iZ n ~ l + + oo, where oo, ai, . . . , a n are constants and n is a positive integer. If a n 7* 0, the function is said to be of the n* 11 degree. A rational fractional function is the quotient of two rational inte- gral functions having no common factor and hence it is of the form a n z n + a n -iZ n ~ l + + ao where m and n may be equal or different positive integers. If a n ^ 0, |3 m 7* 0, and m = n, then this common value is called the degree of the function; if m 7* n, then the larger of the two is called the degree. ART. 12.] LIMITS 23 All functions which are not rational are classified as irrational functions. Another important classification of functions is into algebraic and transcendental. We say that w is an algebraic function of z when w and z are related by an irreducible equation of the form ) =0, where/ (2),/i(z),/ 2 (2), . . . , f n (z) are rational integral functions of z. We say that this equation is irreducible if it is not possible to write the left-hand member as the product of two polynomials, neither of which is a constant. It will be seen that all rational functions, for example, are algebraic functions. All functions that are not algebraic are called transcendental functions. Such functions include the trigonometric, exponential, and logarithmic functions. 12. Limits. From the study of elementary mathematics, and particularly from the study of the calculus, the student is familiar with the general notion and properties of limits. We shall recall some of the fundamental properties by way of emphasis and extend the notion of a limit to the realm of complex numbers. If we have given, for example, the sequence of numbers 9 1i ^i 1i ^) - 1 2> 31 >*,,! it is at once seen that by taking n sufficiently large the terms of the sequence can be made to ultimately differ from unity by as little as we please. We express this fact by saying that the sequence has the limit 1. Likewise, the sequences 111 J_ *>>- I 2n> ' ' ' *l * I ^2> ... have the limit zero. If we may assign at pleasure to a number values which are numerically as small as we may choose, then the number is said to be arbitrarily small. We shall usually denote such a number by e. We may now define the limit of a sequence more exactly as follows: Suppose we have given an infinite sequence of real numbers {a n \ = 0,1, 02, a s , . . . , a n , . . . . 24 DEFINITIONS CONCERNING FUNCTIONS [CHAP. II. If there exists a definite number a, and, corresponding to an arbi- trarily small positive number e, a positive integer m such that for all values of n > m, we have | a n - a | < c, then a is called the limit of the sequence, and we write L a n = a. n = oo If we have given an infinite sequence of complex numbers, the moduli of these complex numbers form a sequence of real numbers, and so do the moduli of the differences between these complex num- bers and any complex constant. We say that a sequence of complex numbers ot\, az, 03, ... .,,... has the limit a or converges to the limit a, if the moduli of the differ- ences between these complex numbers and a form a sequence having the limit zero; that is, if corresponding to an arbitrarily small posi- tive number c, it is possible to find a positive integer m such that we have | a n a | < e, n > m. (1) We then write L a n = a. n=oo Since the relation given by (1) holds for all integral values of n > m, it likewise holds for a particular set of values of n > m, say for the even values of n > m. In other words, any subsequence selected from the given sequence will have the same limiting value as the given sequence. The foregoing definition of the limit of a sequence may be expressed in terms of a and 6, where a = a + ib, a n = a n + ib n ', for, we have the following theorem. THEOREM I. The necessary and sufficient condition that the se- quence of complex numbers ai, a 2 , as, . . . , a n , . . . converges to a limit a = a + ib is that L a n = a; L b n = 6. (2) n=oo n=oo ART. 12.] LIMITS 25 We have a n a = a n + ib n a ib = (a n a) -f i(b n V), whence | a n - a | ^ | a n - a \ + | b n - b \, (3) and . I a n a | = | a n a \ , \ b n b \ = | a n a \. (4) The condition stated in the theorem is necessary; for, if the given sequence has the limit a, we may write | a n a \ < f for n sufficiently great. Hence, from (4) we have also | a n - a | < e, \b n -b\ < e; that is, L a n = a, L b n = b. n=oo n=o The given condition is also sufficient; for, if the two limits (2) exist, we thus have for n sufficiently great. | o n a | < , I 6 n 6 I < e, and hence from (3) it follows that | a n - a | < 2 e. Therefore, the given sequence has the limit a as the theorem requires. Suppose the variable z takes a set of values dense at a, that is, a set of values such that in every neighborhood of a, however small, there are an infinite number of points representing values of z. We express this relation between z and a by saying that a is a limiting point of the variable z. Under these conditions the variable z may be said to approach its limiting value a; that is, it may so vary that | z a | decreases indefinitely. When z varies in this manner, we write z = a, which is to be read "as z approaches a." In the discus- sions to follow the given set of values of z will usually include every value in the neighborhood of a, and unless otherwise stated this will be understood to be the case. As the variable z takes different values, any single-valued function of z, say /(z), has by definition a definite value for each value z is allowed to take. The values of /(z), corresponding to the values of z in a suitably small deleted neighborhood of a limiting point a, may likewise differ from some number A by amounts whose numerical values are less than an arbitrarily small positive number. We then ' 26 DEFINITIONS CONCERNING FUNCTIONS [CHAP. II. speak of the number A as the limiting value of the function /(z) corresponding to the limit a of z. We may also say that f(z) approaches A as z approaches a; for, under the conditions just stated, no matter how 2 may approach a, /(z) will at the same time approach A. We may now formulate the definition of the limit of a function as follows: If a is a limiting point of z, and if corresponding to an arbitrarily small positive number e there exists a positive numbersuch that for all values of z entering into the discussion for which | z a \ < 6, with the possible exception of z = a, we have !/()- A|<, (5) then f(z) is said to have the limiting value A corresponding to the limit a of z. We indicate the existence and the value of this limit by writing L f(z) = A. (6) z=a The limit of a function does not depend upon the value of the function for the limiting value of the variable, but only upon the values that the function takes in the deleted neighborhood of such a point. Frequently the value of the function at the point is quite different from its limiting value. So far as the mere existence of the limit is concerned, it is not essential that the function be defined for the limiting value of the variable. The general laws of operation with limits as developed in the algebra and used in the calculus of real variables hold equally well for complex variables, as they are developed without reference to any particular domain of numbers. As we have already seen, the existence of the limit of a function involves the condition that the same limiting value of f(z) is obtained whatever be the set of values through which z is permitted to ap- proach the critical value a. As z may be written in the form x + iy, where x and y are independent real variables and a is a number of the form a -\- ib, \i will be seen at once that the limit given in (6) is related to the double simultaneous limit L F(x, y}, discussed in x=a y=6 connection with functions of two jeal variables,* the existence of which requires that the same limiting value be obtained by all pos- sible methods of approach of the variable point (x, y) to limiting position (a, 6). * See Townsend and Goodenough, First Course in Calculus, Arts. 101 and 102. ART. 12.] LIMITS 27 A necessary and sufficient condition for the existence of the limit (6) may be stated as follows: THEOREM II. Given z = x + iy, a = a + ib, /3 = A + iB; the nec- essary and sufficient conditions that /(z) = u + iv approaches $ as z approaches a are that L u(x, y) = A, L v(x, y] = B. (7) x=a x=a V=b y=b To prove that the conditions stated in the theorem are necessary, we have given L /() = ft (8) z=a to show that the two limits (7) exist. From (8), we have |/()-/3| fulfilling the conditions of Theorem III; for, each circle lies within the preceding circles and their radii, which are * 3 41 ' 2 | ABT. 13.] CONTINUITY 33 respectively, have the limiting value zero. Hence, the sequence of circles defines a definite limiting point, which we may designate by a. The number a is then the limit of the sequence a\, a 2 , as, .... 13. Continuity. A function of a complex variable is said to be continuous at a point if the value of the function at that point is equal to the limit of the values assumed by the function in every neighborhood of the point. There are three things involved in con- tinuity; first, the function must be defined at the point in question; second, the function must have a unique limit as the variable ap- proaches the critical value; and third, the value of the limit must be equal to the value of the function at the point. If either one of the two latter conditions fails, the function is said to be discontinuous. If the first condition is not satisfied, then we can not discuss the continuity of the function at the point in question; for, the function does not exist at that point. This definition does not differ in form from the definition given in calculus for the continuity of a function of a real variable, in which we say that a function f(x) of a real variable x is continuous at x = a, if L/Cr)=/(a). (1) z=a This definition requires that the same limiting value /(a) be obtained by every possible approach to the point x = a, that is, from either the right or the left and through any set of values dense at the point a that may be chosen from those values that x may take in the neighborhood of a. It is necessary that we take into consideration all such values of x in the neighborhood of x = a, in determining the existence or non-existence of the limit. In a similar manner, we say that a function of two real variables /(re, y) is continuous at the point (a, 6) with respect to the two variables taken together if we have L f(x, 0. (5) By taking t < =-, we have by combining (4) and (5) m l/(*)-0|>f, \z-zo\<8. But as A is greater than zero, this relation establishes the theorem. L.)-d '. ART. 13.] CONTINUITY 35 The continuity of /(z) = u-\-iv with respect to z may be seen to depend upon that of u(x, y), v(x, y) with respect to x, y, as stated in the following theorem. THEOREM II. The necessary and sufficient condition that /(z) is continuous at z = a is that u(x, y), v(x, y) are both continuous in x, y at the corresponding point (a, b), where a = a + ib. This result follows as a direct consequence of Theorem II of Art. 12 and the definition of continuity. It has already been pointed out that when /(z) is continuous at z = a, we may write I/GO -/()!< , |z-a|<, (6) where e is an arbitrarily small positive number and 8 is another posi- tive number depending for its value upon e and a. In other words, e is first selected as small as we choose and then 6 is so determined that the condition given in (6) is satisfied.' For any fixed value of , the value of 8 may change when some point other than z = a is taken as the limiting point. Moreover, for any particular value of z, various values may be assigned to 5. If z is any point in a given region, denote by 5(z ) the value of 8 corresponding to the previously assigned value of e. Then, for this given e, 5(z) is a function of z, made up of the totality of all the values of 8(z) as z takes the various values in the given region. Consider now the function 8(z) for all values of z in the given region. If for an arbitrarily chosen e > 0, 5(z) has a lower limit 5' different from zero, we say that the function /(z) is uniformly con- tinuous in the given region. From what has been said, it will now be seen that an essential characteristic of uniform continuity is that the relation (6) is satisfied by any definite value 8, where < 8 < 8', regardless of the value of a. It is shown in the theory of functions of a real variable that any function that is continuous throughout a closed interval is uniformly continuous in that interval. The corresponding theorem for com- plex variables may be stated as follows: THEOREM III. If a function /(z) of the complex variable z is con- tinuous in a finite closed region S, then it is uniformly continuous in that region. To prove this proposition, we shall assume the contrary to be true and show that this assumption leads to a contradiction. The 36 DEFINITIONS ('ONCKIININC, FUNCTIONS [CHAP. II. assumption is then that the function /(z) does not satisfy the defini- tion for uniform continuity in the closed region S. Since /(z) is continuous at every point within S or upon its boundary, we know from the foregoing discussion that for a fixed but previously assigned value of e there is associated with each point z of the region a defi- nite number 5(z), which, however, may vary with the point. The function 5(2) is fully defined at all points within S or upon its bound- ary. By the assumption that /(z) is not uniformly continuous, we have the condition that the lower limit of 5(z) is zero. Inclose the region S in a rectangle by drawing lines parallel to the two axes. Divide this rectangle Ri into four equal parts by again drawing lines parallel to the axes. In the part of S lying in at least one of these subdivisions, say R 2 , d(z) must have the lower limit zero. Divide in the same way Rz into four equal parts; in one of these divisions, say R 3 , d(z) has the lower limit zero. Con- tinue this process indefinitely. In the limit the sequence of rectangles Ri, Rz, Rz, . . . defines a definite point z' (Art. 12), which may be a point within or upon the boundary of the region S. In any case, since S is a closed region, z' is a point of S. We may then say that there is, under the assumption as to uniform continuity, at least one point z' of the given region such that in every neighborhood of z' the lower limit of the 6's is zero. The given f unction /(z) is, however, continuous for z = z', and hence for the point z' there exists a 8 different from zero, where 5 = S(z'), such that for any two values zi, Z2 of the variable, for which we have | 2l - Z' | < So, I Z2 - Z' I < S , , $ are real functions of a real variable t and possess contin- uous first derivatives with respect to t. We then have from the definition of a differential dx = <'(0 dt, dy = t'(f) dt. (1) The connection between the real functions (f), \f/(i) and the com- plex variable z is given by the equation z = x + iy- We have therefore dz = D t z dt Replacing #'() dt, $'() dt-by their values as given in (1) we have dz = dx + i dy. ART. 14.] ANALYTIC FUNCTION 45 The higher derivatives and higher differentials follow the same laws as in the calculus of real variables and the same symbols are used to represent them, As we have already pointed out (Art. 11), the general definition of a function does not impose upon the functional correspondence such special properties as continuity, differentiability, etc. To say that /(z) is a function of the complex variable z asserts nothing further than that /(z) depends upon z in such a manner that for each value given to z there is thereby determined a definite value or set of values of the function /(z) . We make a substantial advance when we can ascribe to a function the properties of continuity and differentia- bility. The functions to be considered in this volume possess for the most part both of these properties. If a given single-valued function /(z) has a uniquely determined derivative at the point a and at every point in the neighborhood of a, then z = a is called a regular point of /(z). By some authors, the function is said to be analytic at z = a and by others it is called holomorphic at this point. We shall, however, reserve these terms for other uses. A point in every deleted neighborhood of which there are regular points but which is itself not a regular point is called a singular point of the given function. If every point of a given region S is a regular point of a single- valued function /(z), then /(z) is said to be holomorphic in S. It should be borne in mind throughout this and the succeeding chapters that we have defined a region to be a continuum of inner points; hence, it is understood that a region does not include its boundary points unless so specified. We shall speak of a function /(z) as being an analytic function of 2 if it is holomorphic in at least some region S with the possible exception of certain singular points which do not interrupt the continuity of S. It is always possible then to join any two regular points of S by a continuous curve which lies wholly within S and which does not pass through a singular point. A more pre- cise definition of analytic functions will be given in Chapter VII. From the definition of a function which is holomorphic in a region, we have at once the following general properties. Given two func- tions /(z), <(z), each holomorphic in a region S; then it follows that inS: ! /( 2 ) + 0(2) is holomorphic, 2. /(z) (z) is holomorphic, 46 DIFFERENTIATION AND INTEGRATION [CHAP. III. f( z \ 3. -7 T is holomorphic, except for those values of z for which (z) = 0. (z) 4. If w is a regular point of f(w), and ZQ is a regular point of w = 0(z), where (zo) = w , then ZQ is a regular point of the function f \(z}l considered as a function of z. From these properties, it follows that every rational integral function of 2 is an analytic function, holomorphic in the finite region of the complex plane. Since every rational function is holomorphic, except at most at a finite number of points where the denominator is zero, it also is an analytic function. 15. Line-integrals. It was pointed out in the last article that the definition of the derivative of a function of a complex variable involves a more complicated limit than the corresponding definition in the case of functions of a real variable. A similar generalization is necessary in the discussion of integration, in that we must in gen- eral take into account the path along which the integral is to be taken. In the case of functions of a real variable, the independent variable x can pass continuously from any value Xi to some other value 2 along only one path, namely, by passing through the inter- mediate values along the X-axis. In the case of functions of a com- plex variable, the independent variable z can pass from a value z\ to another value 22 by any number of different paths. Consequently, ' the definition of a definite integral between two values of 2 can have a significance only when we consider the path by which 2 passes from the one value to the other. As the subject is not always considered in elementary text-books on calculus, we shall now define a line-integral and discuss some of the more general properties of such integrals. Among other things, we shall show that the integral of a function of a complex variable taken over a given path may be expressed in terms of line-integrals of functions of the real variables x, y taken over the same path. In the calculus of real variables a definite integral is defined as the limit of a sum; that is b f(x)dx= L i*/(fe)A*. (1) f b /a Stated in words: the portion of the X-axis between a and 6 is divided into n parts, the length A*a; of each division is multiplied by the value of the function at some arbitrary point & in that division and the limit of the sum of these products is taken, as the number of such divi- ART. 15.] LINE-INTEGRALS 47 sions increases without limit while the length of the divisions simulta- neously approaches zero. It is of importance to observe that the n divisions between the limits of integration are taken along the X-axis and each of these divisions is multiplied by a value of the func- tion at a point on the same axis. Suppose instead, these divisions and the points at which the functional values are to be used as mul- tipliers are taken along some curve C, called the path of integration, the function with whose values we are concerned now being a func- tion of the two variables x, y. Let the functional value at a point on the curve in each division be multiplied by Ax, which is the orthog- onal projection upon the X-axis of the division of the curve. The limit of the sum of these products as Ax approaches zero, that is as the number of divisions increases indefinitely, is a line-integral of the given function along the path C. This curve may lie in the XF-plane, or in case we have a function of three real variables, the path of integration may be a curve in space. In the particular applications to be made of integrals in the present volume the path of integration will always be a plane curve. Any rectifiable curve, that is any curve having a definite length, may be taken as the path of integration. However, as there is a certain element of arbitra- riness in the choice of the path of integration, we shall avoid certain complications in the discussions to follow by taking as that path a curve that may be broken up into a finite number of divisions, each of which is either a rectilinear segment parallel to one of the coordi- nate axes or else has the property that it is determined by a function y = (x), where 0(x) and its inverse function x = \f/(y) are single- valued and have first derivatives that are continuous except at most at the end points, at which points they may become infinite. Such" a curve is monotone by segments and for convenience will be desig- nated as an ordinary curve,* whenever a special name is necessary for the sake of clearness. However, in the present volume only ordinary curves will be employed. Let AB be one of the finite number of divisions of which an ordi- nary curve C (Fig. 15) is composed. Let the coordinates of the points A, B be (XQ, y ) and (x n , y n ), respectively. Divide the arc AB into n parts by the insertion of n 1 points p\, p 2 . . . , Pk . . . , p n -i, whose coordinates are (xi, yi), (&, y 2 ), . . (xk, Vk), . (XH-I, y n -i), respectively. * Compare: Pringsheim, Encyklopddie der Math. Wiss., II, A 1, p. 22; also Dodd, Bull, of Univ. of Tex., No. 222, March, 1912. 48 DIFFERENTIATION AND INTEGRATION [CHAP. III. Select at pleasure a point (*, TQA) upon each arc (pk-i, Pk), k = 1, 2, . . . , n. Having given a function F(x, y) which is continuous in x and y together along the curve C, form the sum of the products of the subintervals Xk- 1 a f k X FIG. 15. and the values of the given function F (x, y) at the points (*, T)*). Fi- nally, consider the limit of this sum as the number of subintervals AkX between A and B is increased in- definitely, while at the same time the length of each interval ap- proaches zero, namely, the limit (2) If this limit exists, it is called a line-integral * of F(x, y) along the given curve C between the limits A and B. Such an integral is represented by the symbols f F(x, y) JC dx, F(x, y) dx, C F(x, y} dx. 'JAB In defining a line-integral, we took the limit of a sum of products formed by multiplying F(k, TQ*) by the projection of the arc (pk-i, Pk) upon the X-axis. By taking the projection of this arc upon the F-axis, we may define in an analogous manner the line-integral F(x, y) dy. It will be observed that the ordinary definite integral is merely a special case of a line-integral, namely, where one of the axes of co- ordinates is taken as the path of integration. The existence of the limit defining a line-integral may be made to depend upon that defining an ordinary definite integral. Let us assume that F(x, y) is a, continuous function of the two variables x, y together along the path of integration. Let an arc AB of the curve y = (x) be selected as the path of integration (Fig. 16). For the present, we shall also restrict the discussion to the case where Sometimes called also a curvilinear integral. AST. 15.] LINE-INTEGRALS 49 no line parallel to the F-axis cuts this arc in more than one point. We may replace y by (x) and write F(x,y)=F(x,(x))dx, (4) Ja JAB X FIG. 16. (5) in where a, b are the projections of A, B, respectively, upon the X-axis. Consequently, the line-integral / F(x, y) dx not only exists when Jc F(x, y) is continuous in x, y, but we may write ( F(x, T/) dx = ff(x) dx. JC Ja It will be observed that the integral I F(x, y) dx depends i Jc general upon the curve C as well as upon the function F(x, y} ; for, taking x, y, z as the space coordinates of a point, the integral I f(x) dx is represented by the shaded area (Fig. 16) under the curve z = /(z). This area is the projection upon the XZ -plane of the area upon the cylinder perpendicular to the XF-plane through the path of inte- gration y = 4>(x), and underneath the curve of intersection of this cylinder and the surface z = F(x, y). As the path of integration y = (x) changes, the cylinder changes and of course the projected area may change. In the discussion thus far we have considered only the case where the curve y = <(z) is cut by a line parallel to the F-axis in but a " * See Townsend and Goodenough, First Course in Calculus, p. 177, Art. 80. 50 DIFFERENTIATION AND INTEGRATION [CHAP. III. single point. It may happen that such a line may meet the given arc AB in more than one point, say at the points y\, y 2 , . . . , as shown in Fig. 17. In such a case the given arc should be divided into several portions such that each por- tion satisfies the required condition. As the path of integration is an arc of an ordinary curve, the number of such subdivisions is always finite. In the case shown in the figure, the arc AB may be decomposed x into the arcs AD, DE, EB where each satisfies the necessary condi- FlG - 17 - tion. We may therefore write f F(x,y)dx = f F(x,y)dx + f F(x,y)dx + f F(x,y}dx. (6) JAB J AD JDE JEB If P, Q are two real functions of x, y, we shall understand by the line-integral f I/** * Pdx + Qdy, or I Pdx + Qdy, 'c L (7) the sum of the two line-integrals S*X , y /^'ni Vn J Pdx, J Qdy. From what has been said, it follows that these integrals exist if along C the functions P, Q are continuous in x, y taken together. It is frequently convenient to change the independent variables x, y so as to express the equation of the path of integration in a para- metric form. For example, suppose we have x=tyi(f), y =%(), (8) where ^i(0, >^(0 are continuous functions of the real variable t having continuous single-valued first derivatives. As the point (x, y) varies from A to B along the given path of inte- gration, suppose t varies from o to t n . Corresponding to the divisions (Pk-i, Pk) of the arc AB, we have the increments A* = tk fo-i, k = 1, 2, . . . , n. By the law of the mean, we have then from (8) x k - x k -i = ART. 15.] LINE-INTEGRALS 51 where t h ' lies between t k -\ and t k . Corresponding to t k f there is a point (&, r,k) of the arc (pk-i, />*) Hence, if we have given a continuous function P(x, y), we may write t=i t=i Passing to the limit as n becomes infinite, we have from the definition of a line-integral f P(x, y) dx = f tn P\*i (0, *i(0 J*/(0 <& (9) /^B t/t In a similar manner, we may show that if Q(x, y) is continuous in x, y together, we have f t/^ do) For the general form of the line-integral as given in (7), we have then f t/^- (11) The integrals in the second members of (9), (10), (11) are ordinary definite integrals. From the relations expressed in these equations, the laws of operation with line-integrals may be deduced from those of ordinary definite integrals. The following consequences of these relations are to be especially noted. 1. The law for the change of variable in ordinary definite integrals applies likewise to the more general case of line-integrals. 2. The integrals f Pdx + Qdy, f Pdx JAB JBA have the same numerical value, but are opposite in sign. 3. If ZQ is any point upon the path of integration AB, then f Pdx + Qdy = f Pdx-\-Qdy+ f Pdx + Qdy. JAB JAZ O JZ<,B The function to be integrated may involve a parameter in addition to the variables x, y. It is sometimes desirable to be able to differ- 52 DIFFERENTIATION AND INTEGRATION [CHAP. III. entiate such an integral with respect to the parameter. Suppose we have given the line-integral / P(x,y,d)dx, JAB where P(x, y, a) is continuous in x, y, a taken together and where the 3D path of integration AB is independent of o. Suppose also that OtZ exists and is likewise continuous in x, y, a. We may then show that 4. -I f P(*, y ,a)d*. (12) da JAB JAB oa This result follows as a consequence of (9). We have J- f P(x, y,a)dx = - r>f*i(ft *,(), a} */(*) *. (13) da JAB aa Jt* dP Since -- exists and is continuous in (t, a), we have * da . (14) This last integral is by (9) equal to f /4 aP(a ?> a ) Hence, we have from (13) and (14) the required relation as stated in (12). The path of integration may be a closed curve, that is, it may be the boundary of a given region, in which case the limits of integration are represented by the same point of the plane. There is still a choice of direction in which the integral is to be taken. We say that it is taken in a positive sense with respect to the region bounded, if it is so taken that this region lies always to the left of the observer as he proceeds along the curve in the direction in which the integral is taken. Often the boundary of a region consists of two or more closed curves. For example, the region S in Fig. 18 is bounded by the curve M and the circles 1, 2, 3. We may speak of M as the outer portion of the boundary and of the circles 1, 2, 3 as inner portions of the boundary. A region is said to be simply connected if every closed curve in it forms by itself a complete boundary of a portion of the given region. A region that does not satisfy the definition of * See Goursat-Hedrick, Mathematical Analysis, Art. 97. ART. 15.] LINE-INTEGRALS 53 a simply connected region is called a multiply connected region. The region S shown in Fig. 18 is a multiply connected region, since it is possible to have a closed curve surrounding one of the circles that does not of itself form a complete boundary of a portion of the region O FIG. 18. -X FIG. 19. inclosed. In a simply connected region, a curve between two points may always be changed by continuous deformation into any other curve between those points, both curves lying completely in the re- gion considered; this is not true for a multiply connected region. A finite, multiply connected region may be changed into a simply connected region by drawing from each inner portion of the bound- ary a line connecting it with the outer portion. A line joining two points of the boundary is called a cross-cut, and we shall so choose the line that it will neither intersect itself nor other cross-cuts. For example, the multiply connected region S shown in Fig. 18 may be changed into a simply connected region by connecting the circles 1. 2, 3 with the boundary curve M by cross-cuts as indicated in Fig. 19. It serves the purpose equally well to connect each of the circles with some one other circle by means of cross-cuts and one of them with the outer boundary M . Thus in Fig. 19 we might have joined circle 2 to circle 3 and to circle 1 and then have joined any one of the three circles to the contour M . The boundary points connected by a cross-cut may be distinct, as in the case of the cross-cuts joining the separate curves constituting a portion of the boundary in Fig. 19 or the cross-cut a, b in Fig. 20; or we may have the two boundary points coincident as in the case of the cross-cut drawn from the point c and returning to the same point, in which case the cross-cut is a closed curve. 54 DIFFERENTIATION AND INTEGRATION [CHAP. III. The notion of a path of integration may now be extended so as to include the boundary of a multiply connected region; for, having made the region simply connected by the introduction of cross-cuts, the integral may be taken along the contour of this simply connected region including the cross-cuts. For example, in Fig. 19 the arrows indicate the positive direction of the integral with respect to the region S'. It will be noted that in any such case the integral is taken twice along each of the '-X cross-cuts, once in either direction. This portion of the integral van- ishes in accordance with the second property of line-integrals already stated. We may then say that the integral taken over the contour of a multiply connected region, that is, the sum of the integrals taken over the several closed curves constituting the boundary, with the proper signs attached, is the same as the integral taken over the boundary of the simply connected region formed by inserting cross-cuts in the given region.* 16. Green's theorem. One of the important theorems associ- ated with line-integrals gives, under certain conditions, a relation between such integrals and ordinary double integrals. This theorem, known as Green's theorem, may be stated for functions of two real variables as follows : THEOREM I. In a given finite region S let C be the complete boundary of any portion of the plane such that C lies within S and incloses only points of S. If in the given region P(x, y) and Q(x, y) are continuous real functions of x and y together, having the continuous partial deriva- .. dQ dP ., tives > -T . then dx dy' where the double integral is to be taken over the region bounded by C. * Cf. Osgood, Lehrbuch der Funktionentheorie, 2d Ed., Vol. I, Chap. IV, Art. 4, Chap. V, Art. 7. ABT. 16.] GREEN'S THEOREM 55 Let us consider the double integral 'dP dxdy, taken over the region bounded by the curve C. We shall first con- sider this curve to be a single closed curve such that any parallel to the F-axis cuts it in at most two points as indicated in Fig. 21. Let A and B be the points of C at which x has a minimum and a maxi- mum, respectively. A parallel to the F-axis lying between A a and Bb cuts the curve C in two points whose ordinates may be denoted by 2/1, 2/2, respectively. dP O Because of the continuity of -r- , dy the following integrals exist, and we may write * FIG. 21. Performing the integration in the second member with respect to y, we get from this equation the following relation: ff^dxdy = ]P(x, 2/2) - P(x, yjldx. (2) However, the two integrals / P(x, 2/2) dx, \ P(x, 2/1) dx *J a *J a are line-integrals taken along the paths Ay 2 B, AyiB, respectively. Hence, instead of the integral in the second member of (2) we may write a single integral taken in a negative direction around the curve C, or what is the "same, we may write the negative of that integral taken in a positive direction around C, and thus have (3) *. See Hobson, Theory of Functions of a Real Variable, Arts. 314 and 315. 56 DIFFERENTIATION AND INTEGRATION [CHAP. III. In a similar manner, we can deduce the relation By subtracting (3) from (4) we have as the theorem requires. To extend (3) and (4) to any finite region bounded by an ordinary curve C, all that is needed is to divide the region into subregions, each of which satisfies the condition that a line parallel to the F-axis, in case of (3), or to the X-axis, in case of (4), cuts the boundary curve in not more than two points, or in a segment as p, q, Fig. 22. This is always possible as indicated in Fig. 22, since the given contour by hypothesis has only a finite number of maxima and minima with respect to x and with respect to y. If the boundary C consists of several closed curves, the region is multiply connected. By . the proper introduction of cross-cuts this region may be made simply connected, and the foregoing argu- ment then holds. As we have seen, however, the value of the integral along C becomes in this case the sum of the integrals taken in the proper direction along the several closed curves composing C. We shall now consider the following theorem, which is important in subsequent discussions. THEOREM II. In a given finite region S let C be the complete bound- ary of any portion of the plane such that C lies within S and incloses only points of S. If in the given region P(x, y), Q(x, y) are continuous real functions of x and y together, having the continuous partial deriva- dQ FIG. 22. tinea dx dy , then the necessary and sufficient condition that the integral Pdx + Qdy (5) ABT. 16.] GREEN'S THEOREM 57 vanishes for every such curve C is that W_P dx~~dy~ for all points of S. That this condition is necessary may be established as follows. We have given the condition that the line-integral (5) vanishes to show that the condition (6) follows as a consequence. The function f)Q dP T: -- is continuous in S. If this function is not identically zero for all values of x, y in S, then it is possible to find a subregion R j/") r)P sufficiently small such that is of the same sjgn for all values ox ay of x, y in R. From Green's theorem, we have r)/~) /IP If in R the function - --- is always of the same sign for all values dx dy of x, y, then the double integral in the second member of (7) can not vanish and consequently the line-integral I P dx + Q dy can not 9/C equal zero when taken around the contour of R. Hence, in order that the integral (5) taken around every complete boundary C in S shall vanish, the condition (6) must be satisfied identically for all values of x, y in S. That the condition stated in the theorem is also sufficient follows at once from equation (7) ; for, if (6) holds for all values of x, y in S, then the integral (5) taken along any complete boundary C in S vanishes as the theorem requires. We are now in position to establish the following proposition. THEOREM III. In a given finite simply connected region S, let L be any ordinary curve joining two points of S and lying within S. If in the given region P(x, y), Q(x, y) are continuous real functions with dO respect to x and y together, having the continuous partial derivatives, ox j r> -TT-, then the necessary and sufficient condition that the line-integral I P dx-\- Q dy is independent of the path L is that dx dy for all points of S. 58 DIFFERENTIATION AND INTEGRATION [CHAP. III. This theorem follows directly from the conclusions of Theorem II. Let (XQ, y ), (xi, r/i) be any two points in the region S (Fig. 23). Let Li, Lj be any two non-intersecting ordinary curves joining these points and lying wholly in S. The lines LI, L 2 taken together con- stitute a closed curve C lying in S; dQ = dP = dy S, then by Theorem II the integral / UK. 11. I L* , ... U<*,). ox Jx lt yi In a similar manner, we may show that Since Xi, y\ is any point of S, we may write dF dF , . K =P(x ' y} > -dy = Q(x ^> for all values of x, y in S. * See Goursat-Hedrick, Mathematical Analysis, p. 151. 60 DIFFERENTIATION AND INTEGRATION [CHAP. III. 17. Integral of /(). We shall now consider the integral of a function of a complex variable. Let /(z) be a continuous single- valued function of z along a given ordinary curve C joining the two points a, |3. Between the points a = z a and /3 = z n insert n I division points of the curve, z\ y 22, . . . , z n -i. Form the sum of the products /($ A) A*z, where A*2 is the difference z k z k -i and /(ft.) is the value of the given function /(z) at some point f & on the curve C between z*_i and zjt. The integral of /(z) along C between the limits a and /3 is defined as the limit of this sum as the number of division points between a and /3 is increased indefinitely and each difference I z* z k -i | approaches zero; that is, f /(z) dz = L 2/(ft) A*z. (1) *J C n = oo j_j The existence of this integral can be established either directly, as in the case of real variables, or by making it depend upon the existence of the line-integrals already discussed. We shall choose the latter method. For this purpose we write /(z) = u(x, y) + iv(x, y}, where u, v are real functions of the real variables x, y, and hence we have by putting f ^"= &+ itit,' t=i Xk-i) - v (&, - x k -i) + M(&, Since /(z) is continuous in z along C, it follows from Theorem II, Art. 13, that both u(x, y), v(x, y) are continuous in x, y together along the same curve. Moreover, as Az = 0, we have Ax = 0, Ay = 0. Hence, upon passing to the limit, we obtain I /() dz = I (udx v dy) +i I (v dx + u dy); Jc Jc Jc (2) for, from the discussion of line-integrals it follows that the two inte- grals in the second member of this equation both exist because of ART. 17.] INTEGRAL OF /() 61 the continuity of u(x, y), v(x, y), and hence the integral / f(z) dz Jc must also exist. In the case of simple functions the integral between two given points on a given curve may be evaluated by direct application of the definition (1). Ex. 1. Evaluate the integral I dz. J a This integral is independent of the path over which it is taken; for, by defi- nition, we have ft A dz = L 2-,(Zk Zfc-l) = L (Zi Z + Z 2 - Zi+ Zj Z2 H + Zn - Zn-l) J a n = oo J~ n=oo 4 = 1 = L (z n Zo) = a, n=o since ZQ = a, z n = ^, no matter what the intermediate points may be. The geometric interpretation (Fig. 24) of this result will enable the student to understand more clearly the nature of a definite integral of a function of a complex variable. Such an integral was defined as the limit of the sum^/(f*)( z * Zk-i)', that is, we consider the limit of a sum of prod- ucts, each of which is the value of the given function at some point f/t on the curve multiplied into the directed chord z k z k -\. In this particular case, the . FIG. 24. value of /(ft) is always unity. Adding Zi ZQ to 22 Zi, we have geometrically the directed chord z 2 ZQ. Adding to this result the directed chordz 3 22, we have the directed chord z 3 z , etc. Finally, we have z n z , which is identically /3 a as we have seen. This result is very dif- fff ferent from taking the integral I | dz \. In this case, we add merely t/a n the chords without reference to direction; that is, we have L ^ | A&2 |. n=oo k = l In the limit we should have in this (&se not /3 a but the length L of the path * of integration from a to /3. * See Townsend and Goodenough, First Course in Calculus, Art. 88. 62 DIFFERENTIATION AND INTEGRATION [CHAP. III. n rff Ex. 2. Evaluate I z dz. This integral is independent of the path over which it is taken; for, jthe/linait defining the integral exists when ft is any point in the interval Zk-i, ft^Zk) We may, therefore, select for ft any convenient point in this interval. If we tslce it to be zt or Zk-\, we have respectively /0 " f& n I zdz= L V z k (zk - Zk-i), or I zdz = L 2 2 *-i( z * - z t _i). Ua = j. = i Ja n=oo j. = i Hence, we have by taking one-half of the sum of these two results fp I zdz J a L -oo j V*'! **W I *%S *! I *"w n-oo ^ L ( Zn 2 - 2o 2 ) ( 2 - a 2 ) no matter what the path is, since as in Ex. 1, 20 = a, z n = /3. In both of these examples the result obtained is the same as that obtained by substituting the limits of integration in a function of which the integrand is the derivative and taking the difference. The definite integral of a function of a complex variable has been defined as a limit. From this definition and the laws of operation with limits, the general properties of such integrals can be deduced ; or, they may be shown to hold as a consequence of the corresponding properties of line-integrals in the calculus of real variables. The proof in many cases is so evident that it is left to the reader to supply. If a, /3 be two points on the path of integration C, we then have among other properties: B rB cf(z)dz = c I f(z)dz. ~ - Jo. 3. P[/(*) +(*)] dz = Pf(z) dz f /o t/a t/a This last property can be readily extended to the case involving any finite number of functions. It can not, however, be extended ART. 17.] INTEGRAL OF /() 63 to the case involving an infinite number of functions without intro- ducing some condition as to the character of the convergence of the series thus introduced. fP /"> f P 4. / f(z) dz = / f(z) dz + / /O) dz, !/ t/a /B! tuyere 21 Ztes upon )-/(<*) (3) Z a which holds for all values of z in T' such that | z a \ < <5. Draw about the point a a circle of radius 5. From some point on in the sequence of regions defining the point a, all of the regions lie within this circle, and consequently if a is taken as the point Zi the values of z upon the perimeter of any one of these regions are such that (1) is satisfied. This conclusion contradicts the assumption that none of the subregions of the sequence satisfies the required condition. From this contradiction the given proposition follows. By aid of this lemma, we may now demonstrate the Cauchy- Goursat theorem, which may be state* 1 as follows: THEOREM I. Let f(z) be holomorphic in a given finite region S and let C be the complete boundary of any portion S' of S such that C lies wholly in S and incloses only points of S; then f Jc /(z) dz = 0. c The boundary C may consist of a single closed ordinary curve or a combination of such curves. We shall first consider the case where C is a single closed ordinary curve and the inclosed region S' is simply connected. Let S' be divided into squares and partial squares satis- fying condition (1) of the foregoing lemma. Let n denote the number of squares Si and m the number of partial squares Ri. If the integral is taken in a positive direction around the perimeter of the various subregions Si, Ri, it will be seen that each side of these regions that is not a portion of C is taken twice as a path of integration, the two in- tegrals being taken however in opposite direction. Considering the sum of the integrals about the perimeters of all of the regions Si, R t , we may therefore write by aid of 4, Art. 17, n n / m f\ J c f(z) dz = ] J f(z) & + 2 J /(*) dz > W where 7*, X, denote the boundaries of Si, Ri, respectively. From the lemma, we have within or upon the boundary of each Si, Ri a point Zi such that f(z)-f(zi) ,_, ^ (5) ART. 19.] CAUCHY-GOURSAT THEOREM This relation can be written in the form where tji is a function of 2 such that - h.- 1 <, 69 (6) (7) when varies along the contour of Si or R i. We shall now consider the integral around the perimeter of one of the squares St. We have from (6), since Zi is a constant for this integration, r-Mfd*-{fto-*ftoi]f*+J'to f zdz+ frn(z-Zi)dz. (8) t/7; l/-yj t/ 7j V/ 7( From Exs. 1, 2, Art. 17, we know that r& rft I dz = p- a, / z dz = %(& - a 2 ). t/ a ty at In the particular case under consideration, as the path of integra- tion is a closed curve, a and /3 are the same point, and hence both of these integrals vanish. From equation (8) we then have U/(z) dz = I 7)i(z - z^ dz .'i *J "Yi (9) Let the length of one side of the square Si be c,-. The diagonal of the square is then c V2. Hence, we have | z - Zi | ^ Ci V2. Making use of this relation and of that given in (7), we may now write by 7, Art. 17, f /(z) dz )] dz Jc Jc' Jc Jc - 0(z Zo)]f dz. (15) See Art. 22 for method of change of variable. 72 DIFFERENTIATION AND INTEGRATION [CHAP. III. But the integrand in the last of these integrals may be written in the form A*) -#[*-(* -*)U-*)] - A*)- A* -(* -*)U- )]+(! - )/[* -(*-*)(!- *)]. Since (1 8) can be taken arbitrarily small, the right-hand member of this equation can be made as small as choose ; that is, it can be taken less than an arbitrarily small positive member ei. We then have I ff(z)dz- ff(t)dt < l f\dz\ = fl 'L, \Jc Jc r Jc where L is the length of the curve C. But as L is finite, the product i L is arbitrarily small. The integral I f(t) dt is zero by Theorem Jc' I. Hence we have f /(*) dz = 0. Jc THEOREM III. Given a finite simply connected region S in which the integral I f(z) dz vanishes when taken along any closed curve C lying wholly within S. Any two paths of integration having the same extremities and lying wholly within S give the same value of the integral ff(z)dz. Let aw/3, ari|8 be any two curves (Fig. 27) connecting the points a, j8 and lying wholly within the region S. We shall assume that these two curves do not intersect each other. The curve aw/3 fol- lowed by the curve /3na constitute a closed curve C. Then by hypo- thesis, we have FIG. 27. ff(z)dz= f . t/C Jamft f A*)r, \t-z<>\>r. By use of 7, Art. 17, we may now write s. Azf(t) dt c (t - zo) 2 (t - Zo - Az) ML r 8 Az where M is the maximum value of | f(t) \ along C and L is the length of the curve C. Hence as Az approaches zero, we have zero as the limit of this integral. ART. 20.] CAUCHY'S INTEGRAL 79 Consequently, passing to the limit as Az = 0, we have from (11) L r /(oc(t or dt ^ i r j(t) 2uij c (e- In the same way, we may show that 2! The existence of these integrals enables us to affirm the existence of the higher derivatives of f(z). Consequently, the derivative f'(z) is continuous and holomorphic as the theorem states. A similar statement may now be made with reference to each of the higher derivatives. Since /'(z) is holomorphic in S if f(z) is holomorphic in S, it follows that both f(z) and f'(z) are continuous in any closed region S' lying within S and hence by Theorem III, Art. 13, both f(z) and f(z) are uniformly continuous in S'. The fact that the continuity of the derivative f'(z) follows from its existence renders the theory of analytic functions of a complex variable in many respects simpler than the theory of functions of a real vari- able; for, a derivative of a function of a real variable may exist at every point in an interval and yet not be continuous throughout the interval. In the next article, it will be shown that the conti- nuity of the partial derivatives of the first order of u, v, where f(z) = u(x, y) + iv(x, y), follow from the continuity of f'(z). When that result has been established, we shall be able to apply to subsequent discussions the results of Green's theorem. 80 DIFFERENTIATION AND INTEGRATION [CHAP. III. THEOREM III. Let f(t) be a continuous function of the complex variable t along an ordinary curve C, which may be closed or not. The integral defines a function of z which is holomorphic for all values of z different from t. It is at once evident that the given integral defines a function of z. We may put Then, by the reasoning employed in the demonstration of Theorem II, we obtain /' /) dt . that is, F(z) has a derivative for each value of z different from t. Hence, the function ^(z) is holomorphic for all such values of z and if F(z) has any singular points, they must be points on the curve C. By aid of the foregoing theorems, we can now prove the converse of the Cauchy-Goursat theorem. This theorem is due to Morera * and may be stated as follows: THEOREM IV. // /(z) is continuous in a given region S and if the integral I /(z) dz is zero when taken around the complete boundary C of any portion of S, such that C lies wholly within S and incloses only points of S, then /(z) is holomorphic in S. If the given region S is multiply connected, let it be made simply connected by the introduction of cross-cuts. Then every closed curve C lying within the new region S' is a complete boundary and by hypothesis the integral taken along such a curve is zero. We shall show that in this simply connected region /(z) is holomorphic and hence holomorphic in S, even though this given region is multiply connected. Let a be a fixed point of S and ZQ any other point of the same region. Denote by ZQ + Az any point of S' in the neighborhood of ZQ. Because CM dz = o, Jc * See Reale Institute Lombardo di scienze e letter e, Rendiconti, 2 series, Vol. 19 (1886), p. 304. ART. 20.] CAUCHY'S INTEGRAL 81 it follows from Theorem III, Art. 19, that the value of the integral between any two points is independent of the path. Since we are at liberty, therefore, to select arbitrarily the path of integration between a and ZQ + Az, without affecting the value of the integral fzv+te J a /(z) dz, we take a path pass- ing through z and rectilinear be- tween ZQ and ZQ + Az, as indicated in Fig. 32. The value of the integral / /(z) dz, where f is a point upon this Ja path, is a function of f , and we may write = P J a dz. FIG. 32. Hence, we have for f = zo and f =z + Az the following values of F(f) : r*<> F(ZO) = / /(*), t/a F(zo + Az) = P **/() dz, t/ whence AF(zo) = F(ZO + Az) - F(z ) = P */(*) dz - P/C^) dz = P + ^/(z)dz. (12) 7zo The given function /(z) is continuous in S, and, therefore, we have for an arbitrarily small positive number c, another positive number 5 such that I /CO /( z o) | < c, for | z z 1 = | Az | < 5. This relation may be written /(*) where | TJ | < e, for [ A z | < 5. Putting /(ZG) + T) in place of /(z) we obtain from (12) /zo+Az /%+Az AF = / /(b) dz + / t/Zo ^2o dz. (13) 82 DIFFERENTIATION AND INTEGRATION [CHAP. III. Dividing by Az, we have * which is equal to Moreover, we have hi- Aw _ 1 Aw> LI = LI LI ( z ) A*=O Az A*=O Ax A=O ^ Ay Since the first limit exists by hypothesis, the second and third limits must also exist. We have then dw _ dw_ _ 1^ dw_ . . dz dx i dy 84 DIFFERENTIATION AND INTEGRATION [CHAP. III. However, we have dw d { ,...,,, du . .dv x. N ^-W**)+M*rir)] - + <, dw d r , . . . f N1 du . .dv Substituting these values in (3), we obtain du . .dv .du . dv ,_s Equating the real parts and the imaginary parts in this equation, we have du _dv du _ dv ,_, dx~dy' dy~ ~dx' By Theorem II, Art. 20, the derivative -7- is continuous. The con- tinuity of the partial derivatives in (7) follows therefore from equa- tions (3), (4), and (5) and Theorem II, Art. 13. We may now show the conditions of the theorem to be sufficient as follows. From equation (2), Art. 17, we have I f(z) dz = I u dx v dy + i I v dx + u dy, (8) Jc Jc Jc where C may be regarded as any path of integration within the given region S. As u, v are continuous, both of the integrals I u dx v dy, I v dx + u dy (9) \J C *J exist. By hypothesis, the equations (1) are satisfied by u, v. Hence, by Theorem II, Art. 16, both of the integrals in (9) are zero, and we have from (8) r f(z) dz = I for every path of integration C forming a complete boundary of any portion of S such that C lies entirely within S and incloses only points of S. Consequently, by Theorem IV, Art. 20, f(z) is holo- morphic in the given region S as the theorem requires. From equations (3), (4), and (5), we have dw_du .dv __dv_ .du dz ~ dx^' l dx~ dy~ l dy' ART. 21.] CAUCHY-RIEMANN EQUATIONS 85 This relation affords a convenient method of computing the deriv- ative of w, when w = /(z) is expressed in terms of x and y. Ex. Given w = x 3 + 3 x*yi 3xy 2 yH. Show that w is holomorphic everywhere in the finite region, and compute D z w. We have u = x 3 - 3 xy*, v = 3 x*y - y 3 , -8* -3* |-6*,, The conditions of the theorem are fulfilled and the function is therefore holo- morphic for all finite values of x and y. For the derivative D z w we have In this particular case w can be readily expressed directly as a function of z, namely w = z 3 . We have then by the laws of differentiation D g w = 3 z 2 , a result which is identical with that already obtained. In the calculus of real variables and in trigonometry, we became acquainted with certain pairs of functions that we called inverse functions. For example, y = sin x and x = arc sin y, y = x z and x = Vy are illustrations of inverse functions. If we have given any function w = /(z), z may be regarded as a function of w, expressed implicitly by this equation in w and z. It is desirable, however, to know the conditions under which z, considered as a function of w, is holomorphic in a definite region when /(z) is holomorphic in a given region. The desired conditions may be stated as follows : THEOREM II. Let w = /(z) be holomorphic in a given region T, which is defined by the inequality \z ZQ\ c* z JK 2 _ = - J* z' dz' = 0. (Th. 5^, Art. 20.) It will be observed that while the first integral must be taken in a clockwise direction, as the region (z) denote any function having the derivative /(z). Such a function is called a primitive function of /(z) . We write as in the cal- culus of real variables and speak of I /(z) dz as the indefinite integral. The primitive function <(z) can differ from F(z) at most by a constant. For, 0(z) is holomorphic in S, since it has a derivative /(z), and hence F(z) <(z) is also holomorphic in S. We have, since both <(z) and F(z) are primitive functions of /(z), (5) /^.\ (6) for all values of z in (z) differs from F(z) by a constant for all values of z along the path of integration between any two points a, /3 of S, we may write 0(^)= f Z f(z)dz + c. i/a For z = a, this relation becomes (a); (9) that is, the law for the evaluation of a definite integral in the calculus of real variables may be extended without modification to functions which are holomorphic in a given finite region. Ex. Given the function f(z) = z m , m 7* 1; find the value of the integral /z+ 1 /(*) dz = - + c. m -f~ 1 Hence, from (1) we obtain 24. Laplace's differential equation. We have the following theorem. THEOREM I. In a given finite region S, let the complex function /(z) = u + iv be holomorphic; then the functions u(x, y), v(x, y) satisfy the partial differential equation ART. 24.] LAPLACE'S EQUATION 93 This differential equation is known as Laplace's differential equa- tion and is of prime importance in theoretical physics. Since f(z) is holomorphic in S, the derivative/' (z) and also the higher derivatives exist and are holomorphic in the same region. It follows that the partial derivatives of u, v with respect to x and y exist and are continuous. This statement holds not only for the partial derivatives of the first order but likewise for those of the second and higher orders. From the Cauchy-Riemann differential equations, du _ dv du _ _ dv_ . . dx dy' dy dx 1 we obtain by differentiation ~dx 2 = dxdy' dy 2 = ~ dydx' As the partial derivatives of the second order are continuous in x, y, together, we have * d 2 v d 2 v dx dy dy dx Hence, by addition of the equations $Q, we have (4) ^ _|_ r_ = o. (5) In a similar manner we can show that dx 2 + dy 2 = ' We shall now consider the converse proposition, namely: THEOREM II. // in a given finite region S, a function u(x, y) has continuous partial derivatives of the first and second order and satisfies Laplace's differential equation, then there exists a function v(x, y} de- termined except as to an additive constant, such that the complex function u + iv = f(z) is holomorphic in S. We have given the condition that u satisfies the differential equa- tion dx 2 dy 2 * See Townsend and Goodenough, First Course in Calculus, Art. 104. 94 DIFFERENTIATION AND INTEGRATION [CHAP. III. We now define v by the relation du , , du , /!. tf v= / - t/io. Vt u r (ftt dy " ' dx ~* This integral exists because the integrand is continuous, since - are continuous. Moreover, the integral is independent of the y path by virtue of Theorem III, Art. 16, because d I du\_ d fdu\ t dy\~dy)~ dx\dx)' that is, dy 2 dx 2 Differentiating v partially with respect to x, we have from (6) by aid of Theorem IV, Art. 16, dv_ = du m dx dy Similarly, differentiating with respect to y, we get dv du , Q s dy'dx' Equations (7), (8) are however none other than the Cauchy-Riemann differential equations, and hence u + iv = f(z) is holomorphic in S. Ex. 1. Given u = x 3 3 xy*. Show that there exists a function v(x, y} such that w = u + iv is holomorphic in the finite region. Determine the function v(x, y). The given function satisfies Laplace's equation; for, we have du du dx dy hence The required value of y is to be determined from equation (6). As pointed out in the discussion of the foregoing theorem, this integral is independent of the path. It can be conveniently evaluated by making the path rectilinear passing ART. 24.] LAPLACE'S EQUATION 95 from (x , yo) to (x, yo), thence to (x, y). From (x , yo) to (a;, y ) we have y y , dy = 0, while x varies from x to x. From (x, y ) to (x, y), we have dx = and y varying from y to y. Hence, from equation (6) we have t; = | 6 xy a dx + f (3 x 2 - 3 y 2 ) dy + C ^ Zo ^ Z/o - 3 x 2 y + 3 x 2 ?/ - y 3 - 3 X 2 y + yo 3 + C = 3 x 2 y - y 3 - (3 x 2 y - yo 3 ) + C. Putting C 3 x 2 yo + yo 3 = c, we have v = 3 x*y y 3 + c, whence /(z) = + w = x 3 - 3xy 2 + i\ (3x 2 y - t/s) + c}. = (x + iy} 3 + ic = z 3 + ic. From the form of /(), it will be at once seen that it is holomorphic in any finite region. Ex. 2. Given u = log (x 2 + y 2 ) . Find a function v(x, y) such that u + * is an analytic function. We have du _ x du _ y dx ~ (x 2 + y 2 ) ' ^ " (x 2 + y 2 ) ' d 2 t/ y 2 x a 2 ^ x 2 - 7/ 2 ax 2 (x 2 +?/ 2 ) 2 ' dy* ( These results substituted in Laplace's equation show that u satisfies that equa- tion. As in Ex. 1, it is convenient to take the path as rectilinear through the intermediate point (x, y ). From equation (6), we then have = arc tan (- arc tan + arc tan - arc tan + C yo yo x x = arc tan - - + arc tan + C. x 2 yo If we now put C ^ + arc tan = c, we have v = arc tan - + c, E and hence get , y u + iv = log (x 2 + y 2 )* + i arc tan - + ic. x The function w = f(z} = u + iv is holomorphic for all values of z in any finite region not including the origin, since for such values of z = x + iy the functions u and t; satisfy the conditions of Theorem II. 96 DIFFERENTIATION AND INTEGRATION [CHAP. III. 25. Applications to physics. A variety of problems in mathe- matical physics are associated with the solution of Laplace's equation. According to Newton's law two bodies in space attract each other directly as the product of their masses and inversely as the square of the distance between them. When one of these bodies is moved with respect to the other, then work is done hi overcoming the attractive force of the second body. The work done in overcom- ing the attractive force of a given mass M so as to move a particle of unit mass from a given point to an infinite distance is defined as the potential of M at that point. It can be shown that the poten- tial is a function of the space coordinates x, y, z alone; that is, that it is independent of the path. A Newtonian potential function u(x, y, z) due to attractive matter is such that Laplace's equation, iPu a (1 cos 0), x = a (6 sin 0) between (0, 0) and (2ira, 0). 10. Show that the integrals of the functions given in Exs. 8, 9, taken about any closed curve lying in the finite region must be zero. 11. Is the integral C 2 , - dz J 3+2 i Z independent of the path of integration? What conclusion can be drawn from the answer as to the nature of the integrand ? 100 DIFFERENTIATION AND INTEGRATION [CHAP. III. 12. Given the function: /(z) = . Does this function converge z 1 uniformly to its values along the circle C having the origin as a center and a radius equal to one? Does the Cauchy-Goursat theorem apply to the integral taken along C? to a circle concentric with C but lying within C? 13. Givenu>=/(z) s3z* + 7z + 4, , ^ ( T ) . ilL_. I 8 w = r T 1 holomorphic for values of | r | < 1? 14. Given the function /(z) defined by the relation X ,,, at, where t takes complex values along the circle C of radius 2 about the origin. Compute the values of /(z) and/"(z) for z = 1 + i. 16. Given the function /(z) = 3z 3 + 4z 2 + 7z + 2. Find the integral of this function along the circle x 2 + y 2 = 1 from the point a = (1, 0) to the point s ( 1,0). Show that this integral taken around the complete circle is un- changed when any regular closed curve is substituted for this circle as the path of integration. 16. Given u (x, y} = x 4 6xV + y*. Find a function v (x, y} such that t* + iv is an analytic f unction /(z). Find the value of /"(z) for z = 2 + 3 i. 17. Given any rational integral function /(z). Show how the value of /(z) for z = 2 + 3 i can be found when we know the values of /(z) on the circle about the origin having a radius p = 4. 18. An incompressible fluid flows over a plane with a velocity-potential u = x 2 - y 2 . Determine a value of v such that w = u + iv = /(z), is holomorphic in the finite region. Find the components of the velocity and the direction of the flow at the point z = 3 + 2 i. . > **$*. {U** r ^ /*+4/' = i CHAPTER IV MAPPING, WITH APPLICATIONS TO ELEMENTARY FUNCTIONS 26. Conjugate functions. In the present chapter we shall dis- cuss certain elementary functions with special reference to the correspondence between certain portions of the Z-plane and the TF-plane, as determined by the relation between the function w and the independent variable z. Before taking up this general discussion, however, we shall consider the significance of u and v in the relation w = /(z) = u(x, y) + iv(x, y), (1) where /(z) is holomorphic in a given region. As we have seen, both u and v satisfy Laplace's differential equation and hence either may be considered as a potential function. They are consequently of im- portance in theoretical physics. Because of the relation that each function has to the other, they are called conjugate functions. We can represent the functions u(x, y), v(x, y) by the two surfaces u = u(x, y), v = v(x, y), where x, y, u and x, y, v are two systems of Cartesian coordinates of points in space. These two surfaces represent then the real and the imaginary parts of the given function w = /(z). Consider, for example, the function w = z 2 . We have then Jj w = u + iv = (x + iy) 2 = x 2 + 2 ly y 2 . By equating the real and the imaginary parts, we get u = x 2 y 2 , v = 2 xy. Each of the surfaces representing these equations is cut by any plane parallel to the .XT-plane hi a rectangular hyperbola. From the w-surface, we get a system of such hyperbolas having the lines y = x as the asymptotes. From the v-surface, we obtain a system of rectangular hyperbolas having the two axes as asymptotes. 101 102 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. The two systems of curves when projected upon the .XT-plane appear as in Figs. 33, 34, respectively. In either case the curves are the projections of the intersections of the given surfaces with a system of equiangular hyperbolic cylinders whose generating lines are per- pendicular to the .XT-plane. We have considered the two surfaces u = u(x, y), v = v(x, y) as related to distinct coordinate axes. Suppose now we think of them FIG. 33. FIG. 34. as being referred to the same system of axes. The projection upon the .XF-plane of the" curves of intersection reveals an important relation between the two systems of curves. It will be shown that these two systems of curves, given by u = v = c 2 , where Ci, Cj are constants, are orthogonal systems in the .XT-plane. To do this, we make use of the slope of the curves, which is given by r-- From the relation u(x, y) = d, we have for -^ 7* (IX fj dy_ _ _dx_ dx du dy *^ in order that the curves given by v(x, y) = cz be orthogonal to the system u(x, y) = c x , the slope of v(x, y) = C2, at points of intersec- tion with the curves u(x, y) = Ci must be the negative reciprocal of ART. 26.] CONJUGATE FUNCTIONS 103 the slope of u(x, y) = Ci at these points. The general condition of orthogonality is then dv du _ &K _ dy_ dv du' dy dx which may be written in the form du dv_ du dv _ ft dy dy dx dx This condition is satisfied by conjugate functions; for, we know that the conjugate functions u(x, y) and v(x, y) satisfy the Cauchy- Riemann differential equations du _ dv du _ dv dx dy' dy dx Multiplying these two equations member by member, we have pre- cisely the condition of orthogonality given above. The fact that these two systems of curves are orthogonal increases the ease with which either curve may be constructed when the other is given. All we need to do is to construct a second system every- where orthogonal to the first. When so drawn, the two sets of curves obtained in the foregoing example are as shown in Fig. 35. If we think of both systems of FIG. 35. curves as projected back upon each of the surfaces u = u(x, y), v = v(x, y), we shall have upon each surface two systems of curves cutting each other at right angles. The curves cut from either surface by planes parallel to the JXT-plane are called the level lines. The curves of the orthogonal system are called the lines of slope, or the curves of quickest descent. In theoretical physics other special names are employed to designate the projection of these two systems upon the .XF-plane. In an electric field or in the field of a gravitational force, a surface such that the potential is the same at all points of it is called an equipotential surface. Hence, any right cylinder through the lines 104 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. of level on the w-surface or the y-surface just discussed is an equi- potential surface. Since one such surface may pass through each level line, we have a system of equipotential surfaces. Curves drawn per- pendicular to these surfaces are called lines of force. The traces of the equipotential surfaces upon the .XT-plane are called equipotential lines. In the applications to be considered, the lines of force as well as the equipotential lines lie in a plane, which we shall take as the complex plane. In the case of the flow of an incompressible fluid or of streaming in electricity the two orthogonal systems of curves are referred to as the equipotential lines and the lines of flow respectively; while in the theory of heat they are called the isothermal lines and the lines of flow. In the case of electric currents the lines of flow are frequently called stream-lines. We shall have frequent occasion in this chapter to return to the properties of conjugate functions. 27. Confonnal mapping. The relation w = f(z) gives a definite association between those points of the complex plane representing the values of z and those representing the values of w. As a matter of convenience it is usual to represent the z-points in one plane, called the Z-plane, and the w-points in another plane, called the JP-plane. These two planes have a relation to each other somewhat similar to that which the two coordinate axes have in the considera- tion of functions of a real variable. As the point P traces any curve in the Z-plane, the corresponding point Q will trace a curve in the W-plane. We express the relation between the two curves by say- ing that the curve in the Z-plane is mapped upon the T7-plane. In discussing the general properties of mapping, it is often convenient to speak of the mapping of the one plane upon the otherrather than of the mapping of some particular configuration from the one plane upon the other. If f(z) is multiple-valued, then to each point in the Z-plane there correspond in general several distinct points in the W-plane. In such cases it is often convenient to map the whole of the one plane upon a portion of the other. From the discussion in Art. 22, we are able to state the conditions under which the TF-plane can be mapped in a definite manner upon the Z-plane. We have seen that if the Jacobian du du dx dy ^~ dv to dx dy ART. 27.) CONFORMAL MAPPING 105 does not vanish within a given region of the Z-plane, which is the, case if f'(z) ^ 0, we can always solve the equations u = *i(z, y), v = V z (x, y) (1) for x, y in terms of u and v. Moreover, there is but one such solu- tion possible. Denoting the result of this solution by x = xiO, v), y = X2), (2) we can by means of these equations map in a definite manner the TF-plane upon the Z-plane. Whether the TF-plane maps upon the entire Z-plane or only upon a portion of it depends in general upon the character of the two functions xi> X2- By means of relations (1), (2), we can, however, establish a one-to-one correspondence between the points of a region of the Z-plane and those of a corresponding region of the TF-plane; that is to say, if T is the region of the Z-plane under consideration and S the corresponding region of the TF-plane, then to each point of T there corresponds one and only one point of S and conversely. Ex. 1. Given w = z 2 . Let it be required to map a given configuration from the Z-plane upon the TF-plane and conversely by means of this relation. As in Art. 25, we have u = x 2 y 2 , v = 2 xy. We may also write z = p(cos0 + isin 0), w = p'(cos 6' + i sin 0') = p 2 (cos 2 + i sin 2 0). Hence, we have p > = P 2, 8' = 20. From the relation between and 0', it will be seen that a half of the Z-plane maps into the whole of the TF-plane, and on the other hand a half of the TF-plane maps into a quadrant of the Z-plane; for example, the upper half of the TF-plane maps into the first quadrant of the Z-plane. To map from the TF-plane upon the Z-plane, suppose we put u = c, a constant. We obtain a rectangular hyperbola given by the equation X 2 _ yt c Regarding c as a variable parameter, we have two systems of rectangular hyper- bolas having respectively the lines y x as asymptotes, according as c is posi- tive or negative. As the upper half of the TF-plane maps into the first quadrant of the Z-plane, the given lines u = c map into those branches of these hyperbolas 106 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. situated in that quadrant, as represented in Fig. 37. For v = c', a positive con- stant, we obtain in the Z-plane a system of hyperbolas orthogonal to the first systems. We see then that the upper half of the TF-plane maps into the first quadrant of the Z-plane and that the orthogonal systems of lines parallel to the r s t^-^ 1 2 3 It 5 FIG. 36. two axes of the TF-plane map into orthogonal systems of rectangular hyperbolas, having respectively the two positive axes and the line y x as limiting cases. Let us now undertake to map certain simple curves of the Z-plane upon the TF-plane. We know from what has been said that a half of the Z-plane will map O ft X FIG. 38. FIG. 39. into the whole of the TF-plane. Consider the line x = of the Z-plane. We have in the TF-plane w = _^j that is, for any value of y, either positive or negative, w has a negative real value. Consequently, the whole of the F-axis maps into the negative /-axis. The ART. 27.] CONFORMAL MAPPING 107 points Bi, Bz (Fig. 39), map into the same point Q in the W-plane (Fig. 38). If 2 describes a semicircle of radius a as indicated, then w describes a complete circle of radius o 2 about the origin O'. If 2 describes the line x = c, then w describes a parabola cutting the C7-axis at c 2 ; for, eliminating y between the equations we have u = c 2 y 1 , v = 2 cy, v 2 = 4c 2 (c 2 - u). The position of this parabola is shown in Fig. 38. In a similar manner any other curve in the Z-plane may be mapped upon the TF-plane. The given function determines an electrostatic field * in the immediate vicin- ity of two conducting planes at right angles to each other. In this field the equipotential surfaces are the system of hyperbolic cylinders determined by the equation v = 2xy. As a special case we have the two planes intersecting at right angles. The relation between w and 2 also determines the field between two coaxial rectangular hyperbolas. The system of hyperbolas v = c are the lines of equi- potential, while the curves of the orthogonal system u = c are the lines of force. Let us now consider the general case where one region of the com- plex plane is mapped upon another by means of a function w = f(z) which is holomorphic for the values of z under consideration. We shall inquire into the effect of such mapping upon the angle that one curve makes with another at their point of intersection. If the magnitude of the angle is preserved, even though reversed in direc- tion the mapping is said to be isogonal or conformal. We shall now demonstrate the following proposition. THEOREM. The mapping of the Z-plane upon the W-plane by means of a function w = f(z) is isogonal, without reversion of angles, in the neighborhood of a regular point ZQ of f(z}, provided f (ZQ) 7^ 0. V\ a U O X FIG. 40. FIG. 41. Let Ci, C 2 be any two curves in the Z-plane intersecting at ZQ. Suppose Ci, C 2 map into the two curves Si, S 2 of the W-plane inter- secting in the point w corresponding to z . We are to show that the * See Jeans, Electricity and Magnetism, p. 262. 108 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. angle that C\ makes with Ct is the same as that which Si makes with *Sj. The relation between w and z is given by the function w = Since Zo is a regular point, the derivative of /(z) exists and is defined by the relation Since /'(ZD), Aw>, Az are all complex numbers and /'(ZQ) ^ 0, we may write /'(ZD) = p(cos 6 + i sin 0), p ^ 0, Aw; = pi(cos 0i -M sin 0i), Az = p 2 (cos 2 + i sin 2 ), (4) where 0i, 2 are taken to be the chief amplitudes of Aw, Az, respec- tively. From (3) it follows that Pi /'(zo) = L - (cos 0! - 2 + i sin 61 - 2 ) Az=0 P2 Pi = L - L (cos 0i - 2 + ^ sin 0i - 2 ), (5) = L - \ cos L (0! - 2 ) + i sin L (0i - 2 ) f > Ar=OP2( As=0 Az=0 ) since cos z and sin z are continuous functions. We have therefore P = L !, e= L (ft -fc). (6) Az=0 P2 Az=0 Denote by <^ the angle that the tangent to the curve Ci at ZQ makes with the positive X-axis and by fa the angle that the tangent to the corresponding curve Si makes with the positive C7-axis. We have then, since Aw approaches zero with Az, L 2 = 2) L 81 = (fr. (7) A2=0 Az=0 The amplitude of /'(ZD) is less numerically than 2 IT, because of the restriction of the values of 0i, 2 to the chief amplitudes of Aw, Az. We may then write = fa - I = 2z. ABT. 27.] CONFORMAL MAPPING 113 The acceleration a is given by the relation hfwS-A Consequently, as the z-point starts at z = i and moves as given by the con- ditions stated in the problem, w starts at the point w = i 2 = 1 and moves along the upper half of the parabola, starting with an initial velocity of v = 2 z = 2 i. At the end of any given tune, say 3 seconds, we have Z = ZQ = 3 -\~ 1, Wo = Zo 2 = 8 -j- 6 t, and PO = 2 Zo = The acceleration of the ic-point remains constantly equal to two centimeters per second per second. The acceleration in this case being a real number its O' O U FIG. 44. FIG. 45. amplitude is zero and it is directed at each point parallel to the positive [7-axis as shown in Fig. 44. The direction of the velocity at any point is determined by the amplitude of 2, since v = 2 z. As the velocity is always measured along the tangent to the path of the moving point, it follows that the tangent to the tw-curve is always parallel to the half-ray from the origin to the point z. The speed of the t^-point at any instant is \D t w\ = \f'(z)\'\Dtz\=2-\z\. At the end of 3 seconds the speed is then 2- |zo| = 7^ cm ^i [ DCO 114 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. 28. The function w = z n . In a previous article we have dis- cussed a special case of the general function w = z n , namely, the case where n = 2. There are some additional properties of the general case that will now be considered. The function w = z 2 is the simplest case that we have of a general class of functions known as linear automorphic functions. By such a function we mean one that remains unchanged when the inde- pendent variable is replaced by any one of a definite set of its linear substitutions such that the set form a group.* In this case the func- tion remains unchanged under the linear substitutions consisting of z = z' and the identical substitution z = z'. It may be shown that the general function w = z n is likewise an automorphic function. To find the particular linear transforma- tions that leave the function unchanged, we make use of the number 2ir . , 2ir co = cos h * sin n n This number is one of the n th roots of unity; for, by DeMoivre's theorem, we have / 2 . . . 2w\ n co n = cos h i sin = cos 2 TT + i sin 2 TT = 1. \ n n ) The other n th roots, aside from unity itself, are CO 2 , CO 3 , ... CO"" 1 . Moreover, since ( w *) = i, ( k = 1, 2, 3, . . . , n - 1), we may write (co fc z) n = z n . Hence, if co fc z', where k = 0, 1, 2, 3, . . . , n 1, is substituted for z in the function w = z n , the function remains unchanged. The sub- stitutions z = coV form a group of linear substitutions and hence w = z 11 is therefore a linear automorphic function. In the discussion of the function w = z 2 , attention was called to the fact that a half of the Z-plane maps into the whole of the W-plane. Let us now consider the general case, where w = z n . We z = p(cos0-Hsin0), w = p'(cos 0' + i sin 6'). * See Fricke, Enq/klopddie d. Math. Wiss., Vol. II 2 , p. 351. ART. 28.] THE FUNCTION Z n 115 From these relations we obtain p = p n , 6' = nS. Here, as in the special case where n = 2, a circle about the origin in the Z-plane maps into a circle in the TF-plane, and a straight line through the origin becomes a straight line through the origin in the /lY h TF-plane. From the relation between 6 and 6' it will be seen that ( - 1 W of the circle in the Z-plane maps into the whole of the circle in the TF-plane; consequently, a sector bounded by any two half -rays drawn from the origin making an angle - - radians with each other maps tv into the whole of the TF-plane. The values of 6' corresponding to the chief amplitude of w belong to the interval v < 8' = IT. The sector bounded by ORi, OR Z (Fig. 47), making respectively the angles - , with the positive X-axis, maps in a continuous, single- FIG. 46. FIG. 47. valued manner upon the entire TF-plane. The lower bank of the line ORi goes over into the upper bank of the negative axis of reals of the TF-plane. The upper bank of the line OR Z maps into the lower bank of the negative axis of reals of the TF-plane as shown in Figs. 46 and 47. 116 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. Any portion of the Z-plane which maps into the entire TF-plane is called a fundamental region of the given function. Thus the region RiOR 2 is a fundamental region of the function w = z n . Likewise any sector bounded by two half-rays from the origin and making an angle of with each other may be taken as a funda- n mental region. In the case of linear automorphic functions, any sub- stitution that leaves the function unchanged maps any fundamental region of the function into another region that does not overlap the first. The second region may be taken likewise as a fundamental region of the given function. It is to be observed that while the mapping of the Z-plane upon the TF-plane by means of the given function is continuous and single-valued, it does not follow that the mapping of the TF-plane back upon the fundamental region of the Z-plane is continuous. As a matter of fact, such a mapping is single-valued but not con- tinuous. In other words, not every continuous curve in the TF-plane maps into a continuous curve in the fundamental region of the Z-plane. Suppose, for example, that we have a curve in the TF-plane crossing the negative axis of reals. As we have already seen, the upper bank of the negative [/-axis maps into the lower bank of the line ORi', while, on the other hand, the lower bank of the nega- tive [/-axis maps into the upper bank of the line OR 2 . Conse- quently, the curve C (Fig. 46), which is a continuous curve in the TF-plane, becomes a discontinuous curve in the Z-plane. We can say, however, that the mapping from the TF-plane to the fundamental region of the Z-plane is a single- x valued process; for, to every point ^ of the TF-plane there is one and FIG. 48. on ly one corresponding point in the fundamental region of the Z-plane. A fundamental region of the function w = z n need not be bounded by straight lines. It serves our purpose equally well to take any con- tinuous curve proceeding from the origin and to revolve it through the angle , as indicated in Fig. 48. The boundary lines of the ABT. 28.] THE FUNCTION 2" 117 sector extend indefinitely from the origin in the case of the function under discussion. It must not be assumed that the fundamental region of all auto- morphic functions can be obtained hi this manner. The function here considered is a special case. A more general case will be dis- cussed in a subsequent chapter, in connection with doubly periodic functions. By means of the fundamental operations of multiplication and addition applied to complex constants and functions of the type w = z n , where n is integral, we obtain the rational integral function It is of interest in this connection to point out some of the appli- cations * that may be made in theoretical physics of the transforma- tion w = z n . For the special case where n = 1, we have 1 w = -> z or u + iv = ; - 1 x + iy whence x ~ y /i\ U = -j j o' V = ~T~i 2' W X? -j- y x -f- y Corresponding to the lines u = c we have x = or c(z 2 + y 2 ) x = 0. This equation is represented by a system of circles having their cen- ters on the X-axis and all passing through the origin. For the or- thogonal system, we have - y = c or c(z 2 + i/ 2 ) + y = 0, * See Jeans, Electricity and Magnetism, p. 261 et seq; Lamb, Hydrodynamics, 3 d Ed., p. 66. 118 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. which is represented by a system of circles having their centers on the F-axis. These two systems of circles are shown in Fig. 49. When a fluid flows over a plane surface, any point P from which the fluid flows out in all directions in a uniform manner is called a source. By the strength of the source is understood the total flow in a unit of time across a small closed curve about the source, that is, FIG. 49. the time rate of the supply of the fluid through the source. A negative source is called a sink. Let P' be a sink and suppose it to be of equal strength with the source P. Denote this common strength by m. If we now think of P as approaching P' in such a manner that the product m PP' remains constant, say equal to n, then we say that we have a plane doublet * of strength n. The velocity-potential due to the doublet is given by nx x z + y 2 ' * See Pierce, Newtonian Potential Function, p. 434, et seq. ART. 28.] THE FUNCTION Z n 119 while the lines of flow are given by By comparing these functions with the conjugate functions u(x, y), v(x, y) given in the equations (1), it will be seen that u, v determine the velocity-potential and the lines of flow respectively of a plane doublet at the origin whose strength n is 1. The X-axis is the axis of the doublet. The lines of flow v = c are, as we have seen, the system of coaxial circles having their centers on the F-axis, while the system of coaxial circles having their centers on the X-axis are the lines of equal velocity-potential. Writing the given function w = z n in the form u + iv = p n (cos 6 + i sin 0)"; we have, upon equating the real parts and likewise the imaginary parts, u = p n cos nd, v = p n sin nd. For n 1 the first of these equations gives, for the flow of an in- compressible fluid, a system of equipotential curves parallel to the i Y-axis, and the second gives as the corresponding lines of flow a system of lines parallel to the X-axis. It has been pointed out that when n = 2, then the equation u = c gives a system of rectangular hyperbolas having the axes of coordi- nates as their principal axes. In the applications to the flow of an incompressible fluid, these curves are the lines of equal velocity- potential of an irrotational fluid having constant density and a steady flow. The curves v = c, that is the lines of flow are likewise a system of rectangular hyperbolas, having in this case the axes of coordinates as asymptotes. The lines 6 = 0, = ^ are parts of the same line of flow corresponding to v = 0; hence, we may take the positive parts of the coordinates axes as fixed boundaries, and thus obtain a case of irrotational fluid motion in an angle between two perpendicular walls (see Fig 37). By the proper selection of the value of n, we may so change the conditions as to represent an irrotational inotion of the fluid be- tween two rigid walls making a given angle < with each other. The lines of flow are given by the equation p n sin n6 = c, 120 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. where the lines 8 = 0, 6 = - are parts of the same line of flow, namely the one by putting c =0. If now we so select n that the angle - shall be the required angle 9, that is if n = - , we get as the required 9 equations of the equipotential curves and lines of flow, respectively, the following: T a * a TTV T . 7TC7 v = fT sin . 9 u = fy cos o FIG. 50. Ex. Find the value of n such that the function w = z n shall determine an irrotational motion of a liquid between two walls making an angle of 60 with each other. Find the velocity-potential and direction of the flow at the point 2o = 2 (cos 20 + i sin 20), assuming the flow to be steady. Trace the equi- potential curve through the given point. We have n = - = 3, and therefore w = z* = u + iv, whence u = y? 3 xy 2 , v = 3 x*y The velocity-potential at the given point is u = p j cos 3 6 = 2 J cos3-20 = 4. The equipotential curve is then given by the equation x* - 3 xy* = 4. ^ UL. ABT. 28.] THE FUNCTION Z n 121 From the definition of velocity-potential, we know that the components of the velocity in the direction of the X-axis and F-axis are , , respectively. ox oy The velocity v is given by du . du , , -A " = -^-^' (8) We can determine graphically the velocity by means of the derivative D z w. As we have seen, Art. 21, du .du , n , w = - l ' By comparison of (8) and (9), it will be seen that the point P' representing the velocity at Zo is the reflection upon the axis of imaginaries of the point P repre- senting D z w. We have D^w = 3 zo 2 = 12 (cos 40 + i sin 40). The point P is found by laying off on a line making an angle of 40 with the axis of reals the distance OP = 12. By reflection upon the F-axis the point P' is found, which represents the velocity at ZQ. Drawing from o a line parallel to OP', we have the direction of the flow at 2o- This flow is in the direction of the normal to the equipotential curve through 2o as we should expect. It should also be observed that if the point o is allowed to move along a curve of flow v = 3 x 2 y - y 3 = k, the point P' varies in such a manner as to describe the hodograph* of the motion of ZQ, and thus the velocity of P' determines the magnitude and direction of the acceleration of o. The relation between the velocity in the Z-plane and the deriva- tive D z w, as brought out in the above exercise, is perfectly general and may be applied to any case, so long as the function w = /(z) is holomorphic in the region under consideration. It should also be noted in this connection that the velocity of a point moving along the line v = c in the TP-plane is always the negative of the square of the speed of the corresponding point moving along the curve of flow in the Z-plane; for, denoting by v w , vz, the velocities in the TF-plane and Z-plane respectively, we have by equation (13), Art. 27, (du .du\( du .du\ Vw = D Z W V z = - -- * - 1 1 -- l T~ ) \dx dy/\ dx dyj _(duV _ (du\ 2 (dx) ' \dy) where r is the speed along the curve of flow in the Z-plane. * See Ziwet, Theoretical Mechanics, Part I, p. 80. 122 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. 29. Definition and properties of e*. We shall now define the exponential function e z , where 2 is a complex variable. The defini- tion and properties of the function e x , where x is a real variable, can not be assumed to hold when the variable is complex. The function e' should be so defined, however, as to include, as a special case, the function e*. It will assist us in formulating this definition to recall the definition and some of the general properties of e x . The number e is defined by the limit = L ( 1 + -V = 2.7182818 . = \ n) where n takes all positive real values. Likewise, the exponential function e* is defined as the limit L ( 1 H 1 . n =w V n) This function obeys the general law, /(*!> /(*)=/(*! + **). (1) Differentiating the function f(x) = e x , we have D,f(x) = = * (2) We shall define the exponential function of a complex variable by the relation, e*=L l + ", (3) where n is a positive integer. We shall show that this limit exists for all finite values of 2 and that the function thus defined has the general properties expressed for e x in (1) and (2). To show the existence of this limit, we proceed as follows. We write 9 f -I '111 1 + = 1 + n n 1 + - = p cos 0, ^ = p sin 6, (5) Putting we get ( 1+ ;)" = p n (cos n0 + i sin n&). (6) l + - = [p(cos + i sin ART. 29.] THE FUNCTION 6' 123 Since n can be taken so large that 1 + - , and therefore cos 6, is n always positive, from (5) we obtain 6 as the principal value of arc tan y n + x , and whence (7) The limit in (3) may then be written / ?\ n ( / r\ n f 2 ~l L (1+*) = L (1+-) 1 + 7-t-vi B =oo V n/ -, (V n/ L ( + *rJ ( cos n arc tan 7 -- h * sin n arc tan w + x = L (1 + X\ n / - -I/ n) =, \ n + cos n arc tan r ) + 27 ) n i L sin n arc tan r W + XJ (8) provided each of these limits exist. These limits, however, do ex- ist and can be readily evaluated. We have from functions of a real variable L 1 + J =e *' n=oo \ / For y = 0, the second limit in (8) is one; for y ^ 0, we have r w 2 i 2 ( r V 2 >H L l+ ? r-J = L \l + - (9) => [_ (n -\- x) j ( n =oo \_ (n -f x; J ) However, we have 1" r 1 ~| 2 n==o"' (n+i) ~ ' = e. ^,2 ~jra L n = co l"l 1 1 1 2 jt co vn~f~zj^ n e (n + *) 2 1 (n + a:) 2 1 e 124 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. Hence, from (9) we get f = 2 J Since the cosine is a continuous function, we may write the third limit in (8) in the form - =*> y / L cos n arc tan - = cos L n arc tan 7 =*> n -\- x = n + x . arc tan , ny n -\- x /-^ = cos L - - - = cos y. (10) B =>n + :c y n-\- x Similarly, we have for the final limit in (8) 11 L sin n arc tan 7 = sinv. (11) n=oo Tl + X Hence, substituting these results in (8) we obtain e* = e^cos y + i sin y}. (12) This result not only shows that the limit given in (3) exists for all finite values of z, but it gives a very convenient form of the defini- tion of the exponential function e*. From this form of the definition, we can deduce a convenient method for writing the complex number z = p(cos 6 + i sin 6) ; (13) for, putting x = in (12), we have e iv = cos y + i sin y, or writing this result in the usual form, we have e ie cos 6 + i sin 0. Hence we may write (13) in the form * = pe'9, (14) a form of expression that is often convenient. The function e 1 is uniquely determined; for, we have from (12) e z = u + iv = e? cos y + ie x sin y, whence u = e? cos y, v = & sin y. (15) AKT. 29.] THE FUNCTION e z 125 From these equations it follows that the conditions that f(z) = e* is an analytic function are satisfied; for, we have for all finite values of x, y du _ dv du _ dv dx dy' dy dx Therefore, e z is holomorphic in the finite region of the complex plane and consequently is an analytic function. Hence, in the finite region e* is continuous and has a continuous derivative. Moreover, e* is a single- valued function of z, and e x appears as a special case. From (12) it can be shown that the general properties of the exponential function of a real variable may be extended to the case where the variable is complex. For example we may deduce as follows the general law expressed in (1), namely, Substituting the values of e* 1 , e* 2 , as defined by (12), we have i sin t/ 2 ) sn The law of differentiation stated in (2) holds also where the vari- able is complex. Remembering that du , .dv D z w = r--r*ali dx dx we have from (12) D z e z = e* cos y + ie* sin y = e x (cos y + i sin y) It is to be observed that e 2 is a periodic function; that is, the function remains invariant when z is replaced by z plus some con- stant, say o>. Such a function satisfies the relation /( + )-/(*). The constant w is called a period of the given function. A periodic function takes all of its values as the variable z takes the values in a definite region of the complex plane, known as the region of peri- odicity, and repeats those values as z varies over another equal portion of the plane. In this particular case the regions of peri- odicity are parallel strips bounded by lines parallel to the axis of 126 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. reals and at a distance of 2ir from each other. We say then that the function has the period 2 in. To show this to be the case, we may write f+**i = g*+i(H-2r) = e 1 {cos (y + 27r) + i sin (y + 2ir) \ = e? I cos y + i sin y \ = e z . Instead of 2 iri, we could have taken equally well any multiple of it as a period. It would not have answered the purpose to have taken a fractional part of 2 in nor any number less than 2 TTI ; that is 2iri is the smallest constant that answers the purpose. We in- dicate this fact by calling 2 in the primitive period of the function. When, as in this case, all of the periods are multiples of a single primitive period, the function is called a simply periodic function. Let us now undertake to map the Z-plane upon the TF-plane, and conversely, by means of the relation w = e z . Since the given function is holomorphic having a derivative different from zero for finite values of z, it follows that the mapping is conformal in the FIG. 51. FIG. 52. finite region; that is, in this region the similarity of infinitesimal elements is preserved. We shall make use of the fact that the function is periodic, having the period 2 id. If we draw through the points id and id in the Z-plane two lines parallel to the X-axis, the region bounded by these lines maps into the entire TF-plane and therefore may be taken as the fundamental region. In V ' ~J ABT. 29.] THE FUNCTION 6* 127 mapping we shall consider the boundary line y = T, but not the boundary line y = ir, as belonging to the fundamental region. What is said of this region may be said of any one of the regions bounded by the lines y = (2fc + l)7r, y = (2 fc -!)*-, k = , -2, -1,0, 1,2, . The line u = is the map of certain lines parallel to the X-axis. To show this, put u = in (15) and thus obtain = e 1 cos y, (16) whence for finite value$of x, we have u = as the map of the lines y=(2fc + l)|, * = - , -2, -1,0, +1, + 2, . - . . (17) Within the fundamental region TT < y = TT, we have the lines cor- responding to k = and k = 1 ; and hence the line u = 0, that is the F-axis, is the map of two lines of the Z-plane lying within this region, namely: y =| and y=-|- (18) 7T 7T For y = ~ we have from (15) v = e?, and for y = = we have v = e x ; t IT so that the positive F-axis is the map of the line y = ~ , while the nega- Zi tive F-axis is the map of the line y = - . Since for x > 0, y = db ~ we 2i Zi have \ e* \ = e x > 1, it follows that the portion of the positive F-axis exterior to the unit circle about the origin is the map of the positive half of the line y = = , and the portion of the negative F-axis exte- Zi rior to the unit circle is the map of the positive half of the line y = - . Likewise, since for x < 0, y= = we have \ e* \ = e 1 < 1, 2. 2i TT 7T it follows that the negative halves of the lines y = r , y = - map Z 2i respectively into the portions of the positive and the negative F-axis lying within the unit circle. In a similar manner we have from (15), for v = 0, = e 1 sin y, (19) 128 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. and hence the line v = 0, that is the t/-axis, is the map of the lines y = for, k = - , -2, - 1, 0, + 1, + 2, . (20) For the fundamental region TT < y = T, we have k 0, 1, and consequently V = 0, T. (21) For these values of y, we obtain u = e x , u = e x , (22) respectively. Hence the positive t/-axis is the map of the X-axis, and the negative t7-axis is the map of the line y = TT. The positive halves of the lines y = and y = TT map into that portion of the t/-axis exterior to the unit circle, while the negative halves of the same lines map into that portion of the t/-axis within the unit circle. Any line parallel to the X-axis maps into a half-ray in the W-plane proceeding from the origin, Fig. 53. This may be shown as follows. V\ FIG. 53. FIG. 54. Eliminating x from equations (15) by multiplying the first of these equations by sin y and the second by cos y and subtracting, we have u sin y v cos y = 0. For constant values of y, this equation gives straight lines of the form v = mu, where m = tan y. Since e x is positive for all finite values of x, it follows from (15) that any line y = c maps into a half-ray from the origin taken along the line v = mu; the portion of this half -ray ART. 29.] THE FUNCTION 6 Z 129 interior to the unit circle corresponds to negative values of x, while the portion exterior to the unit circle corresponds to positive values of x. If successive values of y differ by equal amounts, then the corresponding half-rays in the TF-plane will make equal angles with each other. The map of a line parallel to the F-axis may be easily obtained as follows. Eliminating y from the equations u = e z cos y, v = e 1 sin y by squaring and adding, we have W 2 _|_ V 2 _ e Zx ( C0g 2 y _J_ S j n 2 y) = 6 2 *. For any constant value of x, we have then a circle in the TF-plane about the origin as a center. For x = 0, we have u 2 + v 2 = 1 ; that is, the F-axis maps into the unit circle about the origin in the TF-plane. For x = c > 0, the map in the TF-plane is a circle exte- rior to the unit circle; and for x = c < 0, the map is a circle lying within the unit circle. From what has been said, it will now be seen that the regions a, b, c, d, A, B, C, D, Fig. 52, map respectively into the regions a, b, c, d, A, B, C, D, Fig. 51, the lower bank of the line y = IT, and the upper bank of the line y = IT mapping respectively into the upper and the lower banks of the negative [/-axis. Any line y = mx passing through the origin, other than the axes of coordinates, maps into a curve in the TF-plane that cuts the half-rays from the origin at a constant angle; that is, it maps into a logarithmic spiral about the origin. Since the point x 0, y = maps into the point u = 1, v = 0, all of these logarithmic spirals pass through the point u = I, v = 0. As we have seen the whole of the TF-plane can be mapped into any one of a number of strips parallel to the axis of reals in the Z-plane. Suppose we map it into the fundamental region lying between the lines y = TT, y = TT. Consider the line u = c, c > 0. For this value of M, we have c = e x cos y, e* or y = arc sec (23) 130 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. If we put e* = c, that is x = log c, we have y = arc sec 1=0. The curve whose equation is (23) then cuts the X-axis at the point x = log c. As x increases y increases and approaches asymptoti- cally the line y = ^ . The sign of y is determined by the sign of v in the relation v = e* sin y. (24) Since e 1 is always positive, sin y and therefore y is positive or nega- tive according as v is positive or negative. The curve is therefore FIG. 55. symmetrical with respect to the X-axis and is situated as is indicated in Fig. 56. As c is assigned different values, the point where the curve crosses the X-axis changes. For c > 1, the curve cuts the X-axis to the right of the point x = 0; for c = 1, it crosses at the origin; for < c < 1 it crosses to the left of the origin. It will be remembered that the negative half of the t/-axis maps into the lines y = IT, y IT. That portion of the line u = c, where c < 0, lying above the f/-axis will, as we have seen, map into the TT portion of the fundamental region lying between the lines y = = and y = TT. Moreover, since c is now negative it follows that y decreases as x increases. The form of the curve is indicated in Fig. 56. It is asymptotic to the line y = ^. In the same way it follows that the portion of u = c lying below the t/-axis maps into a curve begin- ning on the line y = IT and becoming asymptotic to the straight line y = ~ , since the ordinate increases with x. The results of map- A AHT. 29.] THE FUNCTION 6* 131 ping the TF-plane upon the fundamental region ir > 2' *' From (15) it will be seen that along these lines, we have u = e, 0, e*, 0, e*. As x decreases without limit through negative values, each of these values of u approaches zero. However, as x increases the value of u remains zero along the lines y = -, ~, but increases without Zi 2i limit along the line y = 0, and decreases without limit along the lines y = TT, TT. Hence, we have a surface that is flat at the extreme left and towards the right has ridges and valleys of increasing magni- tude. These ridges and valleys extend parallel to the axis of reals and their magnitude is readily determined by taking a cross-section of the surface parallel to the F-axis. As an illustration, let us consider the function z = w + e w . As we shall see later, this function is of importance in the consideration of certain problems in mathematical physics. We shall map a given configuration from the TF-plane upon the Z-plane by means of this relation. In order to do so, we must first obtain x and y in terms of u and v. Writing the given function in the form x + iy u + iv + 6"+" = u + iv + e u e iv = u + iv + e u (cos v + i sin v), we have upon equating the real and the imaginary parts x = u + e u cos v, y = v + e" sin v. (25) The axis v = maps into the X-axis; for, we have in this case from the equa- tions (25) x = u + e u , y =0, 132 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. and, consequently, every point on the C7-axis maps into a point on the X-axis, the point w = mapping into z = +1 (Fig. 57). For t> = v, we have x = u e u , y IT. As w moves along the line v = * from u - oo to u = 0, the corresponding point in the Z-plane moves along the line y = * from x = - oo to x = 1. As the value of u continues to increase from u = to u = oo , the value of x passes from U FIG. 57. FIG. 58. x = 1 back to x = oo . We say that the line v v maps into the line y = * in such a manner that the line doubles back upon itself at the point x = 1. In a similar manner, the line v = v maps into the line y = a-, bending back upon itself at the point x = 1. Let us now consider the line w = 75 For this value of v we have m x=u, y = I + e u . For u = oo , we have x = oo , i/ = - . As u increases, y increases until u 2i reaches the value zero, where x = 0, y = | + L ABU increases through positive values, y continues to increase with u as indicated in the figure. For values of v lying between ^ and *, say for (v e), we have 4 x = u e u cos e, y = T e + e u sin . From these equations, we have for u = oo, x = -oo, y = T t. ART. 30.] THE FUNCTION LOG Z 133 As u increases, both x and y increase, although y increases very slowly, until we have DuX = 1 e u cost = 0; that is, until e u cos e = 1. As e u cos becomes greater than 1, DuX becomes negative and x decreases while y continues to increase and that more rapidly. The general form of such a curve is shown in the figure. For values of c lying between and - , the line v c maps into a curve such m that y at first increases very slowly and then more rapidly as u takes on large positive values. In this case, however, x also continues to increase as u increases. For values of v less than 0, the curves lie below the X-axis and are symmetrical as to that axis with those already obtained. The mapping of these curves there- fore presents nothing new. The given function expresses the motion of a fluid from a reservoir of indefi- . = - nitely large size into a narrow, restricted channel bounded by thin parallel walls.*-g^ If the sign of w is changed, the given function represents the flow as taking place 3 hi the opposite direction. As may be shown, the velocity of the flow increases indefinitely in the neighborhood of the points ( 1, T), ( 1, IT}. 30. The function w = log z. We shall now define the logarith- - mic function and discuss some of its properties. In real variables -r the logarithm is frequently defined as the inverse function of the exponential. This property will be used in defining the logarithm of a complex number. The general properties of inverse functions have been discussed in another connection. To determine the in- verse of the exponential function let us consider the relation * = z. (1) We have, as elsewhere, w = u + iv, z = p(cos -\- i sin + i sin <). Equating the real and the imaginary parts in this equation, we obtain e u cos v = p cos , e u sin v = p sin <. (3) * See Lamb, Hydrodynamics, 3 d Ed., p. 70. 134 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. Each number entering into these equations is real, and consequently we can solve the equations for u and v by the means already at our disposal. Squaring each member of these equations and adding, we have (e u ) 2 (cos 2 v + sin 2 v) = p 2 (cos 2 + sin 2 ), whence (e) 2 = p 2 , but as e u and p are always positive numbers, we may write e = P. (4) Making use of this relation, we have from (3) cos v = cos , sin v = sin <, whence v = 0. (5) Since u and p are both real numbers, we have from (4) u = logp = log \z |. It is to be noticed that for z = 0, and therefore p = 0, the equation e u = p has no finite solution. For all other values of z, the equa- tions u = logp = log \z |, v = = amp 2 determine definite values of the coordinates u, v. The correspond- ing value of to is defined as the logarithm of z. We have then as the formal definition of w = logz log z = log p -f i$, (6) which may also be written in the form log z = log [ z | + i amp z. (7) For any particular point of the complex plane, say ZQ, there are an infinite number of values of log ZQ differing from each other by some multiple of 2iri. This result is a consequence of the peri- odicity of the exponential function, which is the inverse of the loga- rithmic function; or, it follows directly from the definition of a logarithm, for since we have the same point z if is replaced by (< + 2 kir), where k = 1, 2, . . . , it follows from the definition that log z has an infinite number of values for this same value of z. In ART. 30.] THE FUNCTION LOG 2 135 the discussions of the present chapter, unless otherwise stated, will be restricted to the chief amplitude of z, and for such a value of tj>, log p + i<(> is called the principal value of the logarithm. In a subsequent chapter we shall discuss the logarithm as a multiple- valued function, thus giving to all possible values. The logarithms of the positive real numbers appear as a special case of those of complex numbers, because for such numbers the value of is zero. The logarithms of negative numbers may now be given a definite significance; for, if z is a negative real number, we have z = P(COSTT -f isinx), and hence we obtain log z = log p + ITT, which is represented by a definite point in the complex plane. Except for z = 0, the logarithmic function is holomorphic in the finite region; for, the Cauchy-Riemann differential equations are satisfied. We have w = u + iv = log p + i = lc Consequently, we obtain = log Vaj 2 + y 2 -f- i arc tan - x v = arc tan - , JU du x dv y whence = =. > -r- = -- 5 : ; x 2 + y 2 x x 2 + y* Hence, with the restriction placed upon its amplitude the function is analytic. This conclusion does not hold, however, for x = 0, y = 0, since the partial derivatives =-, - become indeterminate dx dy dx dy in this case. In putting y amp z s= < = arc tan - , *c it should be noted that care must be taken to distinguish between 6 6 arc tan and arc tan -- . We may not replace these two ex- a a 136 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. pressions by arc tanf J, as one might at first think possible; for, the first expression is the amplitude of z = a ib, while the second is the amplitude of z = a + ib. These two values of z have the same moduli, but their amplitudes differ by TT. For a similar reason, we must distinguish between arc tan and arc tan - ~ a a The function log z obeys the laws of logarithms for real variables. We have, for example, log zi + log Z2 = log (z^), (8) where z\, z* are different from zero. To show that this relation holds, we have log Zi + log Z2 = flog pi + ii\ + {log pa + ifal = {log pi + log p2J + H$i ~l~ ifal = log (ziz 2 ) ; since, in multiplying two complex numbers we multiply the moduli and add the amplitudes. To find the derivative of a logarithm, we have dw _ du .dv dz dx dx ' and therefore x* + y 2 x*- + y 2 x + y z The mapping of the Z-plane upon the TF-plane by means of the logarithmic function gives a conformal representation because the function is analytic. The only finite singular point is the origin. As we have seen the logarithm is the inverse function of the expo- nential function. The general properties of the configuration ob- tained by mapping by means of the relation w = log z follow at once from the earlier discussion of the mapping by means of the functional relation w = e*. The only difference in the two cases is that the Z-plane and the W-plane are interchanged. ABT. 30.] THE FUNCTION LOG Z 137 Because the logarithmic function enables us to map a system of concentric circles and their orthogonal rays into a system of parallel lines and another system of straight lines orthogonal to them, it is for some purposes one of the most useful of the functions thus far discussed. One of the important applications of this function is to that system of map drawing known as Mercator's projection. If we undertake to map the earth's surface upon a plane by means of a projection from the north pole as the center of projection, we obtain a corresponding configuration in the plane. The region about the north pole becomes very much distorted by this process. It is often desirable in navigation so to direct a ship's course as to cut the suc- cessive meridians at a given angle, that is, to keep the ship headed toward a definite point of the compass. The curves indicating the ship's course upon the earth's surface, known as loxodromes, project into logarithmic spirals upon the complex plane, when that plane is tangent to the earth's surface at one of the poles while the opposite pole is taken as the center of projection. By means of the logarith- mic function, the system of circles and orthogonal rays into which the earth's surface maps by this method of projection may be changed into two systems of parallel straight lines orthogonal to each other. In this new system the meridians become a system of parallel straight lines, while the parallels of latitude become a system of parallels orthogonal to the first. All loxodromic curves become straight lines cutting these two systems of parallels at a given constant angle. This result simplifies the problems of navigation by the use of the mariner's compass. The configuration represented in Fig. 59 arises in mathematical physics whenever we have a source at the origin and a sink at an infinite distance. The rays JP IG> 59. are in this case the lines of flow and the concentric circles are lines of equal velocity-potential. If we pass through the origin a straight wire of indefinite length, through which a current of electricity is passed, we have in any plane per- pendicular to this wire an induced magnetic field likewise repre- sented by the configuration in Fig. 59, except that in this case the pencil of rays become the lines of equipotential and the system of concentric circles are lines of force. If the electric current flows 138 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. through the wire in the direction from below to above the plane of the paper, the direction of the magnetic force is then as indicated by the arrow-heads.* Consider the function w = log ; r . Writing the given function in the form z + 1 v + iv, we have upon separating the real and imaginary parts | Z - 1 | ( X - I) 2 u = logf- -. = log . S & 1)' y v = amp (z 1) amp (z + 1) = arc tan ^7 arc tan For u = c, we have v V(z + 1)* + y 2 (X- 1)2 +3/2 (9 + iy + f' or l +1 =0- (10) ~ This equation is represented by a system of coaxial circles having their centers on the X-axis. The centers of these circles are given by the equations / jkvH a V-' e ~t~ 1 vut* &** ' and the radii are equal to 1 For negative values of c the center lies to the right of the origin. The point (+1, 0) lies within all of these circles. For positive values of c the centers of all the circles lie to the left of the origin and inclose the point ( 1, 0). Correspond- ing to c = 0, we have a circle of infinite radius, that is a straight line perpendicular to the segment joining the points (1,0) and ( 1, 0) at its middle point, namely at the origin. For v = c, we have y y c = arc tan ^ 7 arc tan ' z > whence y y x-l x + 1 2y - or * See 3. 3. Thomson, Electricity and Magnetism, 4 th Ed., p. 329. ART. 30.] THE FUNCTION LOG Z 139 This is the equation of a system of coaxial circles having their centers on the F-axis. Each of these circles passes through the points (1, 0) and ( 1, 0). The general form of the configuration in the Z-plane is given in Fig. 60. This configuration may be reproduced in the physical laboratory as follows. Given a glass plate covered with iron filings. Pass long straight parallel wires through the points A = (+1, 0) and B = ( 1, 0) perpendicular to the plate FIG. 60 and allow an electric current of equal strength to flow in opposite directions through the two wires. By jarring slightly, the filings tend to arrange them- selves along the system of circles about the points A, B and having their centers upon axis of reals. These circles are the lines of magnetic force. The direction of this force depends upon the direction of the two currents. If the direction of the current through the plane of the paper at A is downward and that through B is upward, the direction of the force is as indicated in the figure. The orthog- onal system of circles all pass through the points A, B and are the lines of equi- potential. For the flow of incompressible fluids, we obtain the same configuration when- ever one of the points A, B is a source and the other a sink, both being of the same strength. The circles through A and B are then the lines of flow and the circles of the orthogonal system are the lines of equal velocity-potential. If A is the source and B the sink then the direction of the flow is from A to B. For the function w = log (z + 1) (2 1), we have a different configuration. From the given function, we obtain u + iv = log | 2 + 1 [ | 2 1 | + i I amp (z + 1) + amp (2 1) } , 140 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. or u = log 1 2 + 1 1 | z - 1 | = log V(x + I) 1 v = amp (z + 1) + amp (z 1) = arc tan For u = c, we get - 1)* + y 2 , + arc tan X "~ I X "7" 1 + y 4 + 2(z* + 1) y* - 2x 2 + 1 = e 2 ', which f or c ?* is represented by a system of Cassinian ovals as shown in Fig. 61. YL FIG. 61. For c = the equation represents a lemniscate having its double point at the origin. For the orthogonal system of curves, we have arc tan 1 + arc tan y whence tanc = x+ 1 ' z- 1 1 - a; 2 y* 1 x* - 1 _ yi _ tanc = 1. This equation gives a system of hyperbolas passing through the two points (1,0) and ( 1,0) as indicated in Fig. 61. This configuration comes into consideration in theoretical physics whenever the points A and B are sources of equal strength. If we are considering the flow ART. 30.] THE FUNCTION LOG Z 141 of an incompressible fluid, the curves u = c are the lines of equal velocity-potential, and the curves v = c are the lines of flow. If, however, we are considering a magnetic field, induced by passing currents of equal strength in the same direc- tion through parallel straight wires intersecting the plane at A and B, the curves u = c become the lines of force, and v = c are the lines of equipotential. Ordi- narily a line of force does not intersect itself. In the case under consideration one of the lines of force, namely u = 0, does intersect itself, having a double point at the origin. In order that a double point may exist the partial derivatives of u with respect to x and y must vanish.* These partial derivatives are the com- ponents of the force acting, and since both are zero there can be no force at such a point. For this reason such a p<5int is called a point of equilibrium. f In the case of an irrotational fluid motion the components of the velocity are zero at a point of equilibrium and hence no flow takes place at such a point. The same configuration occurs in the discussion of the colored rings in biaxial crystals due to the interference of polarized light. By means of the logarithmic function, we may express the more general case of any number of sources and any number of sinks, each having a given strength. Suppose we have a source at each of the points a\, a.% . . . , a n having strengths of ki, k z , . . . , k n , respectively. Let there be a sink at each of the points ft, ft, . . . , /3 m , each of strength Xi, X 2 , . . . , \m, respectively. Since the sinks are to be considered as negative sources, the corresponding factors appear in the denominator of the function of which the logarithm is to be taken. The corresponding function is then w = log k * As a special case which presents some interest, let us consider the function (z - l) z w = log ! The function w determines, for example, the equipotential lines and the lines of force in a magnetic field about two parallel conductors in which the electric current is passing in opposite directions in the two and is twice as strong in the one as in the other. The wires pierce the complex plane at A = (I, 0) and B s ( 1, 0). The wire through A carries a current twice as strong as the one through B. In order to obtain the two systems of conjugate curves given by the function, put * See Townsend and Goodenough, First Course in Calculus, p. 370. t See Maxwell, Electricity, Vol. I, Chap. VI; Jeans, Electricity and Magne- tism, p. 59; Lamb, Hydrodynamics, p. 17. 142 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. we have w = log^^ = log^ + log e<(* .-) Hence, we obtain For u = c, we have u = log , v = 2 0i - Pa 2 P2 ' or p 2 = ^ = #pi 2 , K > 0. (12) For the orthogonal system, we have c = 2 0i - 2 . (13) To plot any one of the system of curves represented by (12), give K an assigned value and give to pi any convenient succession of values. Compute the corre- sponding values of p 2 by means of (12). With z = -j-1 as a center and the assumed values of pi as radii, draw circles. Likewise, with z = 1 as a center and the computed values of p 2 as radii draw circles. The intersections of corre- sponding circles give points on the required curve. To plot a curve belonging to the system given by (13), give to c any assigned value and from the points z = + l,z = 1, and draw lines making angles 0i and 02 = 2 0i c, respectively, with the axis of reals. The intersection of corre- sponding lines gives points on the required curve. The general form of the two systems of curves is shown hi Fig. 62. To determine the double points of the lines of force, that is, the points of equi- librium, we have p, 2 (x - I) 2 + J/ 2 u = log = log V( X + I) 2 + y 2 The double points are given by putting partial derivatives of u with respect to x and y equal to zero and solving the two resulting equations for x and y. We have then to solve the equations dx s- x - I) 2 + y 1 (* + l) 2 +2/ 2 Equations (14) and (15) are satisfied simultaneously by the values y = 0, x = 3. These values are therefore the coordinates of the point C of equilibrium. To ART. 30.] THE FUNCTION LOG Z 143 determine which one of the lines u = c maps into the particular curve having a double point at (3, 0), we substitute the values x = 3, y = in (12) and determine the corresponding value of c. This substitution gives (? c = 64, or c = log8; that is, the potential function has at each point of this curve the value log 8. FIG. 62. The distribution of matter being confined to the plane, the intensity of the force at any point per unit of strength is equal to the reciprocal of the distance.* Hence, in order that C shall be a point of equilibrium, it follows from the laws of physics that C must lie on the X-axis and that we must have _2-..JL =0 AC BC It will be seen that this equation gives the same values of the coordinates of C as those already obtained. * See Wangerin, Theorie des Potentials und der Kugelfunktionen, Vol. I, pp. 135-137. 144 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. 31. Trigonometric Functions. The definition of the various trigonometric functions may be made to depend upon the expo- nential function e* already defined. From the definition of e*, it was shown that e a = cos + i sin 0, whence e~ ie = cos i sin 0, where in both cases is real. Solving these equations for sin0, cos 0, we get gifl g- iff sin = ^ , 2^ gW _1_ g- M COS = - & In a similar manner, we shall now define sin 2, cos 2 in terms of the exponential function e*, by putting Since the function e* is analytic, it follows that sin 2, cos z are also analytic functions. Moreover, sin x and cos x appear as special cases of the sine and cosine of the complex variable 2. The trigonometric functions of a complex variable satisfy the same trigonometric identities as the corresponding functions of real vari- ables. We may show, for example, that the following relation holds sin (z\ + 22) = sin Zi cos 22 + cos Zi sin 2 2 . We have sin 21 cos 22 + cos z\ sin 22 = 2i-2 2 e- .4t = sin (21 + 22). The remaining trigonometric identities may be established in a similar manner. While the fundamental identity cos 2 2 + sin 2 2 = 1 AST. 31.] . TRIGONOMETRIC FUNCTIONS 145 holds for complex as well as real values of z, it follows from the defi- nitions of sin z, cos z, that by the proper choice of the complex variable z either of the functions sin z and cos z can be made greater than unity in absolute values, thus differing in this respect from the case where the variable is real. Since e iz has the period 2 TT, it follows from the definition of sin z that it likewise has the period 2 TT; for, we have 2i = sinz. In a similar manner it may be shown that cos z has the period 2 IT; that is, that cos (z + 2 TT) = cos z. The remaining trigonometric functions are periodic, having the same periodicity as the corresponding functions of a real variable. As we shall see the lines that limit the fundamental region of sin z and cos z are parallel to the F-axis, while the fundamental region for e* is bounded by lines parallel to the axis of reals. This difference is a consequence of inserting the factor i before z in the definition of sin z, cos z. It is to be noted also that for the exponential functions e z , e" the region of periodicity is identical with the fundamental region; that is to say, no two points z in one of the strips defining the region of periodicity gives the same value of the function. In the case of sin z and cos z the situation is different and each of these functions has the same value for two different values of z in the strip defining the region of periodicity. For example, if we substitute n- z for z in gtZ g~" * z = ~^r~' e i(*-z) e -i(r-z) we have sin (T z) = - ^-. gtVg iz g iTgtz 2i Remembering that e tv _ COS7r _j_ i Sm7r = e -i r COS7r _ 146 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. we have sin (TT 2) = e~ 2i = sin 2. Moreover, the points representing z and TT z both lie within the region of periodicity x < x = IT, as shown in Fig. 63, provided the real part of 2 is greater than zero and not greater than TT. This result FIG. 63. FIG. 64. shows that the region of periodicity of sin 2 can not be taken as a fundamental region of that function. Again, if in gz _j_ g~* z COS 2 = = we replace 2 by 2, we have the same function as before. Hence, we may write COS ( 2) = COS 2. If 2 is represented by a point in the upper right-hand portion of the strip of periodicity, as shown in Fig. 64, then 2 is a point in the lower left-hand portion as shown. The two points lie symmetrically with respect to the origin. This fact shows that the region of peri- odicity IT < x = T does not answer the purpose of a fundamental region for cos 2. Neither the sine nor the cosine can have the same value for mare than two values of 2 in the same periodic strip. For example, for the cosine, we have e iz _|_ g-tf COS 2 = ~ 2e (1) ART. 31.] TRIGONOMETRIC FUNCTIONS 147 which is of the second degree in e"', and hence if we put cos z equal to a constant there are but two values of e iz and hence of z that sat- isfy this equation. As we have already seen, the regions of periodicity can not always be taken as the fundamental region. We shall show that the region bounded by the lines x = 0, x = x may be taken as the funda- mental region for cos 2. In this connection, we shall consider the mapping of the Z-plane upon the TF-plane by means of the relation w = cos z. We have e iz _|_ e -iz cos z w = u-\-w = - = - e ix e~ v + e~ iz e v Hence, we may write u = For x = 0, we obtain e~ v e v (cos x + i sin x) + -^ (cos x i sin x) py e cos x u = sns. rr v = sin x. (2) er* + e v v = 0. If y = 0, we have u equal to +1; for y < 0, we get u > 1. Hence, the axis of imaginaries maps into that portion of the t/-axis that lies to the right of u = 1, as shown in Fig. 65. For x = TT, we have u= v = and the line x = IT maps into that portion of the negative [/-axis that lies to the left of the point 1. If < x < IT, we have for a positive value of y a corresponding negative value of v; for, the value of sin x is positive, while the factor 148 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. is negative. In a similar way, if y is negative and < x < IT, the corresponding point in the TF-plane lies above the axis of reals. It will be seen upon inspection that for x = ~ , y = 0, the value of m w is zero. As x varies from to TT, y remaining zero, w takes all of FIG. 65. FIG. 66. the values represented by points on the real axis between +1 and 1 ; for, in this case we have from (2) u = cosx, v = 0. Lines parallel to the X-axis map into ellipses (Fig. 65) having the points 1 as the common foci. We may show this as follows. From (2) we get 2u -2v cosx = e y -|_ e -v > sinz = e v - Squaring both members of these equations and adding, we have 2u \* 2v e y \* -v) ~ For y = c this equation is that of an ellipse. For various values of c we obtain a system of ellipses having the common foci +1, 1. The lines parallel to the F-axis, that is x = c, map into hyperbolas having the foci 1. To get the equations of these hyperbolas, we divide the members of the first equation in (2) by cos x and those of the second by sin x, thus obtaining u cosx e~ v + e v v sins e -v - e v ART. 31.] TRIGONOMETRIC FUNCTIONS 149 Squaring these results, we have / u y = e~ 2 + 2 + be any point in the Z-plane. Suppose the parallels to the axes map into the particular ellipse and hyperbola shown; then cos z is repre- sented by the intersection of the curves as indicated. From the definitions of sin z and cos z, we can readily obtain ex- pressions for the other trigonometric functions in terms of the ex- ponential function. For example, we have sin z cos z tan z = - , cotz = - , etc. cos z sin z These functions are holomorphic in that portion of the finite plane for which they are defined. The first of these functions, tan z, is undefined for those values of z for which cos z vanishes. From the map of cos z upon the TF-plane, it will be seen that cos z is equal to zero only for the real values z = 5 for, k = 0, 1, 2, . . . . a In a similar manner, it may be shown that cot z is undefined for those values of z for which sin z vanishes, that is for the real values z = db for, k = 0, 1, 2, . . . . The trigonometric functions are therefore analytic functions. 150 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. 32. Hyperbolic Functions. As in the case of circular functions, we shall first define the functions w = sinh z, w = cosh z, and from these definitions deduce the remaining functions by means of the relations, A . sinh z 1 1 tanh = : , coth = - r , sech* = = cosh z ' tanh * ' cosh z cosech z = . sinh 2 We now define sinh z, cosh z in terms of the exponential function as follows: e* e~* e? -\- e~* ,, N sinh = = , coshs = = (1) & A By comparing these definitions with those of the sine and cosine, it will be seen at once that sinh z = i sin iz, cosh z = cos iz. (2) The following useful identities follow at once from the definitions given. cosh 2 z sinh 2 z = 1, coth 2 z cosech 2 z = 1. To deduce the first relation, we have (gz _j_ g-z\2 /gz _ g z\2 - 1 ( I Z I \ 2 I e~ 2 ' 4 4 = 1. The rest of the above identities may be deduced in a similar manner. The hyperbolic functions are analytic functions, since e z is an analytic function. Moreover these functions are periodic; for, as we have seen the function e z is periodic. The formulas for hyperbolic functions of real variables may be ART. 32.] HYPERBOLIC FUNCTIONS 151 extended without change to complex variables. We have, for ex- ample, cosh (21 + Zz) = cos i (zi + z 2 ) = cos izi cos iz% sin iz\ sin iz^ = cosh z\ cosh Zz + sinh Zi sinh z^. Hyperbolic functions of real variables may often be conveniently used to express the trigonometric functions of a complex variable in the form u(x, y) + iv(x, y). We have, for example, sin z = sin (x + iy) = sin x cos iy + cos x sin iy = sin x - cosh y + i cos x sinh y, whence u = sin x cosh y, v cos x sinh y. Similarly cos z = cos x cosh y i sin x sinh T/, and u = cos x cosh y, v = sin x sinh y. To express tan z in the form u -f- w, we have sin z sin (x + iy) tan 7 : r~i cos z cos (x + IT/) sin (x + M/) cos (x iy) cos (x + iy) cos (x iy) sin 2 a; + sm2iy _ sin 2 a; + i sinh 2 y \ v cos 2 x + cos 2 iy cos 2 z + cosh 2 y ' sin 2 a: sinh 2 v whence w = = , n , v = cos 2 x + cosh 2 y ' cos 2 x + cosh 2 y We shall now map the Z-plane upon the TF-plane by means of the relation w = cosh z. The results of this mapping can be readily deduced from those obtained in mapping by means of the relation w = cos z; for, it will be seen from (2) that we have w = cosh z if in w = cos z, z is replaced by iz. Hence, to map any configuration from the Z-plane to the TF-plane by means of the relation w = cosh z, all that is necessary is first to map the given configuration from the Z-plane to an auxiliary Z'-plane by means of the relation z' = iz, which merely rotates each point of the complex plane through a 7T positive angle ^, and then to map the resulting configuration from 152 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. the Z'-plane to the TF-plane by means of the relation w = cos z'. The region in the Z'-plane bounded by the lines x' = 0, x' = -w may be regarded as the fundamental region for the function w = cos z'. This region corresponds to the region in the Z-plane bounded by the lines y = 0, y = IT, which may therefore be taken as the funda- mental region for the function w = cosh z. As may be seen, any one of the regions bounded by the lines y = far, y = (k - 1) *-, k = , -2, -1, 0, 1, 2, . . . can be used as a fundamental region. A system of lines in the Z-plane parallel to either of the coordi- nate axes maps by means of the relations w = cosh z, w = cos z ifinto a system of straight lines in the TF-plane which are likewise par- l| allel to the coordinate axes. The lines that map in the one case into ellipses map in the other case into hyperbolas and conversely. This result is verified by a comparison of the equations for u and v in the two cases. We obtain from w = u + iv = cosh z u = cosh x cos y = ~ cos y, e* e~ x v = sinh x sin y = ^ sin y, which are the same equations as those obtained from w = cos z, Art. 31, except that x is replaced by y and y by x. In other words by the change from cos z to cosh z the lines of level and the lines of slope are interchanged. In a similar manner we may establish relations between the maps obtained by means of the remaining circular functions and the corre- sponding hyperbolic functions. From (2) it will be seen that the introduction of the factor i enables us to express any hyperbolic function in terms of the corresponding circular function. Conse- quently, it follows that the special significance of hyperbolic func- tions is confined to functions of a real variable. Because of the similarity of the configurations obtained by mapping the lines x = c, y = c, by means of the relations w = cos z and w = cosh z, it is to be expected that similar applications may be made in theoretical physics. If we have the case of a liquid flowing about an elliptic cylinder whose intersection by the complex plane is an ellipse having its foci at 1 and +1, respectively, then the ellipses, Fig. 65, are the lines of flow and the hyperbolas are the lines of equal ART. 32.] EXERCISES 153 velocity-potential. As a limiting case we have the flow of a liquid about a thin plate joining the points +1 and 1.* If that portion of the positive real axis lying to the right of +1 be regarded as a line source and that portion of the negative real axis lying to the left of 1 be taken as a sink the ellipses are again the lines of flow and the hyperbolas equipotential lines. If, however, the line joining +1 and 1 is regarded as a source, then the hyperbolas are the lines of flow and the ellipses are the equipotential lines. The definitions of the transcendental functions thus far discussed have been based upon the definition of e z , which in turn was defined in terms of known functions of real variables. Other methods of procedure could have been employed. For example, the logarithm C* dz of z could have been, and often is, defined as the integral I From this definition the properties of a logarithm can be readily developed. Then e z may be defined as the inverse function of log z, and the remaining functions can be defined as in the text. The other transcendental function which we have given may also be defined by means of integrals; for example, we may make use of the following relations as definitions f* dz C* dz arc tan z = \ - ; r , arc sin z = . , Jo 1 + Z 2 ' Jo Vl -z 2 upon which the definitions of the remaining functions discussed in the text may be based. EXERCISES 1. Discuss the conjugate functions determined by the relation tcP-c + 1. Plot the projections upon the .XT-plane of the lines of level and lines of slope. 2. Discuss the mapping upon the TF-plane of a system of concentric circles about the origin in the Z-plane, by means of the relation W = 323+5. 3. Show that the function z = r [ cos \l + i sin \t \ , where t is the independent variable representing time and where r and X are real constants, represents a movement of the z-point such that the velocity v of the z-point is constant in magnitude but varying in direction, and such that the acceleration of the 2-point is always directed toward the origin and is constant 2 in magnitude and equal to , a being the absolute value of v. * See Lamb, Hydrodynamics, 3 d Ed., p. 69; Webster, Electricity and Magnet- ism, p. 319. 154 MAPPING, ELEMENTARY FUNCTIONS [CHAP. IV. 4. Given w = e*. Suppose that the 2-point has a constant real velocity with a varying speed. 6. Any straight line through the origin making an angle different from zero with the X-axis crosses an infinite number of fundamental regions of the function w = e*. Explain the fact that such a line maps into a single continuous curve in the TF-plane. 6. Given u? = { 1 H J , n = 2, 3, . . . . Determine fundamental regions for this function for the various values of n and show how we may obtain a fundamental region for w = e z as the limiting case. 7. Show that e* is an automorphic function. 8. Construct the map of the function w sin 2 similar to the map of cos z shown in Figs. 65, 66. 9. Making use of Figs. 65, 66, and the figures obtained for the function tc = sin 2, construct the corresponding figures for the functions w = cosh 2, v> = sinh 2. j 10. Show that D z sin 2 = cos 2, D z sinh 2 = cosh 2. j 11. Show that sin 22 = 2 sin 2 cos 2; sinh 22 = 2 sinh 2 cosh 2. 12. Show that for w = sinh 2, we have u = sinh x cos y, v = cosh x sin y. 13. Show that for w = cosh 2, we have u = cosh x cos y, v = sinh x sin y. 14. Prove that sinh x cosh x + i sin y cos y tanas = - ...... cos 2 y cosh 2 x -f- sin 2 y sinh 2 x 16. Discuss the mapping of orthogonal systems of straight lines parallel to the axes in the TF-plane upon the Z-plane by means of the relation W = log (Z-1)(Z + !)(-) Discuss the possible applications in theoretical physics. 16. Discuss the function (2- I) 3 W = log ' , and point out possible applications as suggested by the map in the Z-plane of the lines u = c, v = c. Locate the points of equilibrium, if such exist. 17. Suppose a system of equipotential curves to be given by the confocal ellipses Show that the lines of flow are the confocal hyperbolas *Tx + &*~+^ - !. - 2 < x = -V- 4-1 ART. 32.] EXERCISES 155 18. Show, by the method of Ex. 3, that for uniform motion along any curve, the acceleration is always directed toward the centef of curvature and in magni- tude is equal to radius of curvature where a is the speed in the path. 19. Show that for non-uniform motion in any curve, the component of the acceleration normal to the path is _ (^ _ radius of curvature ' where a is the varying speed in the path, 20. The logarithmic function and the hyperbolic functions have been defined in terms of e*. By means of these definitions, show that w -\- 1 w -\- 1 z = log r = 2 arc tanh w. 21. By aid of the Cauchy integral formula, compute the value of sin z at the point P = ~, B = 5- . CHAPTER V LINEAR FRACTIONAL TRANSFORMATIONS 33. Definition of linear fractional transformation. In several of the illustrative examples of mapping thus far considered, the relation between w and z is such that a portion of the one plane maps into the whole of the other plane. For example, in the case of w = 2 2 one-half of the Z-plane maps into the whole of the TF-plane. To each point in the TF-plane there correspond then two points in the Z-plane, symmetrically situated with respect to the origin. We shall now consider the general linear algebraic relation between w and z. This relation may be written in the form az + /3 w = - ~> (1) 72 + 6 where a, /3, 7, 5 are constants, real or complex, and the determinant a ft 7 5 is different from zero. The relation (1) between w and z differs from those mentioned above in this respect that to each value of z there is, with a convention as to the point at infinity to be noted in the following article, one and only one value of w and vice versa. In our discussion of mapping (Chapter IV), we examined the relation between w and z by allowing one of these variables to de- scribe a given configuration and determining the corresponding configuration described by the other variable. The one configura- tion was then said to be mapped upon the other configuration by means of the given functional relation. In a similar manner a given region was frequently mapped upon another region. It was con- venient to represent the w-points in one complex plane and the 2-points in another, employing for this purpose two distinct sets of coordinate axes, one in each plane. In the present chapter we shall study the particular functional relation given in (1) from a somewhat different point of view. We shall represent both the z-points and the ttf-points in the same plane and refer them to the same set of 156 ART. 34.] POINT AT INFINITY 157 coordinate axes. We shall attempt to find a path along which any particular z-point may be regarded as moving in passing into the corresponding w-point. To distinguish this process from that of mapping already discussed, we shall speak of it as a transformation of the complex plane. Since equation (1) expresses w as a linear frac- tional function of z, we shall call the transformation to be discussed a linear fractional transformation of the complex plane. When the relation given in (1) reduces to the form w = az + /3, we shall speak of it as a linear transformation. If we think merely of the results of such a transformation, we may say that a given configuration is mapped upon the resulting configuration by means of a linear frac- tional transformation. 34. Point at infinity. The general linear fractional relation given in (1) of the last article fails to establish a complete one-to-one correspondence between the finite points of the complex plane. For example, there is no finite value of w corresponding to the value * z = . To make the one-to-one correspondence complete, it is 7 customary to assign an ideal point to the complex plane, called the point at infinity. To this artificial point may be associated the artificial number GO . The complex plane is to be regarded then as closed at infinity; that is, the plane is to be considered as having but a single point at infinity. S We set up a correspondence between the point z = and the ideal point w = oo , and likewise between the ideal point z = oo and point w = - . The one-to-one correspondence between the points 7 of the whole complex plane by means of the general linear fractional relation becomes complete by the aid of this convention; that is, to each point that may be assigned to z, there is one and only one point that represents the corresponding value of w, and conversely. This convention concerning the point at infinity is very convenient ia other connections. When we speak of the existence of the limit L /(z), where a is a finite number, it is implied that the function z=a /(z) has the same limiting value as z approaches a through all pos- sible sets of values, that is, a limiting value that is independent of any changes in the amplitude of z a as the modulus of z a de- creases. Similarly, a function f(z) may have a unique limiting value as z becomes infinite through all possible sets of values; that is, it 158 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. may have a limiting value that is independent of any changes in the amplitude of z as the modulus of z increases beyond all finite bounds. It is convenient to denote this limit by L /(z) and to speak of z = oo Z = oo as the limiting point in this case. Let us now consider the limiting value of the function w = 7Z + 5 as z becomes infinite and likewise that of the inverse function yw a as w becomes infinite. We have , az + /3 a. L w = L - r-^ = - , *=, *=, 7Z + 5 7 and -dw + P _ 8 Li Z = Li - = -- tc=, w=> yw a 7 These two results fully justify, in so far as the general linear fractional function is concerned, the convention introduced concerning the nature of the complex plane at infinity. If we think of the values of z as represented by the points of one complex plane, called the Z-plane, and the values of w as represented by the points of another complex plane, called the TP-plane, then of course an ideal point at infinity must be associated with each plane. We may speak of the neighborhood of the point at infinity just as we speak of the neighborhood of any finite point. By such a neighborhood is understood the set of points exterior to any closed curve, for example the set of points exterior to a large circle about the origin as a center. We say also that a given region contains the point at infinity if it consists of all the points exterior to a given closed curve. In this connection the substitution z = - , is of special z importance. By this transformation every finite point except the origin goes over into some finite point of the plane. The neighbor- hood of the origin corresponds to the neighborhood of the point at infinity, and the origin itself may be said to correspond to the point at infinity. This relation between the point at infinity and the origin affords a convenient method of investigating the properties of a function AKTS. 35-36.] W = Z + j8, W = aZ 159 f(z) for values of 2 in the neighborhood of the point z = oo . To do so we replace in /(z) the independent variable z by . and discuss the 2 , properties of the transformed function 0(0') in the neighborhood of the point z' = 0. If the limit L 0(2') exists and is equal to A, then 2'=0 we say that f(z) has the value A at the point at infinity, and we write A = L /( Z ) =/(). Z=oo If 0(2') is continuous for z' = 0, we say that f(z) is continuous at infinity. If z' = is a regular point of 0(2'), we say that 2 = oo is a regular point of f(z). As 2 becomes infinite the function f(z) may also become infinite and in such a manner that 7p- approaches the j( z ) limiting value zero. We say then that /(oo) = oo. 35. The transformation w = z + p. Before discussing the gen- eral transformation (1) of Art. 33, we shall consider some special cases that are of particular importance and first of all let us consider the transformation w = z + 0. To obtain this function from the general case, put a = 5 = 1 and 7 = 0. This relation indicates that to each number 2 there is added another number /5. From the geometric interpretation of addition it will be seen that the 2-points are transformed into the correspond- ing w-points by moving each 2-point in the direction of the line joining the point /3 with the origin and to a distance equal to [ /S | . In what follows, it frequently will be convenient to describe a trans- formation of the complex plane as a continuous motion, meaning thereby that all of the points of the plane are considered as moving continuously from their initial to their final positions along a system of curves. The motion just described is called a translation. It will be seen that a translation of the complex plane leaves the form and size of any configuration unchanged; that is, it transforms any curve into a congruent curve. 36. The transformation w = az. As already stated a may be a real number or a complex number of the form a = p (cos 6 + i sin 6). 160 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. We shall understand more easily the full significance of this trans- formation by considering first some special cases. Let us suppose for example that 6 = 0; that is, let a be considered as a real positive number. The result is simply a multiplication of the modulus of z by the number p = \a\. Geometrically, this special form of the gen- eral transformation may be regarded as moving every point along the line passing through the given point and the origin, that is, along the half-ray from the origin on which it lies. The point moves out or in along this half-ray according as p is greater than or less than unity. This change affects every point of the complex plane, and we may re- gard the transformation as representing a motion of the points of the plane. We shall refer to such a motion as an expansion or stretching. We are concerned here with the path by which the variable point may be regarded as passing from its initial to its final position, rather than the velocity with which this motion takes place. The number p is called the modulus of expansion. The significance of p may also be seen from a consideration of the derivative. As we have seen, | D z w \ gives the ratio of magnification , that takes place in infinitesimal ele- ments as z varies. We have \D,w\ = \D.(az)\ = \a\ = P ; that is, any configuration in the com- plex plane is magnified in this ratio. Suppose that we have a system of concentric circles about the origin and a pencil of rays passing through the origin. Each half-ray remains un- changed as a whole, although any particular point upon it is moved out or in according as the ratio of expan- sion is greater or less than unity. By this transformation, any portion of the plane inclosed by two half-rays and two concentric circles is transformed into another portion bounded by the same two half -rays and the concentric circles into which the first two are transformed; for example, the region (2), Fig. 67, goes over into (3). Each dimen- sion of the region has been multiplied by the number p. Suppose we now allow 6 to vary, while p remains constantly equal to one. Then by the laws of multiplication already established, we FIG. 67. ART. 36.] THE TRANSFORMATION W = aZ 161 obtain from any value z the corresponding value of w by adding to the amplitude of z the angle 0. Inasmuch as p = 1, no magnification takes place and the resulting configuration is obtained by revolving each point z about the origin counter-clockwise through the angle 6. Considered from the standpoint of the geometry of motion, this transformation may be regarded as a rotation of points of the plane. Such a motion converts the region (3), for example, into the region (4), as indicated in Fig. 67. The concentric circles about the origin then become the lines of motion. Each ray is converted into another ray at an angular distance 6 from it. Let us now consider the general case where a is any complex num- ber. Both p and 6 may have any constant values. We have then a combination of the two special cases already considered; that is, the point is rotated about the origin through the angle 6 while it is at the same time moved along the ray on which it is rotated; for, if we have a = p(cos 6 -\- i sin 6), z = r(cos < + i sin <), then by multiplication we get w = rp(cos 6 -f- + i sin 9 + ); that is mod w = r p, amp w = + . (1) The result of the transformation w = az may be obtained, as we shall now show, by regarding each point as moving along a logarithmic spiral whose asymptotic point is at the origin. For this reason it is convenient to describe the given transformation as a logarithmic spiral motion about the origin. Let w = r'(cos (j> r -f- i sin <') be the point into which the point z = r(cos + i sin 0) is mapped by means of the given transformation w = az. From (1) we then have / = r - p, $' = + 6. A logarithmic spiral passing through z may be written in the form r = ce 1 *. 162 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. In order that this same curve shall also pass through w, it is sufficient that r ' = ce**' that is, it is sufficient that T = T p= re , or k= ~e~ Hence, if the logarithmic spiral whose equation is logp r = ce * ' (2) passes through the point z it also passes through the point w into which z is mapped by the given transformation. Consequently, the given z-point may be regarded as passing into the corresponding w-point by a motion along this spiral. As p and 8 are both determined by a, it follows that a determines the particular system of spirals given in (2) along which any z-point may move into the corresponding 10-point. The arbitrary constant c is the parameter of the system, and to each point z there corre- sponds one and only one value of c and hence one and only one logarithmic spiral of the system. The entire plane is filled by this system of curves, coiled up within each other. 37. The transformation w = az + p. The significance of this transformation may be most readily seen by regarding it as a com- bination of the two preceding transformations. Let a logarithmic spiral motion take place about the origin, and then let the result be translated by adding the number ft. Analytically, this result is equivalent to introducing the auxiliary variable z', defined by the equation / ,-,\ Z = + r v _ x + { y - x ~r m 9 i .o 168 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. or equating the real parts and the imaginary parts, we get _/ - x Solving these equations for x and y, we have ) ~ 3/2 _|_ y'2 > z'2 + y'Z These are the values of x and y which, if substituted in the equation of a given curve, give the equation of the inverse curve. Ex. i. Find the curve into which a straight line not passing through the origin is mapped by geometric inversion. The equation of the given line is Ax + By + C = 0, C * 0. Substituting for x, y their values from (2) we obtain Ax' By' , 2 T X >2 + y'Z x C(x'* + j/' 2 ) + Ax' + By' = 0. This is the equation of a circle passing through the origin. The equation of the tangent to this circle at the origin is Ax + By = 0, which is a line parallel to the given line. Therefore, a system of parallel lines in- verts into a system of circles having a common tangent at the origin. For C = 0, we have the special case of a line through the origin already discussed. Ex. 2. Find the curve into which a circle not passing through the origin is changed by inversion. The equation of the given circle is of the form C = 0. Substituting the values of x, y from (2), we have ** y'* 2gx' 2fy' ( X 't _|_ y>,) t T (X >2 + y'ly "I" X '2 + y>2 + ^ + y , 2 t or COc" +Y 2 ) + 2 gx' + 2fy' + 1=0, which is the equation of a circle not passing through the origin. In the special case where C = 0, we have a straight line not passing through the origin. If we now think of a straight line as a circle of infinite radius, we may then make the general statement that by geometric inversion every circle is converted into a circle. ART. 38.] THE TRANSFORMATION W = Z 169 FIG. The angle at which two curves cut each other is preserved by geometric inversion, but the direction of the angle is reversed; that is, the angle is measured in the opposite direction after inversion. We shall first show that this statement holds when one of the given curves is a straight line passing through the center of inversion. Let A,A',B,B' (Fig. 69) be two sets of points which are inverse with respect to 0. Lines through A, A' and B, B' pass through 0. Moreover, we have OA OA' = OB - OB' = 1. The angle at is common to the two triangles OAB and OA'B', and as the sides of the common angle are proportional, the two triangles are similar. Conse- quently, AOAB = /.OB' A'. (3) If now we think of the points A and B as situated on some curve, the points A' and B' will lie upon the inverse curve. Let the point B approach A as a limit. Then the point B' approaches the point A' along the corresponding curve. The lines AB and A'B' become the tangents AT and A'T' to the two curves at the corresponding points A and A', respectively. In the limit, therefore, the angle OB' A' becomes the angle vertical to A A'B' and hence equal to it. From (3) we then have in the limit LOAT = LAA'T'. The line OA is its own inverse since it passes through the center of inversion, and by hypothesis the curve A'B' is the inverse of the curve AB. By inversion the angle that the tangent to the curve AB makes with OA, measured in a clockwise direction, namely /.A' 'AT, is changed into the equal angle /.A A'T made by the tangent to the inverse curve A'B' with OA, measured in a counter-clockwise direction. Suppose we now consider the case of any two curves intersecting in a point A. The inverse curves will intersect in a point A' which is the inverse of A. To extend the argument to this case, draw a straight line through A, A'. It will pass through the center of hi- 170 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. version. Consider the angle made by the tangent to each curve with this line at the point of intersection. From the foregoing discussion this angle is preserved in magnitude but reversed in direction by in- version. By combination of these angles we have the desired result; that is, by geometric inversion angles are preserved in magnitude but reversed in direction. Inversion is therefore a conformal transformation but with a reversion of any given angle. Reflection upon a straight line is likewise a process that involves a reversion of angles. As we have seen, the transformation w = - is made up of a geometric inversion z and a reflection upon the axis of reals. When we combine these two processes we have a process in which these two reversions annul each other. Hence, we can say that the transformation w - is conformal z without reversion of angles. We have thus far confined our discussion to geometric inversion with respect to the unit circle, because of the fact that inversion with respect to this circle is involved in the transformation w = -. This 2 restriction, however, is not essential to the geometry of inversion. We may define inversion with respect to a circle of radius k by merely replacing the above condition pp' = 1 by the more general one pp' = k 2 . To show that the same geometric properties hold for the general case suppose we think of the whole plane as being so expanded or contracted about the origin that the unit circle changes into the required circle of radius k. Any two corresponding points P, P' with respect to the unit circle become two corresponding points Q, Q' with respect to the required circle. If pi, pi are the radii vectores of the points Q, Q' , then we have Pi PI = kp - kp' = *v -* as was required. Two corresponding points with respect to a circle of inversion are called conjugate points. The conjugate of any particular point with respect to a given circle may be found geometrically as follows. Draw two tangents from the given point A to the given circle (Fig. 70). Join the given point with the center of the circle. Connect the points of tangency B and C. The intersection of the chord BC ART. 38.] THE TRANSFORMATION W = Z~ l 171 and the line OA gives the required point A'. For, from the figure, the triangle AOB is a right triangle, having a right angle at B, and hence we have _ = A'O-AO, showing the two points A and A' to be conjugate points. The foregoing construction holds when the given point A lies without the circle of inversion. If the given point lies within the circle of inversion the conjugate point may be found as follows. Connect the given point A with the center of the circle; through A draw a line perpendicular to AO, and at the points where this FIG. 70. FIG. 71. perpendicular intersects the circle draw tangents to the circle. The point of intersection of these tangents gives the required point A'. The proof is similar to that in the previous case. The following theorems give additional important properties of inversion. THEOREM I. If a given circle cuts the circle of inversion in two points A and B, then its inverse cuts the circle of inversion in the same two points. The truth of this theorem is seen at once from the fact that the points of intersection of the given circle with the circle of inversion are points on the circle of inversion and therefore necessarily invert into themselves. It does not follow, of course, that the given circle as a whole inverts into itself. THEOREM II. If a given circle cuts the circle of inversion at a given angle, then its inverse cuts the circle of inversion at the same angle. 172 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. The theorem follows from the fact that the magnitude of an angle is preserved by the process of inversion, and hence the angle at which the given circle cuts the circle of inversion remains unchanged in magnitude. It is, however, reversed in direction. COROLLARY. // a given circle cuts the circle of inversion at right angles, then the given circle is identical with its inverse and any straight line through the center of inversion cuts the given circle in two conjugate points, If the given circle cuts the circle of inversion at right angles, then by Theorem II its inverse also cuts the circle of inversion at the same angle, and since through two points on a circle but one orthog- onal circle can be drawn, the given circle must be identical with its inverse, as the theorem requires. The only change that takes place in the given circle is that the portion of the circle without the circle of inversion becomes after inversion the portion within the circle of inversion. Since the given circle inverts into itself, it follows that any straight line passing through the origin cuts the given circle in conjugate points. THEOREM III. Given a pair of conjugate points with respect to a fixed circle. Any circle through these points inverts into itself with respect to the fixed circle and cuts that circle at right angles. One of the two conjugate points must lie within and the other without the circle of inversion. Consequently, the given circle cuts the fixed circle and the two points of intersection invert into them- selves. These points of intersection and the two given conjugate points make together four points that the given circle and the in- verted circle have in common. Hence, the two circles must coincide. By Theorem II the given circle and the inverted circle cut the circle of inversion at the same angle but reversed in direction. But as the inverted circle is identical with the given circle each must then cut the circle of inversion at right angles. THEOREM IV. Given a system of circles such that each circle passes through two given points and intersects a fixed circle at right angles. The two given points of intersection of the system of circles are then con- jugate points with respect to the fixed circle. Let the circles of the system be inverted with respect to the given fixed circle M. By the corollary to Theorem II, each circle of the system inverts into itself. It is sufficient for our purpose to con- ART. 39.] GENERAL PROPERTIES 173 sider two circles C\, Cz of the system. The point P of intersection lies on both C\, and C 2 . After inversion with respect to M, the point P must go into a point within M which likewise lies upon both Ci and Cz- It must, therefore, invert into the second point of intersec- tion of these two circles, namely P'. Hence the theorem. THEOREM V. Given two conju- gate points with respect to a given circle. If the circle is inverted with respect to a fixed circle, the given conjugate points invert into conju- gate points with respect to the in- verted circle. F IGt 72. Let M be the given circle and P and P' two conjugate points with respect to it. Suppose the circle M inverts into the circle M' with respect to the fixed circle C, and the points P, P' invert into Q, Q', respectively. It is required to show that Q, Q' are conjugate points with respect to M'. Draw any two circles through the con- jugate points P, P'; these circles cut the given circle M at right angles. These angles are pre- served by inversion. Hence, the circles through the given conju- gate points and cutting M at right angles invert into circles cutting M' at right angles. Since the inverse points of P, P f , namely the points Q, Q', must lie at the respective intersections of these inverted circles, it fol- lows from Theorem IV that the points Q, Q' are conjugate points with respect to the circle M'. With this our theorem is demon- strated. as -f 8 39. General properties of the transformation w = . We shall now consider the general case of a linear fractional transfor- FIG. 73. 174 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. mat ion. We impose the condition upon the four constants a, ft, y, 8, that a ft y d 0. (1) If this determinant were equal to zero, we should have = - and the given relation between w and z would then reduce to a W = 7 and all points in the Z-plane would correspond to the same point - in the TF-plane. By imposing the condition (1), we are able to . set aside this trivial case. The general linear fractional relation may be decomposed into the three following special cases, namely: (1) Z' =2+-, 7 (3) Wa= This statement can be easily verified by making the substitutions indicated and thus obtaining the general linear fractional relation between w and z. Geometrically, we may then consider the general linear transformation as made up of the following: (1) a translation, (2) a geometric inversion followed by a reflection on the axis of reals, (3) a rotation and a stretching followed by a translation; or what is the same thing, a logarithmic spiral motion about the point left invari- ant by the third of the foregoing transformations. As we have already considered each of these operations, we can now formulate some of the general properties of a linear fractional transformation. Among these properties are: THEOREM I. Conjugate points with respect to a given circle are transformed by the general linear fractional transformation into conju- gate points with respect to the transformed circle. ART. 39.] GENERAL PROPERTIES 175 We have seen (Theorem V, Art. 38) that conjugate points with respect to a given circle remain conjugate points by inversion. Since reflection upon the axis of reals does not disturb the relative position of points of a given configuration except to reverse the direction of the angles,- we may conclude that the theorem holds for the special transformation w = - ; that is, it holds for the transformation (2) z given above. It also holds for the transformations (1) and (3) since by both these transformations the similarity of the configuration is preserved. As the general linear transformation is decomposable into these three special transformations, for each of which conjugate points remain conjugate points, the theorem follows as stated. THEOREM II. Any given configuration is mapped conformally, with- out reversion of angles, by means of a linear fractional transformation. This theorem follows from the fact that each of the three simple transformations into which the general linear fractional transforma- tion may be decomposed is such that the conclusions stated in the theorem hold. Since w is holomorphic for all values of z in the finite region except for z = -- , this same result may be obtained independently by the consideration of D z w. We have a8 0y But by hypothesis a8 - 07 ^ 0. Hence, by the theorem of Art. 27, the desired result follows. THEOREM III. By the general linear fractional transformation, circles are converted into circles. It is here understood that a straight line is to be considered as a circle of infinite radius. The truth of the theorem follows from the fact that it holds for each of the three special transformations into which the general linear transformation may be decomposed. THEOREM IV. The general linear fractional transformation leaves two points in the complex plane invariant. To establish this theorem, we proceed as follows. If any point 176 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. 2 of the complex plane is transformed into itself by means of a linear fractional transformation, then we must have that is 7 2 2 + (6 - a) 2 - |8 = 0. (1) This equation is a quadratic and has therefore two roots, namely : - (a- 5) - V(a-3 ,~ Zl= ~27" ~'^ = ~27~ The two points z\, z* remain unchanged by the general linear frac- tional transformation, since each is transformed into itself. These invariant points may be finite and distinct, finite and co- incident, one finite and the other infinite, or finally, both may be infinite. The analytic conditions for these various cases may be expressed in terms of the coefficients of (1). If the discriminant vanishes, that is if the two roots of (1), that is the two invariant points, are coincident. If in addition we have 7 = 0, it will be seen from (1) that both roots of (1) become infinite; that is, both invariant points coincide at the point infinity. If 7 5^ the two points 21, 22 lie in the finite region of the plane. If we have (a -6) 2 the roots of (1), that is the invariant points, are distinct. If hi addition 7 = 0, one of the roots of (1) becomes infinite and hence one of the invariant points is at infinity. It will be observed that when 7 = the linear fractional transformation reduces to the general linear transformation. The general linear fractional transformation contains four con- stants; but as we may divide both numerator and denominator by one of these without affecting the transformation, we have only three independent constants. We may state the following theorem. THEOREM V. There is always one and only one linear fractional transformation that transforms any three distinct points into three given distinct points. ART. 39.] GENERAL PROPERTIES 177 Let 21, z z , 2 3 be the three distinct points that are to be transformed into the given distinct points w i} w z , w 3 . We must then have the three relations Wk = + 5 ' k = 1, 2, 3; that is, 4- 5v>k az k /3 = 0. (1) We have given three linear homogeneous equations in the four unknowns a, /3, 7, 5. The condition that these equations have one and only one solution other than a = /3 = 7 = 5 = 0is that the matrix of the coefficients, or its equivalent matrix, w\z\ Wi z\ 1 10222 Wl 22 1 10323 w 3 z 3 1 (2) shall be of rank three; * that is, that not all of the determinants formed from this matrix by dropping one column shall vanish. We shall show that this condition is satisfied by showing that the two determinants Wi 22 1 Wz 2 3 1 Ao = 22 1 W323 2 3 1 can not vanish simultaneously. Expanding each determinant in terms of the elements of the first column, we have Ai = Wi(Zz 2 3 ) 102(21 2 3 ) + U> 3 (2i 22), A 2 = t0i2i(Zz 2s) ^222(21 2>) + W32 3 (2i 22). Multiplying the first of these identities by z\ and subtracting the second from that result we get 22) (Zi Zs) + tt> 3 (2i 22) (2i 2 8 ) A 2 = = (w 3 22) (21 - 23). (3) Since the points Zi, 22, 2 3 and Wi, w 2 , w 3 are distinct, it follows that (3) can not vanish. Hence, we have A 2 7 0, and consequently the two determinants Ai, A2 can not vanish simul- taneously. * See B6cher, Introduction to Higher Algebra, Art. 17. 178 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. Since the equations (1) have one and only one solution other than a = (3 = y = 8 = Q, it follows that any three ratios of these un- knowns are uniquely determined. Consequently there is one and only one transformation of the required type which transforms the three distinct points Zi, 22, z$ into three distinct points w\, Wz, ws. Hence the theorem. Remembering that three distinct points definitely determine a circle we may now say that any circle can be transformed into any other circle, or into itself, by means of a linear fractional transfor- mation. Since the three points upon the given circle can be selected in an infinite number of ways, it follows that the required transfor- mation can be made in an infinite number of ways. If it is desired to transform four points into four points, we must have an additional condition satisfied. If we have given any four points Zi, Z2, zs, z\ the ratio Zi 2g . Z\ Z4 23 22 23 24 is called the anharmonic ratio or cross-ratio of these four points. The following theorem gives the condition which must be satisfied in order that any four distinct points Zi, 2 2 , 23, 2 4 may be trans- formed into four distinct points w\, w 2 , wz, w* by a linear fractional transformation. THEOREM VI. The necessary and sufficient condition that any four distinct points of the complex plane may be transformed by a linear fractional transformation into any other four distinct points of the plane is that the anharmonic ratio of the two sets of points is the same. Let the four given points be Zi, 22, 2 3 , 2 4 and let it be required to transform these points into the four distinct points w\, w*, w s> w*. If the four given 2-points are transformed by a linear fractional transformation into the four given w-points, then we must have the four relations or &Wk az- j8 = 0. The necessary and sufficient condition that these four equations have ART. 39.] GENERAL PROPERTIES 179 a solution other than a = /3 = 7 = 5 = 0is that the determinant of the coefficients shall vanish; that is, that we have * = 0. Expanding this determinant in terms of the last two columns by Laplace's development, we have 10l2i r - Zi 1 WiZi Wi 2i 1 10222 102 2 2 1 10222 Wz 22 1 10323 103 2 3 - 1 10323 W3 2 3 1 1042 4 104 - 2 4 - 1 1042 4 104 2 4 1 2 3 1 2 4 1 21 1 22 1 10310.4 Making use of the identity 21 1 2 2 1 2i 1 ^ 22 1 2 3 1 2 4 1 2 4 22~1 I 2 ' 1 2 4 1 [23 1 1 3 y * 2i 1 2 3 1 2 2 1 24 1 + ? 24 + 21 1 t 24 I 22 1 ft 1 22 1 ., 23 1 2i 1 z, 1 = 0, w z w 3 :-*-. we may write the foregoing relation in the form (Zi 22) (2 3 2 4 ) \ (l0 3 104 + WiWz) (l0 2 104 + 101103) I + (2i 2 4 ) (22 2 3 ) { (l0 2 103 = (2i 2 2 )(2 3 2 4 )(l0 X + (Zi 2 4 )(2 3 22)(l0i 10 2 )(l03 Wi) = 0, whence we get 2i 2 2 . 2i 2 4 _ 10i 102 . 101 104 . 23 2 2 * 2 3 2 4 103 10 2 ' 103 W t ' that is, the anharmonic ratio of the four points z\, z?, 23, 2 4 is the same as the anharmonic ratio of the four points w\, 102, 103, 104. As this result presents the necessary and sufficient condition that the equa- tions (4) have a solution other than a = /3 = 7 = 5 = 0, it follows that this result also gives the necessary and sufficient condition that the one set of four points may be mapped by a linear fractional transformation into the other set. Consequently, the theorem fol- lows as stated. It may be remarked that as a consequence of the foregoing theorem a linear fractional transformation has the property that it leaves the anharmonic ratio of any four points invariant. If the order in which the four given points are taken is changed * See B6cher, Introduction to Higher Algebra, Art. 17, Theorem 3, Cor. 2. 180 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. then the anharmonic ratio may be changed. Of the twenty-four ways in which four points may be selected, only six give distinct anharmonic ratios. If we denote any one of these ratios by X, then the six are given by * I _x -1- -A_ x ~i X' 1-X' X-l' X That X is in general a complex number follows from the fact that it is defined as the ratio of such numbers. It may therefore be repre- sented as a point in the complex plane. Since any two of these six anharmonic ratios are linearly related, the geometric interpretation of these relations furnishes an interesting exercise in the application of the principles developed in this chapter. If X describes a circle in the complex plane, then the points repre- senting respectively the various ratios likewise describe circles. Moreover, if X is represented by points within a given region bounded by a circle, it follows that the other ratios are represented by points within regions bounded by circles. It is possible to so choose the region for X that the entire complex plane shall be filled by the regions of the six ratios without overlapping. The relative position of these regions may be found as follows. With x = and x = 1 as centers describe two unit circles (Fig. 74). These circles intersect at the points whose coordinates x, y satisfy the two equations x 2 + y z = 1, (x - I) 2 + y 2 = 1. By solving these equations, we have The points of intersection are, therefore, w 2 and o>, where - 1 + i \/3 TT is one of the cube roots of unity. If X takes the values in the unshaded region (0, 5, 2 ), Fig. 74, then- is confined to the region found by A inverting this region with respect to the center and reflecting the result upon the real axis. The numbers X and 1 X are symmetrical with respect to the point ~- The region for ^ - - may be obtained A L A * See Scott, Modern Analytical Geometry, p. 37. ART. 39.] GENERAL PROPERTIES 181 from that for 1 X by inverting this region with respect to the circle about the origin as a center and reflecting the result upon the axis of reals. From the region for - we may find the region for A A be- FIG. 74. cause these two numbers are symmetric with respect to ?. In this way we find the regions described by the various complex numbers as shown in the figure. If X is represented by the points of a shaded region, then the points representing the other five anharmonic ratios are confined to the shaded regions. There are two important special cases of anharmonic ratios. One of these cases is obtained if X has such values that X = - and hence A X = 1. For X = 1, the four points are said to be harmonic. The six ratios are then coincident in pairs. When X is a complex number, as in the present discussion, it is possible for three of the anharmonic ratios to be equal. For ex- ample, it wfll be seen from the figure that the three ratios X, - _ > - may become equal at the common point, 182 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. The reciprocals of these values, that is -, 1 X, - - r, then become A A ~~ 1 equal at 1 -i This equality leads to the second special case of anharmonic ratios: for, putting 1 X - 1 x = r^x = ~Y"' we have X 2 - X + 1 = 0, whence which are the two imaginary cube roots of 1. When X has either of these values the four points are said to be equianharmonic.* If the variables and constants involved in a linear fractional trans- formation are all real, the property that anharmonic ratios are pre- served is commonly spoken of as a protective property; in fact this property may be made the basis of projective geometry. The rela- tion between anharmonic ratios and linear fractional transformation, as established in Theorems V and VI, suggests the extension of projec- tive geometry to the field of complex numbers. In the one case the single variable x takes the totality of real values and the ideal num- ber oo, represented by the points on a straight line including the point at infinity. As a result, we have the projective geometry of a straight line. In the other case the single variable z takes the total- ity of complex numbers and the ideal number oo . Since but a single variable is involved this aggregate is sometimes spoken of in pro- jective geometry as the complex line. This extension of projective geometry to the realm of complex numbers leads to the consideration of the theory of chains, f but, as no use will be made of this theory in the present volume, it will not be considered here. In this connection it is also of interest to point out the general relation between the totality of linear fractional transformations and the theory of groups. We have for example the following theorem. * For a more extended discussion of these cases see Harkness and Morley, Treatise on the Theory of Functions, p. 21, et seq. t For a discussion of this subject, see J. W. Young, Annals of Math., Vol. II, pp. 33-48. ART. 39.] GENERAL PROPERTIES 183 THEOREM VII. The system of linear fractional transformations possesses the group property. The statement contained in the theorem involves the condition that if a linear fractional expression in one variable is subjected to a linear fractional transformation, the resulting expression is a linear fractional expression. Given the relation w = a.\z 7i Suppose z' is associated with z by the relation z' = 722 72 The theorem requires that w be expressed as a linear fractional function of z, where the determinant of the coefficients is also differ- ent from zero. We have w = 722 , T Pi 71 + 722 0i72) 2 + (ai0 2 which is a linear fractional expression in z. The determinant of the coefficients is different from zero; for, we have 0i72 0i 7i 02 72 and each determinant in the second member of this equation is differ- ent from zero by hypothesis. The system of linear fractional transformations possesses the other characteristic properties of a group,* and the relation to the theory of groups is at once established. If a, 0, 7, 5 are integers such that a 7 5 then the transformation w = oz 72 See Bocher, Introduction to Higher Algebra, p. 82. 184 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. defines the modular group.* Many of the properties of linear frac- tional transformations that have been discussed follow also as applica- tions of group theory.f 40. Stereographic projection. Since complex numbers are of the form 2 = x + iy, where x and y may vary independently of each other, two degrees of freedom are necessary for the geometric element used to interpret them and the plane naturally suggests itself for that purpose. Thus far we have restricted ourselves to this mode of representation. There are other ways, however, of representing complex numbers and other surfaces than the plane have been made use of in this connection. It is frequently convenient to employ the sphere for this purpose. In order to do so, it must be possible to establish in some way a one-to-one correspondence between the points of a plane and those upon the sphere. The desired result may be accomplished by assuming the complex plane as before and supposing that we have a sphere tangent to this plane at the origin. We shall refer to the point of tangency as the south pole of the sphere, while the opposite pole will be spoken of as the north pole. If we now take the north pole 0' as the center of projection we can project in a definite manner every point of the plane upon the sphere. Thus in Fig. 75 the point P in the complex plane corresponds to the point P f of the sphere. In this way there corresponds to each point of the plane a definite point of the sphere, and conversely. This method of mapping the complex plane upon the sphere is called Stereographic projection. Since there is a one-to-one correspondence between the points of the complex plane and those of the sphere the values of z and of w = /( 2 ) may be uniquely represented upon the sphere, which we shall refer to as the complex sphere. For example, if z describes a continuous curve in a region of the complex plane in which w = f(z) is holomorphic, then w likewise describes a continuous curve. The projection of these two curves upon the sphere gives the interpreta- tion upon that surface of the relation between w and z. The point at infinity in the complex plane projects into the north pole of the sphere. Hence, to examine the nature of a function for values of the variable in the neighborhood of the point at infinity, it may often be convenient to represent both w and z upon the complex sphere and inquire into the behavior of w as z takes values in the * See Forsyth, Theory of Functions, 2<* Ed., pp. 680, 681. f See Kowalewski, Komplexen Veranderlichen und ihre Funktionen, pp. 30-59. ART. 40.] STEREOGRAPHIC PROJECTION 185 neighborhood of the north pole. The same result, of course, could be obtained analytically. Corresponding to the coordinates x, y of a point in the plane, we may determine the location of a point upon the sphere by means of two coordinates 6, 0, one measured along FIG. 75. some standard meridian and the other along the equator. Such a system of coordinates is a familiar one in the location of a point upon the earth's surface by means of its longitude and latitude. Any given curve can be mapped from the plane upon the sphere by means of the analytic relation between x, y and the coordinates of the corresponding point on the sphere, and the transformed function thus obtained can be studied for values of 6, in the neighbor- hood of the north pole. A closed curve upon the sphere divides the surface of the sphere into two parts. This curve may be regarded as the boundary of either of these regions to suit our convenience. It is desirable to examine somewhat more closely into the effect of stereographic projection upon the character of a configuration. First of all, suppose we have a pencil of rays passing through the origin and lying in the complex plane. Each of these rays projects into a great circle passing through and 0' . They become meridians upon the sphere and one such meridian passes through each point 186 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. upon the sphere. As a special case the axis of reals projects into a meridian of reals and the axis of imaginaries projects into a meridian of imaginaries cutting the meridian of reals at right angles. If we have a system of concentric circles in the plane having the origin as center, they constitute the orthogonal system to the pencil of rays just mentioned. These circles go over into the orthogonal system of circles on the sphere, namely, the parallels of latitude. One of these circles projects into the equator of the sphere. This circle may be conveniently selected as the unit circle in the plane. All concentric circles lying within this unit circle will become parallels of latitude in the southern hemisphere while those lying outside of this unit circle pass over into parallels of latitude in the northern hemisphere. In order to determine the character of a configuration on the sphere and its relation to the corresponding configuration in the plane, we shall now deduce the equations of transformation by means of which the cartesian space coordinates of any point upon the sphere can be expressed in terms of the cartesian coordinates of the corresponding point in the plane. Let , TQ, f denote the co- ordinates of a point on the sphere. Let the -axis and the -q-axis coincide respectively with the axis of reals and the axis of imagi- naries of the plane. Let the f-axis be perpendicular to the complex plane. Suppose the radius of the given sphere to be |. The equa- tion of the sphere is (r-i) f = i, (i) or 2 + T] 2 + f(r-l) = 0. (2) If we now denote by x, y the coordinates of any point P in the plane, the coordinates , r h of the projection P' upon the sphere of the point P are readily found in terms of x, y. From Fig. 75 we have OP 2 = z 2 + y\ (3) cpp 2 = ov 2 + OP' = x* + y* + 1, (4) = 2 + Tj2> ( 5 ) where DP' is drawn parallel to OP. The triangles OPO' and DP'O' are similar, and consequently we have OP_ DP' OP W' ART. 40.] STEREOGRAPHIC PROJECTION 187 By use of (3), (4) and (5), we obtain ~2~~i 2~i T = ~ 5~" (6) x 2 + y 2 + 1 Qip z As OP'O' is a right triangle, we have O'P' = DO' 00' = DO' - 1. Since DO' = 1 f , we have O'P' = 1 - f . (7) From (6) we have then 2 2 * 2 , . z 2 + i/ 2 + 1 1 - We have also ' =DO'-OD, from which we have or r-yfy = x , + [/+ i ( 9 ) Finally, we have t T I - y (10) or = -TJ. (11) y By use of (9) and (11) the equation (2) of the given sphere may now be written :_i x ~^~ y . _! = o or whence TJ = From (11) we get Hence, the general relations between , T), f and a;, T/ are 188 LINEAR FRACTIONAL TRANSFORMATIONS [CHAP. V. From these equations we obtain x = ^f, y = j-L, x* + y 2 = j j;- Ex. i. Find the stereographic projection of a straight line. The equation of the given line is of the form Ax + By + C = 0. Substituting from (13) the values of x, y, we have AS Br, c = , lg . or A + Br, + C (1 - f) = 0. (16) This equation is that of a plane passing through the north pole of the sphere. The curve of intersection of the plane and the sphere is therefore a circle passing (14) FIG. 76. through the north pole. Hence every line in the plane projects into a circle upon the sphere passing through the north pole. Any line parallel, say, to the axis of imaginaries (Fig. 76) goes over into a circle through 0' tangent to the great circle into which the axis of imaginaries projects, but lying wholly in one of the hemispheres into which that great circle divides the sphere. None of these lines, however, other than the one through the origin, projects into a great circle. A system of straight lines parallel to the axis of reals projects into a system of circles likewise passing through 0' but perpendicular to the former system, and all of these circles on the sphere are tangent at 0' to the great circle into which the axis of reals projects. ABT. 40.] STEREOGRAPHIC PROJECTION 189 Ex. 2. Discuss the stereographic projection of a circle in the complex plane whose equation is x 2 + y 2 + 2 gx + 2/y + c = 0. (17) Substituting from (13) the values of x, y, and z 2 + y 2 we have rb + 2 j* 0, the logarithmic spiral motion maps into the loxodromic motion about the finite invariant points; if p ^ 1, < = 0, we have from (5) a pencil of rays and after mapping we get a hyper- bolic motion; if p = 1, 4> ^ 0, the resulting motion is elliptic. It will also be observed, that in case v = 1, the motion represented by (5) is a translation, in which case the invariant points become coin- cident at infinity and after mapping by reciprocation with respect to some finite point the resulting motion is the parabolic motion al- ready discussed. We are now in a position to classify the various types of linear fractional transformations of a plane according to their kinematic properties. For, as a result of the foregoing discussion, it follows that every linear fractional transformation may be interpreted in terms of some one of the motions already discussed. As we have seen, every linear fractional transformation leaves two points of the plane in- variant. These points must be either coincident, or distinct. If they coincide, we have seen that the transformation is a parabolic motion, where a translation appears as a special case. If the in- variant points are finite and distinct, we have in general a loxodromic motion, which reduces to a hyperbolic motion or an elliptic motion according to the particular value taken by v. If the invariant points are distinct, one being at infinity, then our transformation is a loga- rithmic spiral motion, which again reduces to an expansion or a rotation according to the particular values given to v as already noted. ART. 41.] EXERCISES 197 EXERCISES S 1. Given in the Z-plane the two intersecting curves x* + (y 2) 2 = 4 and 2 x + 3y 7 = 0. Map these curves upon the JF-plane by means of the relation w = -. Show that the resulting curves intersect at the same angle as Zr the given curves. 2. Map the orthogonal systems of straight lines u = c, v = c from the TF-plane upon the Z-plane; first, by means of the relation w = z 2 , second by means of the relation w = ^ . How would the two resulting configurations be related z if projected stereographically? y? ifl x? 3. Given the ellipse -r + -f- = 1. Subject this curve to the transformation T: t7 w = 3 z + 10 and find the equation of the resulting curve. Draw the curve. What general principle do the results of this mapping illustrate? -7 4. By applying the Cauchy-Riemann differential equations to the relation ' i w = - , show that w is an analytic function. o 5. A point z moves with uniform speed of 2 cm. per sec. along a circle whose center is at 2 + 3 i and whose radius is 4 cm. Determine the path, velocity, and acceleration of w, where w = 2 z + (1 + 2i). 6. Given the three points 1 + 2 i, 2 + i, 2 + 3 i. Determine the linear fractional transformation that will map these points into the following points: 2 + 2 i, 1 + 3 i, 4. Find the invariant points of the transformation. 7. Let a circle passing through the three points 2 + 3 i, 0, 3 + 2 i be given. Find the conjugate of the point 1 + 2 i with respect to this circle. Construct the curve into which the circle through the points 1+2 i, 2+3 i, 3 + 2i is changed by reciprocation with respect to the circle of radius 2 about the origin. 8. A given translation of the complex plane is represented by the equation w = z + /3, where /3 = 2 + 3 i. Show in two ways how the lines of motion in the complex plane can be changed into lines of parabolic motion upon the sphere having the coincident invariant points at the north pole. Explain why the two methods give the same result. 9. The speed at the point 2 + 4 i of a moving point in a motion of expansion about the point 1 + 2 i is 2 units per sec. At the same instant what is the position and velocity of the corresponding point in the hyperbolic motion obtained through reciprocation of this motion of expansion? 10. When a transformation has the property that on being repeated once it restores every point of the plane to its initial position, it is called an involution. Determine the conditions that must exist between the coefficients a, ft, y, 5, in order that the transformation shall be an involution. 11. Show that every involution is an elliptic transformation. 12. Show how the regions for the anharmonic ratios of four points may be represented on the complex plane in case the four points are harmonic; in case they are equianharmonic. CHAPTER VI INFINITE SERIES 42. Series of complex terms. In this chapter we shall con- sider some of the general properties of infinite series whose terms are complex numbers. We shall assume such knowledge of infinite series as is usually given in elementary texts on algebra and calculus. Special attention will be directed to the properties of power series. Suppose we have given the series 2a n = aj + a 2 + as.+ ++, (1) where a n = a n + #>n, a n , b n , being real numbers. As in series of real terms, we shall denote the sum of the first n terms by S n , that is, we shall put S n = cti + a 2 + + a n . A series is said to be convergent if the sequence 01, 02, . . . , S n , has a limit. If a is this limit, that is if L S n = a, n= then the series is said to converge to the number a. This number is also called the sum of the series and is uniquely determined by the series. We may therefore write 2a n = a. If the limit of S n does not exist as n increases indefinitely, then we say that the given series is divergent. The geometric significance of convergence may be easily seen. All that is needed is to locate in the complex plane the points that represent Si = ai, Sz = cti + 2, S n = ai + 2 + 3 + 198 ART. 42.] SERIES OF COMPLEX TERMS 199 It will be noticed that these points do not necessarily lie along a straight line as in series of real terms, but that they may be distrib- uted over the complex plane. They are, however, so located as to be dense at the limiting point a. In order that the limit shall exist, it is necessary that L \ a n \ = 0. n=oo As in series of real terms, this condition is a necessary but not a sufficient condition for the con- vergence of the given series. As already stated, each term of a series of complex numbers may be written in the form a n = a n + ib n . FIG. 80. O Separating the real and the imaginary parts of the terms of the series, we may put A n = ai + 02 + + a n , B n = 6l + 6 2 + ' ' ' + 6n. We may now formulate the following theorem. THEOREM I. The necessary and sufficient condition that the series of complex numbers ai + 2 + as + + a n + converges to a limit a = a + ib is that L A n = a, L B n = b. n=oo n=o We have S n = A n + iB n , (2) and by Theorem I, Art. 12, the necessary and sufficient condition that the sequence C C Cf Of Olj 02, 03, . . . , o n , . . . has a limit a = a + ib is that L A n n=oo L B n = 6. n=oo Hence the theorem. 200 INFINITE SERIES [CHAP. VI. The point a Is determined upon the axis of reals (Fig. 80) by the series 2a, while 6 is determined upon the axis of imaginaries by the series 26 n . The limiting point a is then identical with a + ib. It will be seen at once, therefore, that if either of the series 2a n , 26 m is divergent, the series is divergent. Ex.1. This series is divergent; for, we have where 2o is divergent although S6 n is convergent. We may also state the necessary and sufficient condition for the convergence of a series of complex terms as follows: THEOREM II. The necessary and sufficient condition that the series converges is that for an arbitrarily small positive number t there exists a definite number m such that I n+i + n+2 + a n+p | < e, n > m, p = 1, 2, 3, . . . . This condition follows at once from Theorem VI, Art. 12. We have the sequence Si, Sz, Ss, . . . , S n , .... The necessary and sufficient condition for the convergence of this sequence is that, for an arbitrarily small e, a corresponding integer m may be found such that I S n+p - S n | < e, n > m, p = 1, 2, 3, .... Replacing the values of S n and S n+p by their values in terms of the a's, we have at once the condition stated in the theorem. THEOREM III. The series 2a n converges, provided the series of moduli of the various terms of the given series converges. AKT. 42.] SERIES OF COMPLEX TERMS 201 We have by hypothesis the condition that the series of absolute values I ail + I <*!+ ' ' + |a n |+ converges. From Theorem II it follows that since the foregoing; series of moduli converges, we have { | On+i !++! m, p = 1, 2, 3, . But we have I n+l + ' * ' + CCn+p | = | CXn+1 [ + ' ' ' + | +p |; whence it follows that I a n+i + + a n+p | < e, n > m, p = 1, 2, 3, . By Theorem II this inequality gives a sufficient condition for the convergence of the given series 2a n . Ex. 2. Let it be required to test the convergence of the series Za n , where p n (cos nO + i sin n6) a n = - n The series of moduli is which converges for p < 1 ; for, we have by the ratio test L :'-% L n + 1 Hence, by Theorem III the given series converges for | j < 1. If the series of absolute values of the various terms of the series 2a n , namely 2 | a n \ = 2 Va n 2 + b n 2 , converges, then the given series 2a n is said to converge absolutely. THEOREM IV. Given the series , where a n = a n + ib n - The necessary and sufficient condition that this series converges absolutely is that the two series 2a n , 26 n converge absolutely. That the absolute convergence of 21 a n and 26 n is a necessary condition may be shown as follows. We have ? I = I On + i6 B 1 = Va n 2 + 6 n 2 . (3) We assume that the given series converges absolutely; that is, that the series of moduli converges. We have then the convergent series S 1 * | = V fll 2 + &! 2 + + Va n 2 + 6 n 2 + -. (4) 202 INFINITE SERIES [CHAP. VI. From (3) it follows at once that the terms of the two series | a n |, 2 [ 6 n | are equal to or less than the corresponding terms of the convergent series (4) and hence both of the series 2a n , 26 n con- verge absolutely. The condition set forth in the theorem is also sufficient. If we write An = | Ol | + | d | + ' ' ' + | a n I; B' n = | 61 | + | 6 2 1 + + | b n I, we have L (A; + B' n ) = L A' n + L B' n ; that is, since | a n |, S | b n \ converge, the series S { | a n | + | 6 | | also converges. We have, however, and consequently the series 2 | a n \ converges; that is, the given series converges absolutely. v~\ t" Ex. 3. Test the series V for absolute convergence. *^4 72- We may write the given series hi the form 2 o ^ o 4 o u from which we have Each of these series is convergent, but neither converges absolutely. Hence the given series can not converge absolutely. In fact we have which is a well-known divergent series. While absolute convergence, as we have seen, gives a sufficient condition for the convergence of 2a n , it is not a necessary condition. It may occur, as we have just seen (Ex. 3), that a series converges even if the series of moduli is divergent. Ex. 4. Given the series 2\ . where z = cos + i sin 0. < < 2 *-. ^ n It will be observed that the series in Ex. 3 is a special case of the series here ART. 42.] SERIES OF COMPLEX TERMS 203 considered, namely, where = ^ . The series of moduli is y. - , which is a ~4 n divergent series. However, the given series 2a n = 2 (o n + ib n ) is convergent, since each of the two series S-^ 1 . cos 20 . cos 30 , a n = 2j ~ cos ne = cos e H 2 ' 3 f" i XV^ 1 , sin 20 . sin 30 , & = 2, - sm n0 = sm 6 H - ' 3 H is convergent for < < 2 ?r.* Absolutely convergent series have certain properties not pos- sessed by series in general. Some of these properties are stated in the following theorems. THEOREM V. The sum of an absolutely convergent series of complex terms is independent of the order of the terms. When we have a finite series the order of arrangement of the terms is a matter of indifference. The foregoing theorem enables us to extend this commutative property to an infinite series, pro- vided it converges absolutely. Let the given series be 2a n = en + 2 + + a n + , (5) where a n = a n + ib n . By Theorem IV the two series 2a n , 26 n converge absolutely. When a series of real terms converges absolutely the sum of the series is not dependent upon the order in which the terms of the series are arranged.! Since we have 2a n = 2(a n + ib n ) = Za n + iS6 n (6) and the terms of the two series 2a n , 26 n can be taken in any order without affecting their sum, it follows from (6) that the sum' of the given series 2a n is likewise independent of the order in which its terms are taken. The associative law of addition may be extended to any convergent infinite series; that is, we can always insert parentheses at pleasure in an infinite series. However, if the series converges absolutely the * See Bromwich, Theory of Infinite Series, p. 159. t Ibid., p. 64. 204 INFINITE SERIES [CHAP. VI. sum is not affected by any arbitrary rearrangement and grouping of the terms; that is, we have the following theorem. THEOREM VI. The sum of an absolutely convergent series remains unchanged if the terms are rearranged and grouped in any arbitrary manner. Let 2a n be any absolutely convergent series haying the sum a. Let us first suppose the terms of this series to be grouped by putting into each group a certain number of consecutive terms. Let AI, AZ, . . . denote these groups, respectively, where' AI = i + + a k , Az = a k +i + + r, We shall now establish the convergence of the series A l + A z + + A n + (7) The sum of the first n terms of this series, namely, is equal to the sum of the first m terms (m > n) of the given series 2a n . As n becomes infinite, m also increases without limit. We have for all values of n S n ' == S m , (8) where S m denotes the sum of the first m terms of the given series 2a n . However, m does not take all integral values, but increases through the values k, r, s, . . . . Since the given series converges, it follows by Art. 12 that the limiting value of S m is the sum of the given series, namely a. As (8) holds for all values of n, it follows that we have also L S n ' = a. n=oo Since the given series converges absolutely the sum of the series is not affected by any arbitrary rearrangement of its terms. Conse- quently, in order to get any desired grouping of the terms of the given series 2a n , all that is necessary is to so rearrange the terms of the series that the terms occur in the required order and then form the series (7) of groups of consecutive terms. Hence the theorem. Convergent series that do not converge absolutely are called conditionally convergent series. The following theorem may be stated with reference to such series, namely: ART. 42.] SERIES OF COMPLEX TERMS 205 THEOREM VII. The sum of a conditionally convergent series de- pends upon the order of arrangement of its terms. Let Za n be the given conditionally convergent series. As in the demonstration of Theorem V, we have 2a n = 2(a n + ib n ) = 2a n + ib n . By Theorem IV we know that at least one of the series 2a n , S6 B must converge conditionally; otherwise the given series would con- verge absolutely. Suppose that 26 n alone converges conditionally. It follows from (6) that the given series 2a n converges to a limit a -\- ix, where x = 21 6 n depends upon the arrangement of the terms of the series 21 b n , and by properly choosing this arrangement x may be made any arbitrarily chosen real number.* This establishes the theorem. In this demonstration we have made use of the fact that in a conditionally convergent series of real terms the series may be made to converge to any arbitrarily chosen real number. This fact would seem to suggest that a conditionally convergent series of complex terms might likewise be made to converge to any arbitrarily chosen complex number as a limit. This, however, is not in general the case. As already pointed out, if 26 n converges conditionally, we may so arrange the terms of this series as to make it converge to any arbitrarily chosen number x. In order that the series Ua n , by a rearrangement of its terms, shall approach an arbitrarily chosen complex number x + iy it is, in general, also necessary that 2a n be conditionally convergent and that the rearrangement of its terms be unrestricted. This, however, is impossible, for when the 6's are properly arranged the a's must be taken in such an order that there is associated with each b h an a k such that ah + ib k is one of the a's. Since the choice of the arrangement of the a's is thus restricted, it follows that while the limit of a conditionally convergent series of complex terms depends upon the order of its terms, it can not, in general, be made to approach any complex number at pleasure by a suitable arrangement of its terms. Some particular conditionally convergent series of complex terms may, however, be made to approach any previously assigned limit by the proper arrangement of the terms. f * See Bromwich, Theory of Infinite Series, p. 68. t See Levy, Nouv. Ann. Math., Vol. 5, 1905, p. 506. 206 INFINITE SERIES [CHAP. VI. To test the convergence of a series of complex terms, we need only to employ the methods of testing series of real terms. We can always make the convergence of any series 2a n depend upon the conver- gence of the two series of real terms Za n , 26 n , where a n = a n + ib n , by use of the Theorem I. Frequently, it is more convenient to test the convergence of the given series by considering the series of moduli. As we have seen, if this series converges then the given series converges. The series of moduli is a series of real positive terms and the tests for the convergence of such series may be at once applied. If the series of complex terms converges condition- ally, we have no general tests; but this is to be expected, since no general tests exist for the conditional convergence of series whose terms are real. In the discussion thus far we have considered only those series whose terms are constants. The terms of the series may, however, be functions of a complex variable. Such a series may be written The convergence of this series for a given value of the variable im- plies that if this value is substituted for the variable, the resulting series of constants converges. The region of convergence may or may not be a closed region. A series of functions of a complex variable defines a function of the complex variable in the region of convergence. 43. Operations with series. In order to make use of series in mathematical discussions, it is necessary to establish the conditions under which convergent infinite series may be combined by the fundamental operations of arithmetic following the formal laws applicable to sums of a finite number of terms. It follows at once from the laws governing operations with limits and from the defini- tion of convergence, that any convergent series may be multiplied or divided termwise by a constant, and that a constant may be added to or subtracted from any convergent infinite series by adding it to or subtracting it from any term of this series, precisely as in the case of a sum of a finite number of terms. We shall now consider the sum, difference, product, and quotient of two convergent infinite series. THEOREM I. // 2a n , 2/3 n are two convergent series having the limits a, /3, respectively, then + . (1) ART. 43.] OPERATIONS WITH SERIES 207 // the given series converge absolutely, then the series (1) converges absolutely. Let S n = i + 2 + + , T n = ft + & + + ft,. Then, we have S n + T n = (x + /Si) + (<* + ft) + ' ' ' + (n + /3n). (2) By hypothesis we have L S n = a, L T n = |8; n = oo n=oo hence, it follows that L (Sn + r) = L Sn + L T^ = + /3, n = oo n = oo 71 = 00 whence a + ft = (i + ft) + (2 + ft) + + (a n + ft,) + . We shall now show that if 2a n , 2/3 n converge absolutely, the series (1) converges absolutely. Let I otn I = Pn, I j8 n | = r n , 2p n = p, 2r n = r, n' = Pi + P2 + ' ' ' + Pn, r n ' = r: + r 2 + + r n . Putting Pn = Pn + r B , we may write ^*n = Pi + & + ' ' ' + Pn. We have then L P n = L (S n f + TV) = p + r; n = oo n=oo that is, P + r = ( Pl + n) + ( P2 + r a ) + + (p + r n ) + . (3) Since we have | a n + j3 n | = | a n | + | j8 n | = P + r n , it follows from (3) that the series of moduli of (2) converges, that is (2) converges absolutely. It is to be noted that the parentheses in (1) may be removed; for the sum of the first n terms of the series 1 + ft + 2 + & + 08 + ' ' differs from the sum of a properly chosen number of terms of (1) by at most the first term in one of the parentheses, this term ap- proaching zero with increasing n. 208 INFINITE SERIES [CHAP. VI. Ex. 1. Add the two series log (l+) ^Z-^Z' + ^Z 3 - *2<+iz 5 - log (1 - z) = -z - z 2 - $ z 3 - i z 4 - i z 5 - . Applying the results of Theorem I, we obtain log (1 - z 2 ) = log (1 + z) + log (1 - z) .-*-! -I*- .... The corresponding theorem for the difference of two series may be stated as follows: THEOREM II. // 2a n , 2/3 n are two convergent series having the limits a and ft respectively, then m. For an arbitrarily chosen e, we may now select a number N m de- pending upon m, such that for n > N m each of the numbers Ri, Rz, . . . , R m is less than We have then ifv I S mn - S m | ^ R, + # 2 + + R m < c , n > N m . (4) However, since the series 21 o^ converges to the limit a', we have for some value mi, \S m -a'\ mi. (5) Combining (4) and (5), we obtain | S mn - a \ < 2 e, TO > mi, n > N m . (6) Since the limit L S mn exists, we have from (6) TO=00 n=oo L S mn = a. m ==o n=< By hypothesis S mn has the limit a as m and n become infinite. Hence, a' must be identical with a and the series 2a m converges to a as the theorem requires. 216 INFINITE SERIES [CHAP. VI. A similar argument shows that when the given double series 2a mn is evaluated by columns, the limiting value is again a, the sum of the given series. Ex. Consider the following double series, which is of importance in the Weier- etrassian theory of elliptic functions : .1 .1.1. 3 T 3 T 3 - (-2a>i+4a> 3 ) 3 (4a> 3 ) 3 (2a>i+4a> 3 ) 3 (4wi+4a> 3 ) 3 (-4a>i+2coj) 3 (-2 Wl +23) 3 (20*) 3 (2wi+2a> 3 ) 3 (4i+2 mi, we have If the value of e is kept constant, it will in general be necessary to select a new integer m? if zi be replaced by some other value 2 2 of S. If, in the selection of m, the least integer that will answer the pur- pose is taken, then with each point z there is associated a particular integer m, namely the first integer for which we have | R n (z) \ < e, where n > m. We may then write m(z) as a function of z. Consider now the totality of all the values of m corresponding to the points of the region S. These values of m may or may not have a finite upper limit. In case a finite upper limit exists, we may say that the series converges uniformly in the region S. Denoting the upper limit of m(z) by M, then for a given e any integer m > M may be associated equally well with each value of z. The definition of uniform con- vergence may now be stated as follows: The given series is said to converge uniformly in a given region S, closed or not, if corresponding to an arbitrarily small positive number t it is possible to find an integer m, which is independent of z, such that for all values of n > m we have simultaneously for all values of z, in the region S, The following example furnishes an illustration of regions of uni- form and of non-uniform convergence of a series of functions. ART. 45.] UNIFORM CONVERGENCE 219 Ex. 4. Given the series 2 _L Z 2 Z 2 Z 2 "*" 1 + z 2 + (1 + z 2 ) 2 + (1 + z 2 )' This is a geometric series; hence, we have 5 + (1 + z 2 ) 71 - 1 We shall consider the character of the convergence of this series for finite values of z = p(cos 6 + i sin 6) in the region defined by the inequalities In this region the series converges and defines the function = 1+Z 2 , = o, for z ?* 0, for z = 0. While the given series converges in the region indicated, that region is not to be understood as the whole of the region of convergence. The remainder after the first n terms is, in the region under consideration, for z T 0, for z = 0. (1 + z 2 )' = 0, For an arbitrarily small e, there corresponds to each value of z an integer m such that | < e, n > m. But there is no integer m, however large it may be chosen, that answers this purpose simultaneously for all values of z; for, suppose we take m = G, chosen as large as we please, then for HI > G we may always find values of z such that I R ni (z) 1 (1 + Z 2 )*!- 1 We need only choose z so that | z | is sufficiently small. It follows then that the given series is non- uniformly convergent in the region selected. Suppose we now restrict the region by excluding a region about the origin; that is, let p = r, where < r < 1. FIG. 82. The lower limit of | 1 + z* | in the new region is then VI + r 4 , which is the value of | 1 + z 2 | when z is at P or Q, Fig. 82. Hence, the upper limit of I Rn(z) 1 (1 + Z 2 ) 220 INFINITE SERIES [CHAP. VI. for a given n is ^ , In order to determine a value of m such that for (1 + r<)~2~ every z of the region we have | R n (z) | < , n > m, 1 put m'-l 2 2 log, whence m' = 1 . log (1 + r 4 ) Then for all values of n > m > m', we have 1 (!+*)*- irrespective of the value of 2 in the finite region where In this region then the given series converges uniformly. It will be observed that the region of uniform convergence may not coincide with the region of convergence. In fact, it is frequently more restricted than the region of convergence. As a convenient test for uniform convergence, we have the following theorem, due to Weierstrass. THEOREM I. Given the series ui(z) + uz(z) + + u n (z) + . // for all values of z in a given region S, closed or not, we have for all values of n \U n (z)\^M n , where 2Af n is a convergent series of positive constants, then Zw n (z) converges absolutely and uniformly in S. The absolute convergence of 2w n follows at once from the fore- going discussion of absolute convergence, since by the conditions of the theorem we have | u n (z) \ = M n and 2Af n is convergent. The uniform convergence of the series may be established as follows. Since the series 2M n is convergent, we can find a number m such that Af. M < 7 M t) = 1 23 ^iu n -^ ^1*1 n , /y i, ^,, o, m+1 m+1 Bm() | = L m+p m+1 < . Since this relation exists independently of the value z may take in the region S, the condition set forth in the definition of uniform convergence is satisfied. It is to be noted that the foregoing theorem gives a sufficient but not a necessary condition for uniform convergence. Other and more delicate tests for uniform convergence may be made to apply to series of functions of a complex variable,* but the one given is suffi- cient to test those series to be considered in the present volume. Uniform convergence gives a useful criterion for the continuity of the function defined by a convergent series of functions. This criterion may be stated as follows : THEOREM II. Given the function /(z) defined by the convergent series Ui(z) + uz(z) + + u n (z) + . If u n (z) is continuous and the series converges uniformly in a region S closed or not, then /(z) is continuous in S, We may write the given function in the form /(z) =S n (z)+R n (z). Since the series converges uniformly in a given region S, it follows that for any value z in S we have, for n > m, I fl(zo) I < e. (1) Suppose z takes an increment AIZ such that z + AIZ lies in S. We then have | R n (z<> + Ai) | < e, whence | AflnCzo) | = | R n (z + Ax2) - #,(*>) | < 2 e. (2) Since S n (z) is continuous in z for all finite values of n, we have 1 AS n (2o) | = | S n (Zo + A 2 Z) - Sn^) I < 6. (3) * See Bromwich, Theory of Infinite Series, Art. 81. 222 INFINITE SERIES [CHAP. VI. By combining (2) and (3), we obtain I A/X^o) I = I A.R n (zo) + AiS n (zo) I < 3 c, | Az | = | A 2 z | = | AIZ | , for all values of Az equal to the smaller of the increments AIZ, A2Z. Hence, since 3 c is arbitrarily small, | A/(ZO) | is arbitrarily small and /(z) is continuous at any point ZQ in the region S of uniform con- vergence. 46. Integration and differentiation of series. We shall fre- quently have occasion to integrate or differentiate a series term by term. The question arises whether the resulting series represents the integral or derivative, as the case may be, of the function defined by the given series. Suppose a function /(z) is defined by the rela- tion /(z) = Wi(z) + M2(z) + ' ' + U n (z) + ' ' ' , where the w's are continuous functions in a region S within which the given curve C lies. Denote by S n (z) the sum of the first n terms of this series. The integral of the function along the path C, if it exists, may then be written f /(z) dz = f L S n (z) dz. (1) *J C *J C n=< For any definite value of n, we may write n / / n /i V^ I / \ j I ^ / \ j lei/ J c .<*>*-J^.*-J c s.< Consequently, the result of term by term integration of the series defining /(z) may be written L^S n (z)dz. (2) It can not be assumed that the two results (1), (2) are equal. The following example furnishes an illustration where (1) and (2) are not equal. Ex. 1. Given the series, the sum of whose first n terms is S n (z) = nze~* a . Consider the term by term integration of this series where the path C of integra- tion is the X-axis from to any point /3, < < 1. The series converges for real values of z. The integral along the axis of reals is an ordinary definite integral for real values of z. We have then L nze^ nzt dz = o n=oo ART. 46.] INTEGRATION OF SERIES 223 f On the other hand, we have 1 ^ J L n n=ao J o B=OO 'o n=oo * Hence in this case we can not integrate the given series term by term. We shall now set up a condition by means of uniform convergence that will be sufficient for the equality of (1) and (2). This condition may be stated as follows: THEOREM I. Lei /(z) be defined by the convergent series ui(z) + uz(z) + + u n (z) + , where u n (z) is a continuous function for values of z along an ordinary curve C, If the series converges uniformly along C, we may integrate the series term by term, thus obtaining f /(*) dz = fu v (z) dz + f2(z) &++ fu n (z) dz + . (3) Jc JC JC JC Each term of the series is continuous and hence the integral / u n (z) dz exists. Moreover, since the series converges uniformly, Jc the function /(z) is a continuous function and the integral / /(z) dz Jc also exists. We shall now show that the relation given in (3) holds. We may write f(z) = ttiCO + u*(z) ++ u n (z) + R n (z), (4) where R n (z) denotes the remainder of the given series after the first n terms. By formula 3, Art. 17, we have f/(z) dz = Awi(z) + u,(z~) + + u n (z) + R n (z)\dz Jc Jc = fui(t)dz+ Iu2(z)dz+ ---- h lu n (z)dz+ /.(*)<&. (5) Jc Jc Jc Jc For n sufficiently large, say n > m, the integral I R n (z} dz becomes J c arbitrarily small. For, we have fR n (z)dz ^ f\R n (z)\.\dz\. (6) Jc Jc As the series converges uniformly, we have for all values of z along C | R n (z) | < e, n > m. 224 INFINITE SERIES [CHAP. VI. Hence, from (6) we obtain I fR n (z)dz < e f\dz\, n>m = e-L, where L denotes the length of the path C of integration and is there- fore finite. Since e L is arbitrarily small, we have L CR n (z) dz = 0. =oo Jc Consequently, when n is allowed to increase without limit, we obtain from (5) the relation given in the theorem. It is necessary also to set up some criterion for the differentiation of a series term by term; for, it can not be assumed that the series formed by differentiating the various terms of a given convergent series is equal to the derivative of the function defined by that series. The following example furnishes an illustration. Ex. 2. Given the series sin 2 z . sin 3 z sin 4 z . T+T: ~r~ +< The series converges for real values of the variable, and defines the function * - for values of z lying between TT and TT. Differentiating the series term by m term, we get cos z cos 2 z + cos 3 z cos 4 z + . 2 This series of derivatives does not represent the derivative of -^; for, it does not t even converge for values of z other than z = 0. For, in order that a series shall converge, we must have, as we have seen, the limit of the n ih term equal to zero. However, in the case under consideration, the limit L | cos m \ does not even n=oo exist for z ^ 0. If the terms of the given series are holomorphic in a given region S, we have the following theorem, which furnishes a convenient criterion for differentiating or integrating term by term such series as we shall have occasion to consider. It also furnishes a condition that the function defined by the series shall be holomorphic in S. THEOREM II. Let /(z) be defined by the convergent series ABT. 46.] DIFFERENTIATION OF SERIES 225 where u n (z) is holomorphic in a region S. If this series converges uni- formly in every simply connected closed region lying wholly in S and bounded by an ordinary curve C, then the series may be integrated or differentiated term by term for values of z in S. Moreover, /(z) is holo- morphic in S. Since the given series converges uniformly and each term is con- tinuous, it follows from Theorem I that it may be integrated term by term along any ordinary curve C lying in S. It is to be noted that as u n (z) is also holomorphic hi S, the integral of each term of the series is zero, since C is the complete boundary of a simply connected closed region lying wholly in S. We have then /(z) dz = 0. Consequently, by Theorem IV, Art. 20, /(z) is holomorphic in the given region S. To show that the given series may be differentiated term by term, we proceed as follows. Consider the series /(O = u t (f) + t(0 + + u n (t) + , (7) where t takes values along the closed curve C. This series converges uniformly, a property that is not destroyed by multiplying the terms of the series by the factor - ., . 2 , where z is any point within C. We have then l /) l Ml (Q l tuft) 1 u n (t) -z) 2 " h 27ri(-2) 2 +*-'" -Af Integrating term by term, we obtain 1 C f(t) dt 1 C Ui(t) dt 1 C Uz(f] dt 2iriJc(t- z) 2 = 2^ Jc (t - z) 2 + 2^ Jc (t - z) 2 _L . . . l * Cu n (f)dt From Art. 20, it will be seen that the terms of this series of integrals are the first derivatives of the terms of the given series. We have therefore for any value of z within C. But C is any closed curve in S', hence (8) holds for any values of z in 5 as stated in the theorem. 226 INFINITE SERIES [CHAP. VI. Ex. 3. Given the series 1 z z 2 * * * -I- ... -I f 1- . . . * /"O 1 \ /O v I 1 'i /O L O\ ^ * i o u o u i u < ^Tl ly v* >l -t" */ \"** T *J This series converges uniformly in any region bounded by a circle about the origin, which is situated within the unit circle. It represents the function * which is given, except for z = 0, by the expression j((l-s)' lo 1 + v; The indefinite integral of this function is readily found by integrating the given series term by term, thus obtaining fi i 2 | Z . F , . Z . r 1-3- 5"*" 2-3- 5-7 "^3- 5-7-9 "*" "*" n (2n - 1) (2n + 1) (2n + 3) ^ Ex. 4. Given the series z 3 z 5 z 7 ~ 3l + 5~l ~ 71 + This series converges uniformly in any finite region. The derived series is 1_* + 1_*. 4 2! ^4! 6! T Consequently, the second series represents the derivative of the function defined by the first. 47. Power series. A series of the form oo + aiZ + 22 2 + ' + a n z n + , where n is a positive integer and n = a n + #>n = pn(cos n + t sin n ), = x + iy = r(cos + i sin <), is called a power series of complex terms. A more general form of a power series may be written oo + ai(z 2o) + a 2 (z 2o) 2 + + <* n (z - Zo) n + . To distinguish the two types, we may speak of the second as a power series in (z zo). For the sake of simplicity we shall confine our discussions for the most part to power series of the first type. In doing so there is no loss of generality, as power series in z ZQ may be readily transformed into series of this type. Because of their importance, we shall consider some of the special properties of the power series. Among these properties is the following : * Schlomilch, Ubungsbuch zum Studium der Hoheren Analysis, 4 th Ed., Vol. 2, p. 239. ART. 47.] POWER SERIES . 227 THEOREM I. If for some positive number G we have for all values of n where ZQ is a constant value of z, then Za n 2" converges absolutely for all values of z for which \ z \ < \Zo\. Denoting the modulus of 2 by r , we have by the conditions of the theorem p n r n = G. The series of absolute values may be written r ro, ~ ( r Ir \ n } sG ! i+ ^+---+y +( The series within the braces converges to the limiting value r if we have r < r . Consequently, for | z \ < | 2 | the series of moduli converges, and hence the given series 2a n 2 n converges abso- lutely as the theorem states. Ex. 1. Test the convergence of the series sinz + ~sin 2 z +^sin 3 z + + - sin n 2 + . O 71 The given series is a power series in sin z. If we put w = sin z, we have up , w 3 w n , This series converges for | w \ < 1; for, we have then < 1 for all values of n. By the foregoing theorem the series converges absolutely within the circle of unit radius about the origin in the FF-plane. To find the region in the Z-plane within which the given "series converges, it is necessary to map upon the Z-plane the circle about the origin in the TF-plane having the radius 1 by means of the relation to = sin z. We have then u + iv = sin(x + /) sm x cos h y + * cos x sinh y, 228 whence INFINITE SERIES [CHAP. VI. tt = sin x cosh y, v = cos x sinh y. The equation of the circle in the TF-plane is u 2 + r 2 = 1. Substituting the values of u, v in terms of x, y, we get sin 2 x cosh 2 y + cos 2 x sinh 2 y = 1, which reduces to the form cosh 2 y = cos 2x + 2, sinh 2 y = cos 2 x. or The portion of the fundamental region - < x = ^ for sin z bounded by the ft & curve given by this equation is shown in Fig. 83 and 84. FIG. 83. FIG. 84. THEOREM II. // the power series Za n z n converges for z = ZQ, it converges absolutely for all values of z for which \z\ < | ZD | . This theorem follows as an immediate consequence of Theorem I. For if the given series converges for z = z , then there must exist some positive number G such that for all values of n \ a n I I *o" | < G, and consequently the series a n z n converges absolutely for values of z for which | z \ < \ ZQ \ as the theorem requires. We have also the following theorem. THEOREM III. // the power series Za n z n is divergent for z = z\, then it is divergent for all values of z for which \z\ > | Zi | . By hypothesis the given series Sa n z n is divergent for z = z\. It must then be divergent for all values of z where | z \ > | Zi | ; for, if ART. 47.] POWER SERIES 229 it is convergent for any such value of z, say 22, where | 22 | > | z\ |, then it must, by Theorem II, be convergent for all values of z whose modulus is less than [z%\ and therefore for z Zi, which is a contra- diction of the given hypothesis. From the contradiction the theorem follows. Theorem II states that if a given series converges for z = ZQ, then it converges within a circle about the origin having a radius equal to | ZQ | ; and Theorem III states that if it is divergent for z = Zi, then it is divergent for all values of z exterior to the circle about the origin whose radius is | z\ \, Nothing is said about the convergence of the series within the region between these two circles, or indeed upon the circles themselves, except at the points z and z\. The question presents itself as to whether it is possible to find a circle about the origin of radius R such that the given power series shall be convergent for all values of z where | z \ < R and divergent for all values of z where \z\ > R. It may be shown as follows that such a circle of radius R always exists, where R may be zero, finite and different from zero, or infinite. Let as before ZQ be a point of con- vergence and Zi a point of diver- gence of the given series. Denote the moduli of 2o, z\ by po, pi, respec- tively. Then we must have p = p\, F If Po = pi, we can take R equal to the common value. If po < pi, lay off upon the X-axis the distances po, Pi + Po Pi. Consider the point ai = For z = ai the given series is either convergent or divergent. Let us suppose that it is convergent. Then by Theorem II the series is convergent for all values of z within the circle about the origin whose radius is OL The region of doubt lies now between the two circles of radii a\, pi, respectively. Consider Pi + the point 02 = For z = 02 the series is again either con- vergent or divergent, say divergent. Then for values of z such that | z | > 02 the series is divergent by Theorem III. The region of doubt now lies between the circles of radii Oi, 02, respectively. Proceeding in this manner we shall obtain upon the X-axis an 230 INFINITE SERIES [CHAP. VI. infinite sequence of intervals each lying in the preceding one. Moreover, the length of the intervals has the limiting value zero. These intervals therefore define a definite number R. If we now describe a circle about the origin having R as a radius, we can say that the given power series converges for values of z for which | z \ < R and diverges for values of z for which [ z \ > R. For z = R the series may or may not converge. This circle whose radius is R is called the circle of convergence of the power series, and R is the radius of convergence. The radius of convergence may be equal to zero, in which case the given power series converges for 2 = only, or it may be finite and different from zero, or it may be infinite, in which case the given series con- verges for all finite values of z. Nothing can be said from the dis- cussion thus far concerning the convergence of the series for points on the circle of convergence itself. As a matter of fact a power series may converge absolutely at every point on the circle of conver- gence, or it may converge conditionally at every such point, or it may converge conditionally at certain points upon this circle and diverge at other points, or finally it may diverge at all points upon this circle.* We shall need methods by which we may determine the radius of convergence of a given power series. It evidently depends upon the coefficients of the given series. A relation between the radius of convergence and the coefficients of the given series is given by the following theorem. THEOREM IV. // the coefficients of the given power series Za n z n are such that the limit L exists, then the value of this limit is equal to the reciprocal of R; that is, it is the reciprocal of the radius of con- vergence of the given series. Put L n=oo = A. To prove that -r is equal to the radius of convergence, it is neces- A. sary to show that the given power series converges for all values of z where | z \ < -r and diverges for all values of z where | z [ > -r * See Encyclopedic des Sci. Math., II 7 , p. 15. ART. 47.] POWER . SERIES 231 We may readily show that the power series converges for values of z where | z \ < -j . As in the demonstration of Theorem I, let a n Pn(cos d n + i sin n ) , z = r(cos $ + i sin } . By Theorem III, Art. 42, the given power series converges if the series of moduli Po + pir + P2 r 2 + + Pn r n + (1) converges. This series converges if we have By the condition of the theorem we have L '- tl= L n=oo Pn n=oo = Hence the condition that (1) converges is that rA < 1; that is, | z | = r < -j- Consequently, the given series converges for all values of z within the circle of radius -r A The given series likewise diverges for all values of z without the circle of radius -r- To show this, suppose it to converge for some A. value ZD without this circle. Let z\ be any point outside of the circle such that \z\\ < \ z |. Then by Theorem II the given series converges absolutely for z\. We have then the convergent series Po + Pin +.-+ p n ri" + . (2) However, we have Pn+l T\ n+l L since n > -j This result contradicts the conclusion that series (2) A. is convergent. From this contradiction it follows that 2a n z" can not converge for any value of z exterior to the circle of radius -j- 232 INFINITE SERIES [CHAP. VI. Since the given series converges for all values of z within the circle about the origin having the radius -7 and is divergent for all A. values of z without this circle, it follows that -r must be equal to R, the radius of convergence, as the theorem states. The application of Theorem IV to the problem of determining the radius of convergence depends upon the existence of the limit L - B=oo Oi n The theorem gives a sufficient but not a necessary condition for convergence. There are convergent series for which this limit does not exist. The following series furnishes an illustration. Ex. 2. Given the series 1_|_i ? j _ -2 _| 3J ^*-4-. . -J yi " 1 -J ?2 n i . ^2 ^2-3 ^22-3 ^2 2 -3 2 ^2 n -3 n - 1 ^2 n -3" This series is convergent for | z | < 1; for putting z 1, we get a series whose terms are less than the corresponding terms of the convergent series The limit L n=oo does not exist since oscillates between - and - , pa, v^, . . . , \/P B , . . . , (3) depending upon whether an even or odd term is taken as the n th term. The following theorem * gives us a means of determining a radius of convergence that is applicable to any power series. 00 THEOREM V. Given the series ^a n z n ; and let p n = \a n \. If A n=0 is the maximum limit of the sequence Pi, then -r is equal to the radius of convergence of the given series. A By the maximum limit of a sequence is understood the largest number that can be obtained as the limit of a subsequence of the given sequence. In the particular case under discussion we are to consider the various subsequences that may be selected from (3) and denote by A the largest number that can be obtained as the limiting value of any of these subsequences. * This theorem was first demonstrated by Cauchy. See his Analyse Alg., p. 286, also Encyclopedic des Sci. Math., II 7 , p. 6. ART. 47.] POWER SERIES We wish to show that the given series converges for 233 that is, within the circle C (Fig. 86), having the origin as a center and R = -j as a radius. Let z' be any point within the circle C. We have then z = I A+e , where < e. There are at most a finite number of ele- ments of the sequence (3) greater than or equal to A + e. Suppose m is the largest of the subscripts of these elements. We have then 1 FIG. 86. = A+t>V Pn , \ z \ or | z' | \/p n < 1, n > m. We therefore have Pn | z' n | = | a n z' n \ < 1, It follows from Theorem I that the series n > m, n > m. a n z n , and hence the given series, converges absolutely for all values of z within the circle whose radius is . As 6 is arbitrarily small it follows that the series Za n z n converges absolutely within the circle C. We wish now to show that the given series diverges for Let z" be any point exterior to the circle C. We have then 1 z" = A- >0. There are now an infinite number of elements of the sequence (3) greater than A e; that is, for an infinite number of values of n we have 1 /7 f ^ ^*- ~"~ ^ ^ 234 INFINITE SERIES [CHAP. VI. or I z" \ Then for | z \ = | z" | we have, for an infinite number of values of n, pn\Z n \ = \a n Z n \ > 1. Consequently, the given series 2a n n can not converge for ) z \ > -r- A. Since 2a0 n is convergent for all values of z lying within the circle of radius -jand divergent for all values of z lying without this A. circle, it follows that A is equal to R, the radius of convergence, A which establishes the theorem. Whenever the sequence (3) has a definite limit as n becomes in- finite, the various subsequences have the same limit and hence the maximum limit is the limit of the sequence. It will be observed also that whenever both the sequence (3) and the ratio have a limit, the two limits are the same, since both are the reciprocal of the radius of convergence. Theorem V often enables us to de- termine the radius of convergence even if the sequence (3) has no definite limiting value. Ex. 3. Determine the radius of convergence of the power series given in Ex*. 2. We have the sequence of positive values 2' V2 3' ' ' ' l The limit of the subsequence in which the odd roots alone are taken is The limit of the subsequence in which the even roots alone are taken is - L J=- ART. 47.] POWER SERIES 235 No other subsequence has a different limit and hence the sequence Pi, "VP2, * P3> , "^Pn, has the limit 7= , and the radius of convergence of the given power series is V&. V6 The following theorem is of importance in the discussion of ana- lytic functions. THEOREM VI. The power series 2 m. Hence the series does not converge uniformly in the open region within the circle of convergence. However, for all values of z within a circle about the origin having a radius R' < 2, there exists a number m such that for n > m, we have Consequently in the closed region bounded by a circle of radius R' < R the given series converges uniformly. The following theorem gives a condition under which a power series is uniformly convergent in the closed region bounded by the circle of convergence. THEOREM VII. // 2a n z n is absolutely convergent at one point on the circle of convergence, then it converges absolutely and uniformly in the closed region bounded by the circle of convergence. If the given series is absolutely convergent at one point on the circle of convergence, say at z = z , we know from the definition of absolute convergence that the series 2 | a n \ R n converges, where R is the radius of convergence. However, any point on the circle whose radius is R gives the same series of moduli. Hence, for values of z such that | z | = R the terms of the given series are not greater in absolute value than the corresponding terms of 53 | | R n . Hence by Theorem I, Art. 45, the given series converges absolutely , and uniformly in the closed region bounded by the circle of radius R. Ex. 6. Given the series 7,-^ , the character of whose convergence is to be examined. This series is absolutely convergent for z = 1, since the series of moduli ^Vis convergent. Hence, by Theorem VII the series converges absolutely and uni- formly in the closed region bounded by the circle about the origin whose radius is 1. AKT. 47.] POWER SERIES 237 The function represented by this series (Theorem II, Art. 45) is continuous in the closed region bounded by the unit circle, and by Theorem VI is holomorphic within this circle. This particular func- tion,* however, is not holomorphic upon the unit circle itself. COROLLARY. // 2a n z n is divergent or conditionally convergent at any point on the circle of convergence, then it can be at best only condi- tionally convergent at any other point on this circle. This proposition follows as a consequence of Theorem VII; for, if the given series is absolutely convergent at any other point on the circle of convergence, then by Theorem VII, it must converge abso- lutely for all values of z for which \z\=R, the radius of convergence, and this is a contradiction of the hypothesis. Hence, if the given series converges at any other points on the circle of convergence, it must converge conditionally. v^ z n Ex. 6. Consider the character of the series 7, ~4 n This series is conditionally convergent at z = 1. It is divergent at z = 1. Hence, in this case, the series can not be absolutely convergent at any point on the unit circle. * For the differentiation and integration of a power series we have the following theorem. THEOREM VIII. The power series 2a n z n may be differentiated or integrated term by term in the open region bounded by the circle of con- vergence. The circle of convergence of the resulting series is the same as that of the given series. That the given power series may be integrated or differentiated term by term in the open region bounded by the circle of conver- gence follows from Theorem II of the last article by the same reason- ing as was employed in the demonstration of Theorem VI. The resulting series in either case has the same circle of conver- gence as the original series. We shall show this to be true for term by term differentiation. A similar argument will establish the truth of the statement for term by term integration. The series of deriva- - tives /'(z) = ui'(z) + ut'(z) + + u n '(z) + (5) is a power series. By the first part of the theorem under considera- tion the series (5) converges for all values of z within the circle of radius R. We must show that it is divergent for values of z exterior * See Picard, Traiie d'ancdyse, 2 d Ed., Vol. 2, p. 74. 23S INFINITE SERIES [CHAP. VI. to the circle of radius R. Suppose it should converge for some value Zi exterior to this circle. Then for values of z such that | z \ < R", where R < R" < \z\\, the series (5) converges uniformly and can be integrated term by term. As a result of integration we should have the original power series, except as to an additive constant, and this power series must then converge for all values of z such that [ z | < R". This, however, is impossible since values of z ex- terior to the circle of convergence of the given series are thus included. From this contradiction it follows that the series (5) can not con- verge for values of z exterior to the circle of radius R. Since the series (5) converges for all values of z within the circle of radius R and diverges for all values of z exterior to it, it follows that R is the radius of convergence of (5) as stated. 48. Expansion of a function in a power series. We have seen that in the open region bounded by the circle of convergence, a power series represents a function which is holomorphic. We shall now show that a function may be uniquely represented by a power series in the neighborhood of any point of a region in which it is holomorphic. Moreover, we shall develop a method for obtaining the required power series. The results may be stated in the follow- ing theorem. THEOREM I. // f(z) is holomorphic in a given region S, then in the neighborhood of any point ZQ in S, f(z) can be represented by a power series in (z ZQ), and that in one and only one way, namely: /CO = Let ZQ be any point in the given region S. About the point ZQ as a center draw the circle (7, having the radius r and lying within S. Let the complex variable t take values corre- sponding to the points of C. For any point z within the circle, we have then FIG. 87. t - i Z ZQ (t - ZQ} (Z - ZQ? i /j _ \ "t (t - 2o) 3 \Z ZQ\ < \t ZQ\ , Or Consider now the series Z t ZQ (t- < 1. (1) AKT. 48.] EXPANSION OF FUNCTIONS 239 This series converges for values of z within C and represents the n _ y function - -- ; for, it is a geometric series having the ratio - t Z t ZQ Considered as a series in the complex variable t, it converges uni- formly upon the circle C; for, | z ZQ \ and | t z \ = r are then both constant and the series of maximum numerical values 1 i |g-3)| , \z-zo\ 2 . . I z - z 1" r ~" r 2 j3 r n+l converges. The uniformity of the convergence is not disturbed if we multiply each term by/(). We thus obtain /(O /(O , (*-*)/(*) , (*-*o) 2 /(0 ^ Since this series converges uniformly, we may integrate it term by term with respect to t, the integral being taken around the circle (7. We thus obtain n + --. (3) Each of these integrals is a constant, and we have by Art. 20, /w(*o_ j_ r /(pa n! ~27rtJc(-zo) n+1> Replacing the integrals in (3) by their equivalent values in terms of the successive derivatives of /(z), we have for z = ZQ the required expansion known as Taylor's series, namely: For 2 = we have for the expansion of the given function in the neighborhood of the origin a special form of Taylor's series known as Maclaurin's series, namely: 240 INFINITE SERIES [CHAP. VI. Within the circle C, that is in a neighborhood of 20, the given function f(z) can therefore be represented by a power series. More- over, within C the given function can be represented by a power series in (z 20) in only one way. For convenience put n\ The series (4) can then be written oo + i (2 - 2o) + 0:2(2 - 2o) 2 + + a n (z - 2o) n + . (5) Suppose it is possible that within a circle Ci, whose radius is equal to or less than that of C, f(z) can be represented by a second power series, say /(*) - /3o + ft(2 - 2o) + ft(2 - 2o) 2 + + ft,(* - 2o)" + ' ' (6) Subtracting (6) from (5) we have = (oo - j8 ) + (i - ft) (z - 20) + (as - ft) (2 - 2o) 2 + + ( n - ft,) (2 - 26)" + . (7) This relation holds for all values of 2 within Cf, hence for 2 = 2o. Putting 2 = 20, we get = o A), or o = j8o- For 2 ?^ 20, however, we have 0=(ai-ft) (2-2o) + (a 2 -&)(2-2o) 2 + +(a n -j8 n ) (2-20)"+ Dividing by (2 20), we obtain 2 )+ + ( - j8 n ) (2- 2 )"- 1 + This series converges uniformly within or upon any circle about 20 as a center and lying within C\. Hence it defines a continuous function, say #(2). Since (z) is continuous and equal to zero for z 7* 20, we have for 2 = 20, 0(2 ) = L (z) = L = 0. z=o *=z<> Consequently, we have Q = 0(2o) = ai - ft = 0, or ai = ft. Continuing in this manner we may show that n = j3 n , n = 0, 1, 2, . . . ; ART. 48.] EXPANSION OF FUNCTIONS 241 hence, in the neighborhood of 20 the given function can be represented by a power series in (z 2 ) in only one way. Since ZD is any point in the given region S, the theorem is established. Ex. Expand /(z) = ^ -- in a Maclaurin's series. J. "~ Z We have /() = 0, /'(z) = (!-*)-*, jT(0) = l, /"( 2 ) = 2 (1 - z)-*, A I =2- 3 (!-)- 3! Wn^ /f\ f M (z)=n\(l-zr (n + 1 \ n l Hence, in the neighborhood of the origin the series z + z* + z* + + z" + (8) represents the given function. It is to be noted that the power series in (z z ) arising by Tay- lor's expansion of a given function which is holomorphic in a region S represents that function for all values of z within any circle that can be drawn about the given point ZD, so long as it lies within the given region S and incloses only points of S', for, it is clear that any such circle can be selected as the curve C along which the inte- grals are taken that determine the coefficients in the expansion. Moreover, if as in the illustration given above, the function is analytic within the entire circle of convergence, the series represents the function giving rise to it within the whole of that circle. We now have shown that every power series defines a function which is holomorphic in the open region bounded by the circle of con- vergence, and conversely that a function can be expanded in a power series in the neighborhood of any point in the region S in which it is holomorphic. It will be seen, therefore, that power series play an important role in the discussion of analytic functions. Indeed, Weierstrass based his entire development of the theory of analytic functions upon power series. 242 INFINITE SERIES [CHAP. VI. EXERCISES 1. Determine the circle of convergence of the series (a) l+ (c) 2. Discuss the uniform convergence of the series z z 2 z 3 z n where | X | > 1. What can be said of the function represented by the second series that can not be said of the function represented by the first? 3. Discuss the behavior of the series +mz for values of z upon the circle of convergence,* (a) for m > 0, (6) for m ^ - i; (c) for - 1 < m = 0. 4. Show that the two series 1 - * | 1- 3 (1 Z) 2 i -^ * 3-5 (1- 2)8 1 1 z r 1 | 2- 1- 4 3 (z 2 I) 2 1 2- I I' 4- 3- 6 5 ( 3 r 1)3 1 1 2- 4 2 1 2- 4- 6 3 r have the same region of convergence. 6. Determine the region of convergence of the series II z- IV , IV, if J+3V Find the derivative of the function represented by the given series. 6. Given the series 1 I * I * i n o 4 i o -i c i * 1-2-3 I- 2-3-4 ' 3-4-5 Verify, by testing the first and second derived series, that they have the same circle of convergence as the original series. * See Goursat, Court \ * * * 3 o 2n 1 For what valuer of cos z is the series convergent? Determine the corresponding region in the Z-plane. Does the series represent a continuous function of z in this region? 8. Given the series z 2 z 2 -2 _l_ _| _1_ . r 1 + Z 2 ^ (1 + 2 2 ) 2 * Show that this series converges for all finite values of z outside the lemniscate p 2 = -2 cos 20. Show that this series diverges at all points different from zero within and upon this lemniscate. 9. Derive the following expansions and determine in each case the region of convergence: *3 *& J (&) (c) (d) log (e) 10. Derive the expansion = 1 - 2 + 2 2 - 1+z and verify the expansion in Ex. 9 (d) by integration of this series. Derive in a similar manner the expansion r* arc tan z = I = z + ^- - J o 1 + z 2 o 5 dz + 11. Making use of the expansions in Ex. 9, derive the following relations: (a) (6) lo (c) cos 2 z = cos 2 z sin 2 z, 5+ . . . > 1 2 tan z=z+5Z 3 + r-?2 (d) (o\ . , e z e~ z .*.+ ' 3!^5! * + * + 4. r>nVi 1 -4- Verify these results by Taylor's theorem. 244 INFINITE SERIES [CHAP. VI. 12. From the expansion derive the expansion 1 1 1 ___,__ 28 13. Making use of the expansions 2 s . 2 s Z 1 81 n Z = z -_ + ___ Z* Z 4 Z 6 derive the expansions , 5^ 14. Derive the expansions C z dz , 1 z 3 , 1 3 2 s (a) arc sin z = 15. Verify the formulae - -i sin (a + 0) = sin a cos + cos a sin , / cos (a + /3) = cos a cos /3 sin a sin /S by means of the power series expansion of the sine and cosine. 16. Given the expansion By aid of this series prove that e and give the reason for each step. CHAPTER VII GENERAL PROPERTIES OF SINGLE-VALUED FUNCTIONS 49. Analytic continuation. In the present chapter we shall dis- cuss some of the general properties of single-valued functions. Let us first consider how a function which is holomorphic in a certain region may be completely represented in that region by means of power series. Consider, for example, the function M-T=t. Expanding this function in a Maclaurin series, we have 1 + Z + Z 2 + Z 3 + + Z + . (1) This series converges within the circle C of unit radius about the origin. Since the given function /(z) is holomorphic for all finite values of z, except z = I, it follows from Art. 48 that the series (1) represents that function for all values of z within C. However, for values of z exterior to C the series (1) does not converge and hence can not be said to represent the given function. We may for con- venience denote the aggregate of functional values corresponding to values of z within C, as given by the series (1), by the symbol <(z). We shall speak of (z) as an element of the function /(z) = - 1 2 A general definition of an element of an analytic function will be given later in this article. Since /(z) can be represented by a Taylor's expansion in the neighborhood of any point of a region in which the function is holomorphic, there is similarly an element (z) correspond- ing to any finite point z of the complex plane, except the point z = 1. The power series defining these respective elements of /(z) converge within circles which may overlap. For example let z be a point within C, so selected that for some values of z upon C we have | z ZQ j < ' FT~^ Expanding the given m function - in powers of (z z ), that is in a Taylor's series, we l z 245 246 SINGLE-VALUED FUNCTIONS [CHAP. VII. get 1 z-Zo , (z - Zo) 2 , i (z - go)"" 1 , , 9 x ++ 4 - This series is a geometric series having the ratio =- , and hence it 1 2o converges for values of z such that | z - 2 1 < 1 1 - 20 | ; that is, it converges for values of z within a circle of C of radius 1 1 ZD | about 20 as a center. Since the point 2 was so selected that at least one point on C is closer to ZQ than one-half of the distance 1 1 20 1, it follows that C must intersect C. In that portion of the plane included within these two circles of convergence, the given function - is represented by either of the i z two series, each giving the same numerical value for any particular value of z within the common region. Consider now an assemblage of power series obtained from , such that the corresponding circles 1 z of convergence cover the entire finite portion of the plane except the one point 2 = 1, which is not a regular point of the given function. This assemblage of power series may be said to completely represent the given function. In the foregoing illustration the function f(z) is given by means of an algebraic expression in z, from which the value of the function can be computed for any value of z except 2 = 1. From this ex- pression we are able to obtain an expansion of the function in a power series in the neighborhood of any point of the region in which the function is holomorphic. We shall now show that had we known merely the values of the function in ever so small a neighbor- hood of any point of the complex plane other than 2=1, say the origin, we should have been able, at least theoretically, to com- pute the value of the function at every point of the region in which it is holomorphic and that without even finding the expression - at all. The unique determination of the values of a function J. Z in a more or less extended region by its values in an arbitrarily small portion of that region is a general property of functions of a complex variable for regions in which they are holomorphic. This property may be more exactly formulated in the following theorem. ART. 49.] ANALYTIC CONTINUATION 247 THEOREM I. If a function /(z) is holomorphic in a given region S, then it is uniquely determined for all values of z in S by its values along any arbitrarily small arc of an ordinary curve proceeding from a point ofS. Let a be a point of the given region S from which the given arc is drawn and suppose j3 to be any other point of S. Let a and /3 be connected by any ordinary curve X lying wholly within S and coin- ciding with the given arc in the neighborhood of a. Let /(z) be holo- morphic in S and suppose its values to be given along that portion of X lying in an arbitrarily small neighborhood of a. We are to show that /(/3) is then uniquely determined. Since f(z) is holomorphic in the neighborhood of a, its derived functions are also holomorphic in the same neighborhood, and hence for z = a the successive derivatives /'(a), /"(a), . . . , /(-)(), . (3) all exist. The existence of the derivative /'(a) implies that the limit Az=0 AZ is the same, when Az approaches zero in any manner whatever. Hence, the value of /'(a) may be found from the given values of /(z) by taking this limit as z approaches a along any curve proceeding from a, for example along the given curve X . The higher derivatives are likewise determined by the given values of /(z). Knowing the value of /(z) for z = a and the successive derivatives given in (3) we may now write out Taylor's expansion of /(z) for values of z in the neighborhood of a, namely, f"(a) fW(a) ) ( 2 -)+J (*-)+ . . . +J-J( f - a )-+ .... (4) This series converges and defines an element Q (z) which is equal to /(z) for all values of z within any circle, drawn about a as a center and lying wholly within the region S. Let C be a circle satisfying these conditions. If the point lies within Co, then the value of /Q3) is already seen to be uniquely determined; for, in order to find this value all we need to do is to substitute /3 for z in series (4). If /3 lies outside of Co, let i be a point of intersection of the curve X with C . Take a point z\ on the given curve arbitrarily close to a\ but within C . The function /(z) is holomorphic in the neighbor- hood of z = zi and the successive derivatives of /(z) for this value of z can be found by successively differentiating (4) term by term and 248 SINGLE-VALUED FUNCTIONS [CHAP. VII. substituting z\ for z in the several derived series. The coefficients of Taylor's expansion of /(z) about the point Zi are therefore uniquely determined. The resulting expansion is 2! (z-zi) 2 + (5) This series in turn defines an element 2 (z) which is holomorphic in an adjacent region *S 2 , having an arc C of an ordinary curve for at least a portion of the boundary between Si and i(z), fa(z) must in this case satisfy the same conditions as at points along the arc C in the case where the regions are adjacent but do not over- lap, namely, they must have equal values and be holomorphic. In both cases we speak of the functions i(z), fa(z) as elements of the function /(). If an element i(z) is continued analytically along a curve from a to /3 if this curve lies wholly within a finite sequence of connected regions Si, S z , . . . , S n such that i(z) has an analytic continuation fa(z) defined for Sz and z(z) likewise can be continued analytically in the region 83, etc. The following theorem * sets forth the necessary and sufficient conditions for analytic continuation. THEOREM II. Given two functions i(z), fa(z) which are holo- morphic respectively in the adjacent regions Si, S z having an arc C of an ordinary curve as that portion of their boundaries common to the two. The necessary and sufficient condition that each of these functions is an analytic continuation of the other is that they converge uniformly to equal values on C. It follows at once from the definition of analytic continuation that the conditions set forth in the theorem are necessary; for, if either is an analytic continuation of the other, then they must have equal values along C, and moreover, for these values they must be holo- morphic and hence continuous with the values at points within Si, Si, respectively. To show that the given conditions are sufficient we define /(z) as follows: /() = i(z) = ^2(2) along C. * See Painlev, Toulouse Annales, Vol. II, p. 28; also Pompeiu, L'Enseigne- ment Mathemalique, July, 1913, p. 305. AKT. 49.] ANALYTIC CONTINUATION 251 We must then show that f(z) is holomorphic in the region S com- posed of S i} S z and the points of C, end points excepted. Let 71, 72 be two ordinary curves joining any two points A, B of C and lying wholly within the regions Si, S z , respectively. By the Cauchy- Goursat theorem, we have f /7i f /7 2 1 (z)dz z(z')dz AB BA (6) (7) Denote by 7 the curve 71 + 72. Combine the integrals in (6) with the corresponding integrals of (7). Since i(z) = fa(z) for values of z along C, we have FIG. 91. / i(2) dz + / ^2(2) dz = 0. JAB J BA By definition /(z) is equal to i(z) for values of z in & and to 02 (2) for values of z in $ 2 . Hence, we have from (6) and (7) ff(z)dz= f 1 (z)dz *J y 1/7! (8) Since 7 is any ordinary curve lying wholly within S and passing through A, B, it follows from Morera's theorem that /(z) is holo- morphic for all values of z in S and hence for all values of the vari- able along the curve C, end points excepted. Consequently, either of the two functions 0i, o (z) which is holomorphic within C . Let Zi be any point within C but lying arbitrarily close to Co. Since (z) is holomorphic in the 252 SINGLE-VALUED FUNCTIONS [CHAP. VII. neighborhood of z = z i} we can compute Taylor's expansion of i(z) satisfy the condi- tions of analytic continuation and consequently define a function /(z) which is holomorphic in this region. Either of these elements may be again continued by the same means. By this process any given element may, at least theoretically, be continued analytically by the method of power series until the whole complex plane, with the exception of certain points or regions excluded by the inherent character of the function /(z) so defined, is covered by overlapping circles of convergence. The character of the exceptional points and regions will be discussed in subsequent articles. ' The importance of the power series method of analytic continu- ation is due largely to its theoretical value. Other methods, although restricted in their uses for theoretical discussions, are of much greater practical importance hi the applications of the theory of functions. We shall now consider a method introduced by Schwarz,* in which he makes use of the principle of symmetry. Let 0i(z) be a function which is holomorphic in a region Si lying in the upper half-plane and having a segment AB of the axis of reals as a part of its boundary. Suppose that as z approaches any point a: of AB along any path whatsoever lying interior to Si, i(z) approaches a definite real value i(z) along the axis of reals. * See CreUe, Vol. LXX, pp. 106, 107; also Mathematische Abhandlungen, Vol. II, pp. 65-83. ABT. 49.] ANALYTIC CONTINUATION 253 In the continuous region S made up of Si, S 2 and the points along the axis of reals between A and B, the functions 0i(z), (z) is uniquely determined and holomorphic. The region S can be so restricted that the function z = ^(T) and its inverse function T = (z) map the regions S, S' upon each other. FIG. 93. Then to any conjugate imaginary points n and fi lying respec- tively in Si and Sz, there are associated two corresponding z-points namely z\ and zi lying respectively in Si and 8%, and conversely. It is to be noted that the particular values of z thus associated depend upon the form of the curve C and not upon the form of the parametric equations (11) of the curve. Suppose, for example, a different para- metric representation of the curve C is obtained by replacing t in (11) by an analytic function of any other real variable r. If we then permit r to take complex values, conjugate imaginary points in the r-plane correspond to conjugate imaginary points in the r-plane, and consequently we get the same corresponding values of z. Of the two z-points corresponding to conjugate imaginary values of a parameter T, either is said to be the reflection or image of the other with respect to the curve (7. Likewise the region Sz f may be spoken of as the reflection of the region Si with respect to C. This definition of reflection with respect to a regular arc of an analytic curve may now be used in developing a method of analytic continuation. Let Si, S 2 f be any two adjacent regions such that S 2 ' is the reflection of Si with respect to the regular arc C of an analytic curve whose parametric equations are x = Vi(C), y = ^ 2 (0- Let i (z) be a function which is holomorphic in Si and defined for values ART. 49.] ANALYTIC CONTINUATION 255 of z along C by its limiting values as z approaches the points of C by any path whatever lying wholly within Si. We can now state in the following form the necessary and sufficient condition that 0i(z) may be analytically continued by reflection with respect to arc C. THEOREM III. The necessary and sufficient condition that 0i(z) can be analytically continued by reflection with respect to the regular arc C of an analytic curve forming a portion of the boundary of the region for which 0i(z) is defined is that 0i(z) converges uniformly to real values along C. If 0i (z) can be analytically continued across the arc C into the region $2', which is a reflection of Si with respect to C, then for the region Sz' a function 02 (z) is determined such that 0i(z), 02 (z) define a function /(z) , holomorphic in the region S' consisting of the points of C and the regions Si, Sz'. Along C the functions /(z), 0i(z), 02(z) take equal values which are continuous with the values taken respec- tively in the regions Si, Sz'. As may be seen, the substitution z = x + iy = ^(r) transforms the functions 0i(z), i(z). Moreover, if FZ(T) is obtained as an analytic continuation of FI(T) by means of Schwarz's method of reflection, then #2(2) is a continuation of i(z) by reflection with respect to the arc C. But as we have seen the necessary and sufficient condition that FI(T) can be analytically continued by reflection upon AB is that Fi(r) takes along AB real values which are continuous with the values taken by this function in Si; that is, that FI(T) converges uniformly toward real values along AB. Accordingly the necessary and sufficient condition that tf>i(z) can be analytically continued across C by reflection is that i(z) of the function /() be defined for the region S bounded by three arcs Ci, 3, Ca of circles cutting the circle C at right angles and suppose that 4>i(z) converges uniformly to real values along Ci, C 2 , Cj. We shall now reflect the region S with respect to one of these arcs, say the arc Ci. In order to accomplish this we shall first show that the reflection of any point of (z) is defined. The function /(z) so defined is called a mono- genie analytic function, or more briefly an analytic function. As it is impossible to further extend this region S, it is called the region of existence of the analytic f unction /(z). The element from which the other elements are obtained by the process of analytic continuation is called the primitive element of the function, and the remaining elements become analytic continuations of it. The region of existence consists of a continuum of inner points, each of which is a regular point of the f unction /(z). The region of existence may extend over the entire finite portion of the complex plane. On the other hand, it is possible in the process of analytic continuation to encounter a closed curve beyond which the function can not be analytically continued. In such a case the curve is called a natural boundary. For example, in the function discussed in con- nection with Fig. 94 of the last article, the curve C constitutes a natural boundary, since it is impossible__tg continue^-the. function analytically across this curve. A pbrtiorT of the complex plane into which the function can not be continued because of a natural boundary is called a lacunary space. Often, instead of a lacunary space, we encounter a set of points, not constituting a continuum, which can not be included in the region of existence. Such points are not regular points of the function, and hence they must be classed as singular points. The various classes of singularities of single-valued analytic functions will be more fully discussed in the following article. Since power series may be used as a means of analytic continuation, it follows that an analytic function may also be defined as one that is developable, except in the neighborhood of singular points, by Taylor's expansion. It is to be noted also that a single-valued analytic function of a complex variable is uniquely determined throughout its region of existence as soon as its values in the neigh- borhood of any regular point of that region are given. The particular method of analytic continuation employed in extending the region from the neighborhood of the given point to the region of existence of the function thus determined is a matter of indifference. Moreover, any two analytic functions are equal for all values of z in this region of existence if they have a common element. An important distinction between functions of a complex variable and those of a real variable may be noted. If a function of a com- ART. 50.] ANALYTIC FUNCTIONS 259 plex variable has a derivative at each point of a given region S, then at all points of S it has derivatives of every order, and in the neigh- borhood of any point of S the function is represented by the Taylor series to which it gives rise. On the other hand, it does not follow that if a function of a real variable has at each point of an interval derivatives of every order that the Taylor's expansion derived from the function represents that function for all values of the variable in the neighborhood of the point at which the derivatives are taken. The following example will serve as an illustration. _ j_ Ex. 1. Given the function /(x) = e *\ x ^ 0, where /(O) = 0. Let it be required to determine the interval of equivalence of this function and the power series obtained from this function by expanding it in a Maclaurin series. We have, for x T 0, /(*)-""*, /<">(*) =G(z)^-', where G(x) is a polynomial in x. For x = we have by use of the limit _ e l2 Cu 2 ) 2 LT vy i ~ ~ L " = L - L A*=0 AX ^i 'J VW T SA A/A^S? n = L ] 4 o (Ax;^ [ . .. = 0. Ai=0 ^ x Ai=0 1 '(Al) See Stolz, Differential-und Integralrechnung, p. 76, also p. 81, Ex. 3. 260 SINGLE-VALUED FUNCTIONS [CHAP. VII. The expansion derived from the given function is a power series, each term of which is zero. This series defines, in the interval of convergence, a func- tion <(>(x) = 0; that is, the function defined by the series is represented geometric- ally by the X-axis. On the other hand, the given function y = f(x) is represented by the curve C tangent to the X-axis at the origin. This curve is symmetrical + 1 O FIG. 95. with respect to the F-axis, and has the line y = 1 as an asymptote as shown in Fig. 95. It follows that the function that gave rise to the series is represented by that series in only one point, namely x = 0. The reason for the distinction pointed out between functions of a complex variable and those of a real variable is, so far as the partic- _i_ ular function discussed is concerned, that while e * is an analytic function, the point z = is not a regular point, since the derivative with respect to the complex variable z does not exist at the origin; although in the realm of real variables the corresponding derivative does exist. Consequently, the point 2 = does not belong to the region of existence in the complex plane. It is of importance to point out in this connection the distinction between an analytic function as defined and an analytic expression. The notion of an analytic function implies a definite correspondence between the z-points and the w-points of the complex plane. This relation may have different forms of expression in different parts of the plane. An analytic expression on the other hand is the result obtained by performing upon the independent variable the analytic ART. 50.] ANALYTIC FUNCTIONS 261 operations of addition, subtraction, multiplication, division, inte- gration, etc., including the general process of taking the limit. It leads to a formal expression of the relation between z and w. This analytic expression, however, may define for different regions of the plane elements of different analytic functions. The following illus- trations will make clear the distinction. Consider the analytic expression This series converges * for all values of z except for values upon the unit circle about the origin. Within this circle the series con- verges to the limit For values of z exterior to the unit circle the series converges to the limit Hence, for \z\ < 1, E(z) may be considered as an element of the 2 analytic function - , and for MB > 1 it may be considered as 1 2 an element of the analytic function ; -- . In either case the ele- J. ~~ 2 ment E(z) may be analytically continued over the entire finite plane with the exception of the point 2=1, but in the one case the result- 2 1 ing analytic function is r -- , while in the other it is ; --- J. 2 A 2 As another illustration, suppose we have two detached regions Si, Sz- Let (1(2), 2 (z) be elements of two distinct analytic functions /i (2), f z (z). Suppose \(z) for values of z within Ci and to 2 (z) for values of z within C 2 . It follows then that E(z) defines an element of the analytic function /i (2) or/ 2 (z) according as z lies within Ci or C 2 . 51. Singular points and zero points. We have defined a sin- gular point of a function (Art. 14) as a point that is not a regular point of the function, but in every deleted neighborhood of which there are regular points. As we have seen in the previous article the singular points of an analytic function are to be considered as boundary points of its region of existence, and they may even form a closed curve constituting a natural boundary of such a function. If a is a singular point of an analytic function /(z), then either the function has no derivative at the point a itself, or there are points in every neighborhood of a at which the function has no derivative. In either case the higher derivatives of the function can not exist at a, and hence the function does not permit of an integral power series development in the neighborhood of a. If we undertake by means of power series to continue analytically an element of an analytic function along an ordinary curve passing through a singular point, the circles of convergence within which the successive elements are defined grow gradually smaller as their centers approach the sin- gular point; for, as the function is always holomorphic within these circles none of them can ever inclose the singular point itself. The singular points of a single-valued analytic function may be classified as poles, or non-essential singular points, and essential singular points. The point z = a is a pole, or non-essential singu- lar point, of the analytic function f(z) if there exists a positive inte- gral value of k such that the product (z - )*/() is holomorphic in the neighborhood of a and different from zero for z = a. The integer A; is called the order of the pole. Thus the point z = 2 is a pole of order 2 of the function f(z} = 3** + ! J() (z - 2) 2 ART. 51.] SINGULAR POINTS, ZERO POINTS 263 for, multiplying f(z) by (z 2) 2 we obtain the function 3 z z + 1, which has the point z = 2 as a regular point and is different from. zero for 2 = 2. If A; is equal to one, the pole is often referred to as a simple pole. If no finite value of k exists such that the singularity of the single- valued analytic function /(z) at a point a is removed by multiplying by the factor (z a) fc , then a is said to be an essential singular point of f(z). If a singular point can be inclosed in a circle, however small, having that point as a center and containing no other singular point of the given function, then the point is said to be an isolated singular point. An isolated essential singular point is one that may be inclosed in a circle containing no other essential singular point. It may, however, have an infinite number of poles in its neighborhood, as we shall see later. A point may, therefore, be an isolated essential singular point without being an isolated singular point. The following theorem due to Riemann is important in establish- ing the character of a function at a point in the deleted neighborhood of which it is limited in absolute value and holomorphic.* THEOREM I. Let f(z) be holomorphic in a given region S except at the point z a, where the behavior of the function is not known. If for all values of z 7* a in S we have \f(*)\(z) which is holomorphic in the neighborhood of z . More- over, (zo) = * is different from zero. Hence we have /(z) = (z - Zo)VO), where () satisfies the conditions set forth in the theorem. The point z = ZQ is said to be a zero point of order k of the analytic function /(z), if there exists a positive real integral value of k such that the product is holomorphic in the neighborhood of z and different from zero for z = ZQ. If z = ZG is a regular point of the analytic function /(z) and if /(z,,) = 0, then by the foregoing theorem z = z is a zero point. By multiplying /(z) by the factor -; rr, where k is the order of (z ZQ) the zero point, the vanishing point is removed. That a theorem does not exist for real variables analogous to Theorem II for analytic functions of a complex variable is illustrated by the function _i^ f(x) = e *, x^O. This function satisfies the conditions of Theorem II, stated with reference to the real domain in the deleted neighborhood of the origin; that is, it has all derivatives with respect to x in this deleted neighborhood, and moreover, L f(x) = 0. z=0 But we have the limit * See Stolz, Differential-und Integralrechnung, Part I, p. 81. ART. 51.] SINGULAR POINTS, ZERO POINTS 267 for all values of k, and hence the zero point can not be removed by introducing the factor -^, no matter how large k may be taken. JU Between the zero points and the poles of an analytic function, * there exists the following relation. THEOREM III. // z = ZQ is a pole of order k of the analytic function. /(z), then 7^- is holomorphic in the neighborhood of ZD and has a zero J( z ) point of order k at ZQ, and conversely. Since z = ZQ is a pole of order k of /(z), we have from the definition of a pole (z - Zo) fc /(*) = <(z), (3) where (z~) is holomorphic in the neighborhood of ZQ and <(zo) =^ 0. Hence in the neighborhood of ZQ, --T-T = ^(z) is also holomorphic (z) and can be expanded in a power series, the first term of which is a constant different from zero. From (3) we have Since both (z z ) fc and ^(2) are holomorphic in the neighborhood of 2o and ^(ZQ) 7^ 0, the last member of (4) is holomorphic in the neighborhood of Zo and can be expanded in powers of (2 20) begin- ning with the k th . Hence ^-r has a zero point of the order k, as J ( z ) stated in the theorem. The converse of the foregoing theorem follows similarly; for, if 77^- is holomorphic with a zero point of order k at Zo, we may write f(z) - = (2 - 20)^(2), (5) where ^(2) is holomorphic and different from zero for z = ZQ. Conse- quently, we have (2 - *)*/(*) = ^, where 7^7^- is also holomorphic and different from zero for 2 = 2 . b(z) Hence z is a pole of order k of the given f unction /(z). In the foregoing demonstration we have made use of the fact 268 SINGLE-VALUED FUNCTIONS [CHAP. VII. that if a function is holomorphic and different from zero in the neigh- borhood of ZQ, then the reciprocal function ;rr^- is also holomorphic and J( z ) different from zero hi the neighborhood of z . If, on the other hand, the point ZQ is not a regular point of the function /(z) , then this point must be a zero point or an essential singular point of the reciprocal function according as it is a pole or an essential singular point of /(z) . _i_ For example, z = is an essential singular point of e * and is also an essential singular point of e **. By aid of Theorem III we can now establish the following theorem. THEOREM IV. // an analytic function /(z) is holomorphic and different from zero in the deleted neighborhood of z = ZQ, and if L /(Z) = 00, Z=ZQ then the point z = ZQ is a pole off(z). Since L/(z) = oo, z=*o we have at once L *=* /(*) As/(z) is holomorphic in the deleted neighborhood of ZQ, it follows that jr-r is holomorphic and different from zero in the same deleted neigh- JW borhood. Hence, by Theorem II there exists a positive integer k such that j-L = (z - Zo)V(z), where (z) is holomorphic in the neighborhood of z and <(z ) ^ 0. The function -^. has then a zero point at z , and consequently by J ( z ) Theorem III, /(z) must have a pole at the same point. Hence the theorem. As in the case of Theorems I, II, the analogous theorem for the realm of real variables does not exist, as the following illustration shows. Ex. i. Show that Theorem IV does not hold for the following function (Fig. 97) of a real variable, namely: l_ /(z) =e*% x 5*0. AKT. 51.] SINGULAR POINTS, ZERO POINTS 269 The conditions of Theorem IV are satisfied for real values of the variable in the deleted neighborhood of the origin. But as x approaches zero the product x k f(x) becomes infinite * for all values of k. Hence, the infinity of the function can not be removed by introducing the factor x k no matter how large k be chosen. + 1 FIG. 97. THEOREM V. The zero points of an analytic function are isolated. Let Z Q be any zero point of the analytic function f(z). We may then write /(z) = (z - 6)**(z), (6) where (z) is holomorphic in the neighborhood of 2 and different from zero for z = z . Since 0(z) is continuous, we can then draw a circle C about ZQ as a center, within which $(z) does not vanish. For any value of z 7^ z within C, (z z ) k is likewise different from zero. Consequently, within C there is no point other than z at which f(z) vanishes. The zero point ZQ is therefore isolated. But ZQ was any zero point of f(z) and hence all such points are isolated. * See Stolz, Differential-und Integralrechnung, p. 81. 270 SINGLE-VALUED FUNCTIONS [CHAP. VII. The corresponding theorem for the poles of an analytic function may be stated as follows: THEOREM VI. The poles of an analytic function are isolated singu- lar points. If an analytic function /(z) has a pole at any point Zo, then by Theorem III TT-T is holomorphic in the neighborhood of ZQ and has the j( z ) value zero at ZQ. But we have just seen (Theorem V) that the zero points of an analytic function are isolated. Consequently, the poles must also be isolated. If the poles of an analytic function /(z) have a limiting point, then the behavior of /(z) in the neighborhood of that point is given by the following theorem. THEOREM VII. // z = Zo is a limiting point of the poles of an ana- lytic function f(z) , thenf(z) has an essential singularity at ZQ. In every neighborhood of ZQ there are poles of the given function. The point z = ZQ can not then be a regular point of the function, and hence must be either a pole or an essential singular point. It can not be a pole, because as we have seen (Theorem VI) every pole is an isolated singular point. It must then be an essential singular point as the theorem states. If an analytic function has an infinite number of poles, they must have at least one limiting point either in the finite region or at infinity. We now see that at this limiting point the function has an essential singularity. It follows then that an analytic function having no essential singularities can have but a finite number of poles. We have seen that if /(z) is holomorphic in the deleted neighborhood of ZQ and L/() -oo, 2=Zo then ZD is a pole of /(z). We shall now show that conversely an analytic function always becomes infinite as the variable approaches a pole; that is, we shall demonstrate the following theorem. THEOREM VIII. // the analytic function /(z) has a pole at z = ZQ, then the function /(z) always becomes infinite as z approaches ZQ by any path; that is, L /(z) = oo . ART. 51.] SINGULAR POINTS, ZERO POINTS 271 Suppose that /(z) has a pole of order k at ZQ, then by the definition of a pole, we have (z - Zo)*/(2) = 0(2), where $(z) is holomorphic in the neighborhood of z and f or z = ZQ is different from zero. For* values of z ^ ZQ, we have then (> - *,)* As <(z) is finite and continuous for z = ZQ, then as z approaches by any path whatsoever - ( u inc (z ZQ) limit. Consequently, we may write by any path whatsoever - ( u increases in absolute value without (z ZQ) I as the theorem requires. Not only may an essential singular point of an analytic function appear as a limiting point of poles, but it may also be the limiting point of other essential singular points or it may appear as an iso- lated singular point of the function. For isolated essential singular points, that is essential singular points that are not the limiting points of other essential singular points, we have the following theorem.* THEOREM IX. // ZQ is an isolated essential singular point of f(z), and /3 is any arbitrary number, real or complex, then z may be made to approach ZQ in such a manner that the corresponding values of /(z) have the limiting value /3. By hypothesis the point z can not be the limiting point of other essential singular points of the function, although it may be the limiting point of poles of the function. Moreover, there can not exist a neighborhood of z , however small, such that at every point of it we have /(z) = /3; for, in this case/(z) would have by Theorem I at most a removable discontinuity at ZQ, and hence this point could not be an essential singular point. Consider the function -jvrn> for values of z in the neighborhood of z . * This theorem, commonly attributed to Weierstrass, was doubtless first demonstrated by the Italian mathematician, Casorati. See Rend. 1st. Lomb., (2) I, 1868; also Vivanti-Gutzmer, Theorie der ein- deutigen analytischen Funktionen, p. 130. 272 SINGLE-VALUED FUNCTIONS [CHAP. VII. Either there exists a finite number M such that for all values of z in every neighborhood of z<> we have \F(zy\ M, however large M may be taken; that is, hi every neighborhood of ZQ there are values of z for which | jP(z) | > - , where e is arbitrarily small. For all such values we have I/W -*_!<* We may then select a set of points Zr, Zj, . . . , Z B , . . . , having ZQ as a limiting point, such that L /CO = ft. *n=Zv The foregoing theorem must not be understood to mean that hi the neighborhood of an essential singular point /(z) actually takes every value. Picard has shown,* however, that in the neighborhood of an essential singular point a single-valued analytic function takes all complex values with the exception of at most two and indeed an infinite number of times. Thus far we have considered only those singularities of a single- valued function that occur at finite points of the complex plane. To determine the nature of the function in the neighborhood of the * See M6moire sur les foncticms entibres, Ann. de 1'Ecole Normale, 1880; also Traitt d'analyse, Vol. II, p. 121. ART. 51.] SINGULAR POINTS, ZERO POINTS 273 infinite point, we subject the variable z to the reciprocal transforma- tion and examine the transformed function (z') for values of z' in the neighborhood of the point z' = 0. The given function f(z) is said to have a pole or an essential singularity at infinity according as z' = is a pole or an essential singular point of <(>(z'). The function (z) is said to have a regular point at infinity if z' = is a regular point of the function (z'). In case the point z = oo is a regular point of the given function /(z), then the transformed function <(z') is holomorphic in the neigh- borhood of z' = 0. We can then expand <(z') in a Maclaurin series and have 4>0') = o + iz' + + a n z' n +. Consequently, the expansion of /(z) in the neighborhood of z = co , when this point is a regular point of the function, is of the form /./ \ i 1 , , <*n , /(z) = a + -+...++.... It has been shown that if /(z) is holomorphic in a given region S, then the integral / /(z) dz taken around any closed curve C lying wholly within S and inclosing only points of S must vanish. It is of interest in this connection to point out that this conclusion does not hold when the given region includes the point at infinity. For this case, we have the following theorem. THEOREM X. // C is an ordinary curve inclosing the point at infin- ity and lying within a given region S which likewise contains the point at infinity, then the integral I /(z) dz vanishes if z 2 /(z) is holomorphic J c in S. Putting z = , we have z f /(z) dz= - f z'~ 2 <(z') dz', (Art. 22) UC Jy where 7 is the curve about the origin into which the curve C is mapped by the transformation z = . The given integral vanishes whenever 274 SINGLE-VALUED FUNCTIONS [CHAP. VII. the integral / z'~* (z') dz' vanishes, that is, if z'~ 2 <(z') is holo- Jy morphic in a region S' about the origin within which the curve 7 lies. However, if z'~ 2 (z') is holomorphic in S f , then z*/(z) must be holo- morphic in the corresponding region S about the point infinity and conversely. Hence the theorem. THEOREM XI. The circle of convergence of a power series passes through at least one singular point of the analytic function determined by the series. In the discussion of Taylor's series (Art. 48) it was pointed out that the power series in (z Zo) resulting from the expansion of a given function which is holomorphic in a region S converges and represents that function for all values of z within any circle that can be drawn about Zo as long as it lies within S and incloses only points of S. That is, the size of the circle C (Fig. 87) within which the series is known to converge is limited only by the region S in which the given function is holomorphic. As we now know, that region S is restricted only by the presence of singular points of the analytic function /(z) of which the given power series defines an element. Consequently, if the circle C is the circle of convergence of the Taylor series, then it must pass through at least one singular point of /(z) ; that is, within every larger concentric circle there must be at least one such point; otherwise, a larger circle than C might be selected in determining a region within which the Taylor series converges and the series would then converge for points outside the circle C. This circle would not then be the circle of convergence as assumed. If a function is holomorphic in a given region, it can be expanded in a Taylor's series for values of z in the neighborhood of any point of that region. The expression obtained for the coefficients of such an expansion enables us to establish the following theorem, due to Liouville. THEOREM XII. A single-valued analytic function which has no singularity either in the finite portion of the plane or at infinity reduces to a constant. If a function /(z) has no singularity either hi the finite region or at infinity, it follows that it is everywhere less in absolute value than some definite number M ; for, otherwise, there would exist a point ZQ, finite or infinite, in every neighborhood of which /(z) would exceed ART. 52.] LAURENT'S EXPANSION 275 in absolute value all finite bounds, that is, would become infinite. The function admits of a Maclaurin expansion about the origin, namely a + aiZ + a 2 Z 2 + ' + a n z n + (6) which converges and represents the function for all finite values of z. The coefficient a n is n 1+1 where C is a circle of radius p about the origin as a center, the value of p being taken as large as we please. Since l/(*)l 0, where e is an arbitrarily small positive number. Consequently, since a n is a constant, we must have a n = 0, n = 1, 2, 3, .... It follows from equation (6) that /(z) = o for all values of z in the finite portion of the plane. Since the point z = oo is a regular point of /(z), we have /(z) = L /(z) = . 2 = 30 It follows from the foregoing theorem that every single-valued analytic function which is not a constant must have at least one singu- lar point either in the finite portion of the plane or at infinity. 52. Laurent's expansion. We have seen that, in the neighbor- hood of a regular point of an analytic function, it can be represented by a power series, but this method of representation does not hold in the neighborhood of a singular point of the analytic function. We shall now show that in the neighborhood of an isolated singular point Zo we can expand an analytic function in a series having also negative powers of (z z ). Such a series is not properly a power series, since a power series was defined as a series involving only positive integral powers. We shall, however, often refer to the series involving nega- tive powers as a power series with negative exponents or a power 276 SINGLE-VALUED FUNCTIONS [CHAP. VII. series in - When the term power series is used without a Z Zo qualifying phrase, we shall as heretofore understand it to mean a series involving only the positive powers of the variable. In the derivation of Taylor's expansion (Art. 48), it was found to be valid within a region bounded by a single circle, provided there v are no singular points of the given analytic function within the circle. Suppose we now consider a region S bounded by two concentric circles Ci, C 2 (Fig. 98) such that within S, f(z) has no singular points and con- verges uniformly to finite values along each circle. There are no restrictions as to singular points exterior to C\ or interior to C 2 . Denote the common center of Ci, C 2 by Zo. To apply this method later to the expansion of a function in the neighborhood of a singular point, it is convenient to take the radius of C 2 arbitrarily small. As in the consideration of Taylor's series, we shall base our dis- cussion upon the fact that we can express the given analytic function /(z) by means of the Cauchy integral formula. Since the given region S is bounded by two curves the integral must be taken over the entire boundary and hence along the two curves in the directions indicated hi the figure. Taking, however, the integral along C 2 in a negative direction with respect to the region S, that is in a counter-clockwise direction, we have for any value of z in S FIG. 98. j_ r/(^_ JL f /) 2TiJ Cl t-Z 2iriJ Cl t- where t is taken along each of the curves Ci, <7 2 in a counter-clockwise direction. Since z is any point in S, then for the first integral we have | z ZQ | < | i z |. By Theorem III, Art. 20, this integral by itself defines a function (z) which is holomorphic for all values of z within Ci and hence can be expanded in a power series in (z ZD) by means of Taylor's expansion. Such an expansion is of the form Zo) + 02(2 Z ) 2 + a n (z - Zo) n + ,(2) ART. 52.] LAURENT'S EXPANSION 277 where we have The second integral also defines a function 1^(2) which is holomorphic for values exterior to C 2 , that is for | z ZQ | > 1 1 z \, the values of t being limited to values on C 2 . To find a form of expansion for \fs(z), we proceed in a manner similar to that used in the discussion of Taylor's series. We shall consider the function - - , which occurs in the t z given integrand. 1 We 1 may (* write z \ -1 1 t z z 1 t Zo -Zo (t- (t Z/ Z ZQ- -Zo) 2 t Zo z Zo (t- z _ (3) z - ZQ (z- Zo) 2 (z- Zo) 3 (z - zo) n This series, considered as a series in t, converges uniformly (Art. 45, Theorem I) for* any constant value of z such that |z ZQ\ > \t 2o|, that is for any value of z exterior to C->. The property of uniform convergence is not destroyed by multiplying each term of (3) by f(t). We have then /(o /(o tz zZo (z Zo) 2 (zZo) 3 (ZZO) H Since this series converges uniformly, we may integrate it term by term, thus obtaining 2iri Ct t-z (t - ^ )/(() dt The integrals in the second member of this equation determine the coefficients of the desired expansion of the second integral in (1). We may therefore write ^) = a- 1 (0-z )- 1 + a- 2 (2-2o)- 2 + ' ' +a-n(2-2o)- n + , (4) where |z-zo|> |*-2o|, a- n =-.(t-zo) n - l f(t)dt, n=l, 2, . 278 SINGLE-VALUED FUNCTIONS [CHAP. VII. Since the series (2) converges for all values of z within C\ and the series (4) for all values of z exterior to Cz, it follows that both con- verge for values of z within S bounded by these two circles. Conse- quently for values of z within S, the given f unction f(z) may be written as the sum of two functions (z), ^(z), the first of which can be ex- panded in a series involving the positive integral powers of (z ZD), and the second of which can be expanded in a series involving the negative integral powers of (z ZQ); that is, we have n=0 n=l We may replace the two circles C\, Cz as paths of integration by a single path of integration. This path of integration may be any ordinary closed curve C lying within S, and inclosing C 2 since each of the circles Ci, Cz may be deformed into C without passing over a singular point of the integrand. The coefficients of the two series (2) and (4) may then be expressed in terms of the integrals taken over the curve C. We have then the following theorem. THEOREM I. // /(z) is holomorphic in the annular region S bounded by two concentric circles about a given point ZQ, then within this region /(z) can be represented by a series of the form where and C is any ordinary curve lying wholly within S and inclosing the inner circle. The series (5) is known as Laurent's series. While there may be an infinite number of terms of the series corresponding to negative values of n, on the other hand only a finite number of such terms may appear in the expansion, the number depending as we shall see upon the character of the function /(z) at the point z . By aid of the fore- going theorem we can now represent a single-valued analytic func- tion in the deleted neighborhood of an isolated singular point by means of a series involving the positive and negative powers of the variable; for, if 2o is such a singular point, then by making the radius of Cz sufficiently small but different from zero we can include in the ART. 52.] LAURENT'S EXPANSION 279 region S any point in the deleted neighborhood of ZQ. Hence, while OO 00 2) n(z 2o) n converges for all values of z within C\, ^a_n(z Zo)~ n n=0 n=l converges for all values of z within C\ except z = ZQ. As has already been pointed out, the nature of a singular point of an analytic function is fully determined by the behavior of the func- tion in the deleted neighborhood of that point. The Laurent expan- sion of the function also determines the character of the singularity. For example, if z = z is a pole of order k of the analytic f unction /(z), then we are able to remove the singularity by multiplying /(z) by the factor (z ZQ)*. Hence there are k terms in the Laurent expan- sion having negative exponents; that is, the expansion is of the form + 2 (z - z ) 2 + +n(z -*)+*. (6) That part of the expansion which indicates the character of the singu- larity, namely r(* - 2 o) r , r=-l is called the principal part of the expansion. In case of a pole of order k, it consists of k terms. The Laurent expansion of a given analytic function in the neigh- borhood of an isolated singular point may be accomplished by direct application of Theorem I, but if the singular point z is a pole of order k, we may write where 0(z) == a-k + a-k+i(z Z ) + ' * * i -* ^ 0, whence Ex. 1. Expand the function in a series for values of z in the neighborhood of the origin. This function can be written in the form 4 z2 _ 4 z -I- 1 1 / \ * ^ -* * i^ * j i -* where (2) = r = 42 + 1-2 '1-2 = 1 -32+2 2 280 SINGLE-VALUED FUNCTIONS [CHAP. VII. Hence, we have 131 The terms -j - + - are the principal part of the expansion of f(z) in the neigh- borhood of the origin. If /(z) has an isolated essential singular point at ZQ which is not the limiting point of poles of /(z), then there is no finite value of k such that (z Zo) k f(z) is holomorphic in the neighborhood of that point, and hence the Laurent expansion has an infinite number of terms involving negative powers of (z ZQ). The expansion is then of the form t + Oo + l(z - Zo) + . . . + On(z ~ 2o) n + . . - ', (7) that is, the principal part of the expansion consists of an infinite num- ber of terms. In case Zo is a regular point of the function, the Laurent expansion has no terms with negative exponents and hence becomes identical with the Taylor expansion. If the point z = oo is a pole of order k of a given function /(z), then the function <(z') obtained by transforming /(z) by the relation z = , must have a pole of order k at the origin. Its expansion is therefore of the form In the neighborhood of z = ob the expansion of /(z) is therefore of the form Z 2" the first A; terms constituting the principal part. As with Taylor's expansion, the question arises as to whether an analytic function is uniquely represented by means of a Laurent series. In this connection we may well consider the following theorem. THEOREM II. // in an annular region S a given function /(z) per- mits of an expansion of the form oo */ \ "X. ' / \ M /(Z) = 2- n(z - So)", ART. 52.] LAURENT'S EXPANSION 281 then the coefficients of this expansion are given by the relation that is, there is but one such expansion possible. As in the discussion of Theorem I, let the region S be bounded by the circles Cj, C 2 , having the point ZQ as a center. We assume the existence of an expansion of f(z) in a series as stated in the theorem. Denote by C any circle concentric with Ci, C 2 and lying between the two, and let the value of the variable along C be denoted by t. The given series converges along C and expressed in powers of (t z ) is \L ZQJ \t> ~~ ZQ) \t ZQ) + oo + ai (t - 2o) + ' ' +ctn(t- Z ) n + . (8) This series converges uniformly (Theorem I, Art. 45) and hence may be integrated term by term along C. Before doing so, however, let us multiply the terms of the series by the factor j- y^ri* Remem- (t ZQ) bering that L (t - z ) n dt = 0, n^-1, (Exs. 1, 2, Art. 18) n = 1, (9) it follows that the integrals of all of the terms of the series vanish except one, namely the term involving We have then as the t ZQ result of the integration I TT^ \n + i = ^ lr ^ a n) whence, we have 1 / * dt, which establishes the theorem. It does not follow from what has been said that /() may not have different Laurent expansions in different circular regions. For ex- ample, suppose we have two regions (Fig. 99) one bounded by the circles CiC 2 and the other by C 2 Cs, where C 2 has upon it a singular point of the given function. In each of these regions there is an expansion in a Laurent series, but the two expansions in such a case are not identical. This condition is illustrated by the following example. 282 SINGLE-VALUED FUNCTIONS [CHAP. VII. Ex. 2. Given the function FIG. 99. This function has two poles, namely, z = 1, z = 2. Within the circle about the origin passing through the point z = 1, the function can be represented by a Maclaurin series. The resulting series is 1 . 3. . 7 , , 15 . .' . 2 n + 1 -l r 2 + 4 Z + 8^ + l6 2 + ' ' ' + ~2^ z + "> which converges and represents the given function for | z | < 1. If we take 1 < | z | < 2, we must use Laurent's expansion. The coefficients of the series are given by where C is any circle about the origin lying between Ci and C 2 . Putting for/(0 its value from (11), the integrand in (12) becomes 1 1 jn+2 _ 2 J+l f+2 Upon decomposing each of these fractions into partial fractions, we have for 1 f 1. 124 2 n + 1 U-2 t P t 3 ft 2 n + 1 \t-l t P t 3 t n + l \ Replacing the integrand in (12) by these partial fractions, since f t n dt = 0, n * - 1, ^0 = 2irt, n = 1, we have 1 C ^ 1 J_r_^L.i / 1 Q^ Vi J c t - 2 2 n + 1 27rtJc-l' i " ' an c t ART. 52.] LAURENT'S EXPANSION 283 But : = is holomorphic within C and hence the integral f vanishes. t A Jc t 2 To evaluate the second integral in (13) we deform the path C of integration into a small circle C' about the point 2 = 1 and put t 1 = pe**, where p is a constant and varies from to 2 TT. We have then c dt c dt - r 2 '// - 9 Jet- l ~ Jc'T^l ~ l Jo Consequently from (13), we have 1 1 n - - ^xi ~ x + L o^+T 2 n+l The terms of the required Laurent expansion corresponding to values of n = are then 111, 1 -~ 2 -* ----- For negative values of n, say n = k, we have J_ r r J*- 1 dt _ r fl-Jdn 2wi \_JC t-2 Jc t- 1 J' But the integrand in the first integral is holomorphic within C and hence the integral vanishes. We have by deforming C into C' and putting as before t - 1 = peje, = I : 2wiJo P&& --5^C*< 1 +^*- 1 ; -x - Hence, each term in that portion of the Laurent expansion having negative ex- ponents has the coefficient 1. The complete expansion is therefore .., i i_i i. ' ,._.... w -. It is evident that the given function has no finite singular point exterior to the circle about the origin and passing through the point 2 = 2. The expansion of the function about the point 2 = co will then hold for this entire region. By putting the entire region exterior to the circle C 2 through 2 = 2 inverts into the region about the origin and lying within the circle C" whose radius is 5. The trans- formed function is z' z' 284 SINGLE-VALUED FUNCTIONS [CHAP. VII. Within the circle C" the function (z') is holomorphic and may be expanded in a Madaurin series, giving *(*') = z' 2 + 3 z' 3 + 7 z' 4 + Heplacing z' by-, we have as the expansion of the given function for values of Z exterior to the circle C t , -+-+-4 2"- 1 - 1 The same result would have been obtained had we expanded the function by computing the coefficients by aid of the formula given in Theorem II, where the path of integration is any circle C 3 about the origin and lying exterior to the concentric circle through z = 2. 53. Residues. We have seen that the integral / /(z) dz van- ishes when taken around the boundary C of a region S, provided /(z) is holomorphic in the open region S and at least converges uniformly to its values along C. Let us now consider the effect upon this inte- gral when S contains one isolated singular point of /(z). Before doing so, we introduce the following definition. If the points of S, with the exception of at most the point z , are regular points of /(z) and C is any closed curve about z and lying wholly within S and aside from ZQ containing only points of S, then the integral J-. if(z)dz taken in the positive direction is called the residue of /(z) at ZQ. Suppose the point z is a pole of the given function. We have then the following theorem. THEOREM I. ///(z) is holomorphic in a given finite region S except at 2o, where it has a pole, then the residue off(z) at z is equal to the coefficient of (z Zo)" 1 in the expansion off(z) in powers of (z ZQ). Let the pole at Zo be of order k. Then the Laurent expansion of the function in powers of (z z ) is of the form = _ -< _l_ "+* -! , ^/V) (z-zo)* + (z-zo)*-^ h z-^ + W' where <(z) is holomorphic in the neighborhood of z , say within and ART. 53.] RESIDUES 285 upon a circle C having ZQ as a center. Taking C as the path of inte- gration, we have //<) dz = a- k f (z- 2o)- fc dz + t/C t/C + _! f ( - zo)- 1 ^ + f () <(z) dz vanishes, since 0(z) is holomorphic in the closed t/C region bounded by C. To evaluate the remaining integrals we make use of the relations / (z-zo) n dz= 0, n^ -1, t/c = 2iri, n, = -1. (2) Consequently, we have from (1) I f(z) dz = 2iria-i, Jc whence a_i = =. I f(z) dz, (3) Zm Jc which establishes the theorem, since by definition the second member of this equation is the residue. The value of the residue of an analytic function at a pole is zero if the coefficient a-\ is zero in the expansion of the function. For example, the function -y has a pole of order three at the origin, yet the residue, JL ffo 2irij c z 3 is zero. The foregoing theorem gives the residue when there is a single isolated pole in the given region S. If there are a finite number of poles in S, we have the following theorem. THEOREM II. Given a function f(z) which is holomorphic in a region S with the exception of a finite number of poles, and let C be any ordinary curve lying wholly within S and inclosing all of the given poles. Then I f(z] dz taken in a positive direction is equal to2wi times J c the sum of the residues of f(z} at these poles. 286 SINGLE-VALUED FUNCTIONS [CHAP. VII. Suppose the poles of /(z) to be z\, z-t, . . . , z*, . . . , z n . About each of these points as a center draw an arbitrarily small circle lying wholly within the region bounded by C. Denote these circles by Ci, C 8 , . . . , C*, . . . , Cn. Then by Theorem VI, Art. 19, we have (4) But as we have seen I /(z) dz is equal to 2 iri times the residue of Jc k /(z) at Zfc and is given by the coefficient of (z z*)" 1 in the Laurent expansion of /(z) in powers of (z z*). The relation given in (4) therefore establishes the theorem. A closed curve C may be regarded as the boundary of either of two regions, one finite and the other inclosing the point at infinity. It is readily seen that the relation (4) holds when C is regarded as bounding the outer region^as well as in the case just considered. Theorem II is still valid therF/ ffiEeip/(z) dz is taken over a curve inclosing the point z = oo. However, when the point z = oo is a pole the residue at that point is not given by (3). For this case we have the following theorem. THEOREM III. // the analytic function /(z) has a pole at z = oo, then the residue of /(z) at that point is the negative of the coefficient of z~ l in the expansion of /(z) for values of z in the neighborhood of z = oo. Putting z=~7, we denote the transformed function by (z'). z As z = oo is a pole, say of order A;, of the given function /(z) then z' = is a pole of the same order of <(z')- Expanding <(z') in a Laurent series for values in the neighborhood of the origin we have '. (5) Replacing z' by - , we have the expansion of /(z) in the neighborhood z of z = oo, namely: " v m ART. 53.] RESIDUES 287 where z z F(z) is holomorphic in the neighborhood of z = oo. In determining the residue of /(z) at the point 2=00 the integral denn- ing a residue is to be taken around an arbitrarily large circle C about the origin in a clockwise direction. The resulting integral is then the negative of the integral taken around C in a positive or counter- clockwise direction. The integral / F(z)dz vanishes by Theorem Jc X, Art. 51. We have then from (6) by aid of the relations given in (2), it being understood that the integral is taken in a clockwise direction around C, / f(z) dz = 2 Trtai, Jc or Consequently, from the definition of a residue, it follows that the residue at oo of an analytic function having a pole at infinity is the negative of the coefficient of z~ l in the expansion of the function in the neighborhood of the point 2 = oo , as stated in the theorem. It is of interest to observe that a function can have the point 2 = oo as a regular point and still have a residue at that point different from zero. For example, the function is holomorphic in the neighborhood of z = oo, yet it has a residue ai at that point. The theory of residues is of value in the discussion of some of the important properties of analytic functions, as we shall see in the succeeding articles. It may also be applied with advantage to the evaluation of certain integrals of functions of a real variable, as we shall now show. First of all, we shall show how we may employ the results of our discussion of residues to evaluate an integral of the form where f(x) is the quotient of two polynomials. In order that this integral shall have a significance, we make the assumption that the denominator is of degree at least two higher than the numerator.* For the sake of simplicity, we shall also assume that the denominator has no real roots. * See Pierpont, Theory of Functions of Real Variables, Vol. I, Art. 635. 288 SINGLE-VALUED FUNCTIONS [CHAP. VII. Let us consider then a function /(z) which is holomorphic along the axis of reals, and with the exception of at most a finite number n of poles, holomorphic in the finite upper half of the complex plane. Consider now a region S bounded by a curve consisting of a seg- ment of the axis of reals and a semicircle C about the origin, lying in the upper half plane and having the radius p. We select the value of p so that the poles of /(z) already mentioned shall all lie in S. Denoting the residue of the given function at each pole z k by R k , then by Theorem II, we have upon integrating about the contour of S /(z) dz (7) where / denotes the integral along the axis of reals between p J-P and p. We shall first consider the limit L ff(z)dz. = 00 UC dz Put z = pe te and hence = i dO. We have, therefore, Z f f(z)dz=i r #(*)&. Jc /o But, from 5, Art. 17, we have \f c f(*)dz (8) Let M denote the upper limit of | z/(z) | upon the semicircle C. From (8), we have then I j /(z) dz < M^'dB = *M. If as p increases without limit, M approaches zero, we have L p=oo (9) We now regard /(z) as the quotient of two polynomials, where the denominator has no real roots and is of degree at least two higher than that of the numerator. Then the required conditions are satisfied, for /(z) has a zero of at least order two at infinity and L M = that is, the result expressed by (9) follows. ART. 53.] RESIDUES 289 Moreover, under the conditions set forth in the statement of the problem each of the limits L f"f(x)dx, L C~"f(x)dx p=oo y p = ao i/O exists, and by passing to the limit we obtain from (7) the value of /00 the integral I f(z) dz, namely *J 00 i=l Xoo fix , ' ,..- -oo (X* + I) 3 The function /(z) = has a pole at z = i. Expanding /(z) in powers of (z i) we have t 3 3t 8 (z - i) 3 16 (z - i) 2 16 (z - i) Hence, the residue of /(z) at z = i is ^ We have then i Mfc vj-- /o - dx oo (Z 2 + I) 3 ~ 16 ^ ^ v n == 7^ " * ITt = ~?P' The method of residues may be applied also in evaluating certain integrals of trigonometric functions when the integrals are taken between the limits and 2 w, as the following example will illustrate. Ex. 2. Evaluate the integral { jr : - - Jo 2 + cos Putting z = eM, we have 1 dz & + tr* Z z whence we obtain JUL, , = ' 2 r (z f ) has at z' = 0, and therefore f(z) has at z = oo, a pole, a regular point which is not a zero point, or a zero point according as n > m, n = m, n < m. The given conditions are also sufficient. To show this let us assume that the poles of /(z) in the finite region are at ft, /3 2 , . . . , /3 m , and let us denote their orders by ki, /c 2 , . . . , k m , respectively. Then from the definition of a pole, we have G,(z) = (z- ft)*'(z - ft)* 2 ...(*- &,)*-/(), (3) where (?i(z) is holomorphic over the entire finite region and is differ- ent from zero for 2 = ft, ft, . . . , /3 OT . However, if G\(z] is not a constant, it must have at least one singular point, and since this can not be a finite point, it must be the point at infinity. More- over, this singular point must be a pole, for otherwise f(z) would also have an essential singularity at infinity. It follows from Theorem I that Gi(z) is either a constant or a rational integral function. From (3) we have _ _ (Z - ft)*'(2 - ft)*' ... (2 - ft.)*. ~ G 2 (2) ' where 6^1(2) and G%(z) have no common factor. Consequently, f(z) is a rational function as stated in the theorem. A single-valued f unction /(z) having a finite number of poles but no essential singular points in a given region S is said to be meromorphic in S. Thus a rational fractional function is meromorphic in the en- tire complex plane. The function w = tan 2 is meromorphic through- out the finite part of the plane, the poles being the zeros of cos 2. As has been pointed out, a rational function may be either integral or fractional. We may now combine the results of Theorems I and III into one theorem for the unique characterization of rational functions. That theorem may be stated as follows: THEOREM IV. The necessary and sufficient condition that a single- valued analytic function is a rational function is that it has no essential singularities. We can now establish the following theorem. 294 SINGLE-VALUED FUNCTIONS [CHAP. VII. THEOREM V. A rational function is definitely determined, except for a constant factor, by its zero points and its poles in the finite portion of the plane. Let f(z) be the given rational function. There can be but a finite number of zero points and a finite number of poles. Let the poles in the finite region be ft, ft, . . . , /3 m having respectively the orders &i, 2, . . . , k m . These points must then appear as the zero points of the rational integral function Or 2 (z) in the denominator of the given function. We may then write (? 2 (z) = C 2 (z- ft)*'(* - ft)*. . . . (z - ft.)*-. The finite zero points of /(z) appear as the zero points of the rational integral function G\(z} of /(z). If we denote these zero points by 01, az, . . . , a n and their orders respectively by ri, r 2 , . . . , r n , we have Gi(z) = Ci (z - 01)^(2 - 2 ) rs . . (z - a n ) r . We have then <*(*) (Z - ft)X* - A)*' (* - &.)*-' /^ where C = 7^. In case Gz(z) reduces to a constant, the function C/2 /(z) is a rational integral function having but one pole, namely z ="o. Any rational function /(z) can also be expressed as a constant or a rational integral function plus a finite number of rational fractions of the form where a, /3 are real or complex constants and v is a positive integer. To express a given rational f unction /(z) in this form consider the prin- cipal parts of the expansion at the various poles. Let the poles of /(z) in the finite region be ft, ft, . . . , /3 OT , having respectively the orders &i, 2, . . . , k m . Let the principal part of the expansion at ft be c'_ t , c'-h+i i c'-i (z - ft)*'" 1 " (z - ft)*- 1 "* ^(z-ftf Subtracting this from/(z) we have left a function that is holomorphic in the entire finite portion of the plane except at ft, . . . , ft. Proceeding hi the same manner with the remaining poles just enu- merated, we finally have a function that is holomorphic in the entire ART. 54.] RATIONAL FUNCTIONS 295 finite plane and that has at most a pole at infinity; that is, we have a rational integral function, say of the form C + CiZ + C2Z 2 + ' + C n Z n . We then have /(z) = C + CiZ + C 2 Z 2 + + C n Z n I C'-*. [ c'-trfl | . . . j C'_! ft // // + ^ *; i C t;+l . . C 1 /*. /D \ I / iO \t. 1 I I / n \ () (TO) \Z Pm) '" (Z p m ) m (Z /3 m ) The terms in the first row of this expansion, aside from the constant term c, constitute the principal part of the expansion of /(z) in the neighborhood of infinity. Consequently, we have the following theorem. THEOREM VI. A rational function is definitely determined except for an additive constant by the principal parts of its expansion at its various poles. The following examples furnish illustrations of Theorems V and VI and are left as exercises for the student. Ex. 1. If a function has no other poles than simple poles at 2 = 1, 1, and its only zero points are of the first order and located at z = i, i, show that it is necessarily of the form Ex. 2. Show that a function having no singularity either in the finite part of the plane or at infinity, other than a pole at the origin with the principal part 3,2 2* + Z*' is of the form THEOREM VII. The sum of the residues of a rational function is zero. Let /(z) be the given rational function. This function will have no essential singular points, but will have a finite number of poles, one of which may be at infinity. Let C be any circle in the finite region not inclosing any of the poles of /(z) and not passing through 296 SINGLE-VALUED FUNCTIONS [CHAP. VII. any such points. This circle, described in a clockwise direction, may be regarded as the boundary of the region exterior to it, that is the region containing all of the poles of /(z). The value of the integral taken in a clockwise direction, is by Theorem II, Art. 53, equal to the sum of the residues of /(z). The value of the integral is unchanged except as to sign if it be taken in the counter-clockwise direction in- stead. But since /(z) is holomorphic within and along C this latter in- tegral vanishes. Hence taking the integral in the clockwise direction it is zero also. It follows then that the sum of the residues of /(z) is neces- sarily zero as stated in the theorem. Some of the general properties of single-valued analytic functions that lead to special properties of rational functions may be readily de- duced from a consideration of the logarithmic derivative, namely that is the quotient of the first derived function by the function itself. If the point z = ZQ is a regular point of /(z) and if /(ZQ) is different from zero, then this point is a regular point of the logarithmic deriv- ative. We shall see, however, that if /(z) has a zero point at z , the logarithmic derivative is not holomorphic in the neighborhood of ZQ. We have in fact the following theorem. THEOREM VIII. // /(z) is holomorphic in the neighborhood of the point z = ZQ and has at this point a zero point of order k then the loga- rithmic derivative has at the same point a simple pole and a residue k. The given function /(z) has a zero point at ZQ of the order k, and hence it can be written in the form /(z) = (z - Zo)V(z), (5) ART. 54.] RATIONAL FUNCTIONS 297 where 2 is a regular point of 0(2) and this function is different from zero for z ZQ. We have, therefore, /'() = k (z - 2 )*- 1 0(2) + (z- 2o)V (2), and hence obtain /(z) 2-200(2) I It \ The function ^lAJ is holomorphic in the neighborhood of ZQ. The 0(2) f'(z) Laurent expansion of -sr-r in powers of (z 2o) contains but one term JW k having a negative exponent, namely - ; the rest of the terms Z ZQ have positive exponents, since they arise from the expansion of the ,// \ holomorphic function ~T-T. Hence, the residue is k and the point v*/ z = z is a pole of order one, as required. THEOREM IX. If f(z) has a pole of order k at z = ZQ, then the loga- rithmic derivative has at the same point a simple pole and a residue k. We may write (Z - 2o)*/(2) = 0(2), where #(2). is holomorphic in the neighborhood of the point z = z , and is different from zero for z = ZQ. We have then for z 9* z tff\f \lc 7 f \ k 1 -t / ^ j fit \ v \ z )( z %o) K(Z Zo) . If X = k > 0, /(z) has a pole of order k at infinity, and if X = k, /(z) has a zero point of order k at infinity. We have then for the logarithmic derivative /'(z) Xz*-V(z) + zV(z) /GO " 2 X *(*) f'(z) By Theorem III, Art. 53, the residue of j^~ is X, provided J \~) .//_\ z 2 , . is holomorphic in the neighborhood of z = oo . This condi- (z) tion is easily seen to be satisfied by substituting for z and examin- z ing the transformed function in the neighborhood of z' = 0. This discussion leads to the following theorem. THEOREM XI. A rational function is just as often zero as infinite, when the entire complex plane including the point at infinity is considered, each zero point and each pole being counted a number of times equal to its order. ART. 54.] RATIONAL FUNCTIONS 299 This theorem follows at once from Theorem VII concerning the sum of the residues of a rational function. It was shown that this sum is necessarily zero. Hence, from Theorem X the number of zero points must equal the number of poles, each being counted a number of times equal to its order. The foregoing theorem admits of a generalization as follows: THEOREM XII. A rational function of degree X takes any given value, real or complex, exactly X times. If f(z) is rational, then F(z) = f(z) C is also a rational function, where C is any constant. By Theorem XI the function F(z) must be as often zero as infinity. Hence, we may say that f(z) takes any arbitrary value C'as often as F(z) becomes infinite. We shall now show that F(z) becomes infinite a number of times equal to the degree of the f unction /(z), thus establishing the theorem. The degree of F(z) is the same as that of f(z). The function F(z) may be written in the form where the degree X of F(z} is the larger of the two numbers n, m, or equal to either m or n in case m = n. In any case F(z) has a pole at every point where the denominator vanishes; for, by hypothesis the numerator and denominator have no common factor. For n = m = X there are X poles in the finite portion of the plane corresponding to the X zero points of the denominator. We can show as follows that the point 2 = oo is a regular point of F(z). Putting 2 = -,, the transformed function 0(2') is a\ a x -0 + ^7+ ' ' ' +-r ^/x + aiZ 'x-i + . . . _|_a x 1 -, # which is for z' = 0. Since 0(2') is holomorphic in the neighbor- Px hood of the origin, the point z = oo is a regular point of ^(2). In this case then the number of poles of F(z) is equal to the degree of the function. For n < m, there are m poles of F(z) in the finite region corre- sponding to the m zero points of the denominator. In this case also 300 SINGLE-VALUED FUNCTIONS [CHAP. VII. the point z oo is a regular point of F(z), in fact F(z) has a zero point at infinity; for, we have ao z PO T 2 / 2 / m which is equal to zero for z' = 0. The total number of poles of F(z) is therefore m = X, the degree of F(z). If we have n > m, then in addition to the m poles corresponding to the zero points of the denominator, F(z) has a pole of order n m at infinity; for, we have I 1 , , n *' ^ *" 2'" 002'" + which has a pole of order n m at 2' = 0. Hence F(z) has a pole of the same order at infinity. In this case therefore the totality of poles is equal to n = X, the degree of ^(2). Hence the theorem. 55. Transcendental functions. As we have seen, an analytic function that is not algebraic is a transcendental function. As all single-valued algebraic functions are necessarily rational, it follows that any single-valued analytic function that is not rational is transcendental, and hence we can now readily identify such func- tions by means of their singularities. A single-valued transcendental function must have an essential singularity; for, otherwise by Theorem IV, Art. 54, it would be a rational function. If a single- valued analytic function G(z) has no singularity in the finite region and has an essential singularity at infinity, it is called a transcendental integral function of 2. The expansion of such a function in a Mac- laurin's series gives GOO = oo + i2 + <* 2 2 2 + + a n z n + , (1) which converges for all finite values of 2. If in (1) we replace 2 by , we have a transcendental integral function of - , namely 2 2o 2 2o From the form of the expansion it will be seen that this function has but one singular point, namely an essential singular point at 2 = ZQ. ART. 55.] TRANSCENDENTAL FUNCTIONS 301 Conversely, if a single-valued analytic function has an essential singular point at z = ZQ and has no other singular points, then it can be expanded in the form given in (2) and hence is a transcendental integral function of -- z ZQ As we have seen, a power series may be integrated or differentiated term by term for values of the variable within the circle of con- vergence, and the resulting power series has the same circle of convergence as the given series. Consequently, the integral or the derived function of a transcendental integral function G(z) is represented by a power series that converges for all finite values of z and hence is itself a transcendental integral function. The following functions are transcendental integral functions: Z 3 2 s 2 7 3!~ l ~5~!~7P " 2 2 . Z* Z 6 . 2-j + r! - r! + - |l + 5l + f! + ' ' ' 2 2 2 4 2 6 A transcendental integral function differs from a rational integral function, in that it may have no zero points or it may have an infinite number of zero points. For example, the function /CO = * is different from zero for all finite values of 2. In fact we may state the following theorem. THEOREM. Any transcendental integral function /(z) having no zero points may be written in the form /CO = G(Z \ where G(z) is an integral function. Since /(z) is a transcendental integral function,/' (2) is also a trans- cendental integral function, and as /(z) has no zero points, then 302 SINGLE-VALUED FUNCTIONS [CHAP. VII. is an integral function, as is also the function defined by the integral I ' F(z) dz. We have then the integral function (z) defined by the relation (z) dz = (*& dz = log/ (z) - Putting 4>(z) + log/(zo) = G(z), we have log/(z) = G(z), whence /(z) = e G(z) , where G(z) is an integral function. The function /(z) = fl(z) e^\ where R (z) is a rational integral function and G(z) is an integral function, is a transcendental integral function having a finite number of zero points. As an example of a transcendental integral function having an infinite number of zero points, we have /(z) = sin z, which is zero at the points Z = 0, 7T, 2 7T, . . . , kw, . . . . A transcendental integral function differs from a rational integral function hi still another way. As the rational integral function has a pole at infinity, there always exists a circle about the origin such that for all points exterior to this circle we have | /(z) \ > M, where M is an arbitrarily large number. Since a transcendental integral func- tion has an essential singularity at z = oo, the given function may be made to approach any value as z becomes infinite. Consequently, there are always values of z exterior to any circle however large about the origin for which | /(z) \ > M and also values of z for which | /(z) I < e, where e is an arbitrarily small positive number. A transcendental function having poles but no essential singular points in the finite portion of the plane is called a transcendental fractional function. This distinction between transcendental inte- gral and transcendental fractional functions is suggested by the corresponding distinction between rational integral and rational fractional functions. In both cases a function is called integral when it has but one singular point and that is at z = oo. Likewise in both cases we call a function fractional if it has poles and no other singular points in the finite region. ART. 56.) MITTAG-LEFFLER'S THEOREM 303 The functions 2 , tan z. sec 2 Sin 2 are illustrations of transcendental fractional functions, for they have an essential singular point at infinity and poles but no other singular points in the finite region. Each may be written as the quotient of two integral functions. It will be shown later (Art. 57) that every transcendental fractional function can be written as the quotient of two integral functions. We have pointed out that any rational function can be expressed as the sum of a rational integral function and fractions of the form (2 - 2o)" It is not difficult in that case to set up the function when the prin- cipal part of the expansion is known for each of the various singular points, which for rational functions consist of a finite number of poles. The corresponding problem for transcendental functions, namely, the problem of setting up a function with arbitrarily chosen singular points and with corresponding arbitrary principal parts, is much more difficult. The question of the existence of an analytic function having a given infinite set of singular points, with given principal parts, will be considered in the following article. * 56. Mittag-Leffler's theorem. Suppose we have given any in- finite set of numbers 2i, 22, . . . , z k , . . . , all different, having the property that and suppose that L 2 fc = 00. t= Mittag-Leffler was the first to show* that there always exists a single-valued analytic function having these points and no others as singular points, with given principal parts of the form G k (^\ k = 1,2,3,. (1) where Gk is an integral function of , rational or transcendental. 2 Z k If, as in rational functions, we add together the functions (1) we have * See Encyklopadie d. Math. Wiss., Bd. II 2 , p. 80. 304 SINGLE-VALUED FUNCTIONS [CHAP. VII. in this case an infinite series. Whether the function defined by this series is everywhere holomorphic except at the points z k depends upon the nature of the convergence of the series. Mittag-Leffler showed that by associating with each principal part Gk ( ) a \2 Zk/ suitably chosen polynomial the series can be made to converge uni- formly and hence define a function having the desired properties. His theorem may be stated as follows : THEOREM. Given an infinite set of points Zl, 22, 2 3 , . . . , Z k , . . . , such that , L Zk = oo. There exists a single-valued analytic function which is holomorphic for all finite values of z 7* z k and which has an arbitrarily chosen integral function of - , namely Gk\- -}, as the principal part of its ex- Z Zk \Z Zk/ pansion in the neighborhood of 2*. Let + The three functions (14), (15), (16) are made use of by Weierstrass in the theory of elliptic functions. 57. Expansion of functions in infinite products. In addition to expanding an analytic function by means of an infinite series, another method is often employed, namely infinite products. Suppose we have the infinite sequence The continued product of the first n elements may be denoted by II u + *). (i) t=i If at most a finite number of the factors (1 + a*) are zero, that is if there exists a positive integer m\ such that for k ,= mi all of the factors are different from zero, then the product (1) is said to con- verge if the limit L f[ (! + *), f| - t = m, * Borel has shown that it is sufficient to put v > log k (Lecons sur Us functions entibres, p. 10). ART. 57.] INFINITE PRODUCTS 309 exists and is different from zero. If the product tends toward zero, or becomes infinite, or if for any reason it does not have a limit as n increases indefinitely, then the infinite product is called divergent. The necessary and sufficient condition that an infinite product converges may be stated as follows : THEOREM I. The necessary and sufficient condition that an infinite product converges is that corresponding to an arbitrary positive number t there exists a positive integer m such that for all values of n > m we have n a + o - 1 < e, P = i, 2, (2) i=n+l Consider the sequence of the products TOj+2 U> II > V IL , n>mi. (3) By Theorem VI, Art. 12, the necessary and sufficient condition that this sequence converges is that for every positive number ei there exists an integer m such that n+p n n-n in\ ni\ n>m>mi, = 1, 2, 3, .... (4) We shall show that this condition is equivalent to the one given in the foregoing theorem. If the given infinite product converges, then we have L = A ^ 0. There then exists a number M > 0, such that for all values of n > we have ' > M. (5) n Dividing (4) by (5) we get n+p n ja._! n ' n>m, = 1,2,3, .... 310 SINGLE-VALUED FUNCTIONS [CHAP. VI. Putting -^ = e, this result may be written +P < e, n > m, p = 1, 2, 3, . . . , which is the condition given in the theorem. Conversely, suppose we have given the inequality (2). We may write n+p n m t 1 2= n+p n nti i n n mi n ' n mi But from the condition (2), we have by aid of the above relation n+p n that is, we have n < , n > m, p = 1, 2, 3, . . . ; n+p n mi (6) or (1-6) n m, n+p n n n n>m, p = l,2,3, .... (7) Suppose we now give to n any definite value greater than m, say n = mi + v. Since e may be chosen arbitrarily, it follows from (7) n that each element JJ of the sequence (3) is in absolute value less mi than a positive number N, which is the largest number in the finite sequence n tn,+2 n n n where p is any constant greater than zero. Likewise each element of (3) is larger in absolute value than a positive number Ni, which is the smallest number in the finite sequence m,+2 n m\ n n n mi ART. 57.] INFINITE PRODUCTS 311 From (6) we have n+p n n-n < n , n > m, p = 1,2,3, .... But since we have n < N, it follows that by putting where e' is arbitrarily small, we have n+p n-n Hence, the condition given in (4) is satisfied and the limit n exists. Suppose the limit is A. Since n is always greater than the positive number Ni, it follows that A 3* 0. Hence the given sequence converges and the demonstration of the theorem is com- pleted. If the product J^[ (1 + | a* j ) converges, then the. product JJ (1+a*) is said to converge absolutely. If an infinite product converges but does not converge absolutely, it is said to con- verge conditionally. As a condition for absolute convergence, we have the following theorem. THEOREM II. // the series V a k converges absolutely, then the in- t=i finite product converges absolutely. From the convergence of the series So:*, it follows that only a finite number of the factors (1 + ak) can be zero. We assume as before that for k = mi the factors are all different from zero. We may then write for n > m\ TT = e log(l+a mi )+log(l+a mi+l) + +log(l+a n ). 312 SINGLE-VALUED FUNCTIONS [CHAP. VII. The given product will then converge absolutely if the series |a*l)-|- - (9) converges. But this series converges, if the series | ai | + I 2 | + + | a* | + - (10) converges; * for, the ratio of the general terms of the two series, namely log(l + |<**|) [Sn has the limit 1 as fc becomes infinite. Since the series (10) con- verges by hypothesis, that is 2 a* converges absolutely, the theorem follows. In the discussion of rational integral functions we saw that such functions are determined except as to a constant factor when the zero points are known. In the case of transcendental integral functions we saw that the given function might have no zero points, or on the other hand it might have an infinite number of such points. If we have given an infinite set of points 2i, 22, 2s, , 2*, . . . , having 2 = oo as a limiting point, it is of interest to see whether an integral function can be set up having these points and no others as zero points. Clearly such a function must be a transcendental func- tion, since a rational integral function can have but a finite number of zero points. Weierstrass has shown how the desired function can be represented by an infinite product. It is at once clear that if $(2) is such a function and G(z) denotes an integral function, then F(z) = *()e c is also such a function, since as we have seen e G(z) can have no zero points. The function e G(z) plays the same role that the constant factor does in the representation of a rational integral function as the prod- uct of a finite number of binomials. We may now state the following theorem. THEOREM III. Given a set of points 4, , ... * ,... (11) not including the origin such that | Zi | S | 22 | S ... | Z* | . . . , L Z k = 00 . t-ao * See Bromwich, Theory of Infinite Series, Art. 9. ABT. 57.] INFINITE PRODUCTS 313 There exists a transcendental integral function of the form having the points z k and no others as zero points. Moreover, the function F(z) = e G W-3>(z) is the most general function having this property. Consider the infinite product where fa [ )is a rational integral function of ( ] as yet undetermined. \ZkJ \Zk/ The factor is called a primary factor. This factor has one and only one zero, namely 2 = 2fc. We shall show how the functions <*( ), k = 1, 2, ... . , \Zk/ can be determined so that the infinite product (12) will converge in an arbitrarily large circle and define a function having the required properties. Let Ck (Fig. 101) be the circle about the origin as a center having | Zk | as a radius. Let Ck be a circle concentric with C* and having the radius p k = e | z k |, < e < 1. Denote by R the radius of any circle C about the origin. Then there exists an integer r such that C lies within r ' r' W , ^ r+l > > and that within and upon C we have at most the points Zl, 22, ... 2 r _i. (13) The product of the corresponding primary factors is an integral function having the points (13) as zero points. The remaining factors of (12) give rise to the product 314 SINGLE-VALUED FUNCTIONS For any integer q we have [CHAP. VII. (15) Hence, the convergence of the infinite product (14) will follow from the convergence of the series fl_J + 0VlM. (16) Since all of the points z k , k ^ r, lie outside of the circle C, it follows that if the right-hand member of (15) converges, it defines a function that does not vanish for z within or upon C. Consequently, the same may be said of the infinite product (14), provided this product is convergent. Expanding log (1 ) in a power series, we have V Zk/ f z\ z 1 / z\ 2 1 / z\ 3 1 i z\ n (z\ - ) be of degree m k 1, where m k 1 is to be de- z k / termined later. We may then put 1 z k From (16) we now have (18) We have R<0\z k \, k = r. For all values of z within or upon the circle C, we have then By aid of this relation -we obtain z k 1 - Hence, from (18) we have 2* (19) ART. 57.] INFINITE PRODUCTS 315 It follows then that the series (14) converges uniformly if the series p ffr ** converges uniformly, which it does if m* = k, as we have seen (Art. 56). Each term of the series (16) is holomorphic for values of z within and upon the circle C, and since the convergence is uniform, it follows that (16) and hence (14) defines a function which is holomorphic within and upon C. But C is any circle about the origin as a center, hence for m* equal to k the product e** defines a transcendental integral function $(z) having the required properties. If we desire the most general function of this type, all we need to do is to introduce as a factor the most general function that has no roots, namely the function e G(z) , where G(z) is an integral function. In the foregoing discussion the origin was not included in the set of zero points. If it is desired to include that point as a zero point, say of the order X, all that is needed is to add the factor z x and write the function F(z) =e G( *z*3>(z), (20) which is also a transcendental integral function. It is evident that by varying the function G(z) in (20) we may obtain an infinite num- ber of transcendental integral functions having the points given in (11) as zero points. We have seen that there may exist an integer p independent of k which causes the series I Zk to converge and hence also the infinite product defining $(2). We may then put ra* = p and have =i (21) In the discussion of integral functions it is desirable to introduce what is known as the class of the function. For this purpose let us 316 SINGLE-VALUED FUNCTIONS [CHAP. VII. suppose that p is the smallest integer that makes the series " converge. Let us also suppose G(z) to be a polynomial, say of degree g. Then the class* of the integral function F(z) given by (21) is defined as the larger of the two integers p 1 and g. Since the degree of G(z) can be changed without affecting the zero points of F(z), we may so choose the polynomial G(z) that g is less than p 1 and hence p 1 is then the class of F(z). The class of any rational integral functions is zero, as is also that of the transcendental integral function On the other hand, the function is of class one. We have seen, Art. 44, for example, that the series converges. Hence there exists a transcendental integral function of the form having the ft points and also the point 2 = and no others as zero points. This function is the ^-function of Weierstrass and is of great importance in his development of the theory of elliptic functions. It is evidently a transcendental integral function of class two. It has been seen that every rational fractional function is the quotient of two rational integral functions. Such a function is uniquely characterized by the fact that it has at most a pole at 2 = oo and only a finite number of poles in the finite region of the complex plane. In a similar manner, we have defined a transcenden- tal fractional function as one that has an essential singularity at 2 = oo and only poles in the finite region. These poles may, how- ever, be dense at the point 2 = oo. We can now demonstrate the following theorem. * The term doss is here used as the equivalent of the French word genre in- troduced by Laguerre, and the German word Hohe, introduced by V. Schaper. See Borel, Lemons sur les fonctions entieres, p. 25; also Osgood, Encyklopadie d. Math. Wins., Vol. II 2 , Part I, p. 79. ART. 58.] PERIODIC FUNCTIONS 317 THEOREM IV. Every transcendental fractional function can be ex- pressed as the quotient of two integral functions. Let the points ,,...,*,..., (22) where | Zl | = | 22 | = ' ' = | Zk | = ' ' , L Z k = 00, i=oo be the poles of the given transcendental function f(z). If any of the poles is of an order higher than one, we shall regard a number of the points z k equal to the order of the pole as coincident. By The- orem III there exists a transcendental integral function Gz(z) having the points (22) and no others as zero points. Moreover, at each point the order of the zero of Gz(z) is the same'as the order of the pole of f(z) ; for, in each case the number of coincident points Zk is the same. It follows that the product Gi(z) f(z) has no singular points in the finite region. Consequently, we have (&(*)/(*) = G,(z), where G\(z) is an integral function. It follows then that G t (z)' as the theorem requires. 58. Periodic functions. In the discussion of elementary func- tions in Chapter IV, attention was called to the fact that certain of those functions are simply periodic; that is, the function remains invariant under a translation of the plane by means of the relation z' = z + no), where n is an integer and w is the primitive period of the function. If /(z) is the given function, we then have /(* + n) =/(*). (1) In the illustrations considered the period-strips were taken parallel to the axes of coordinates. It is not necessary, however, to choose the strips in that manner; for, if we locate the points where z is any point, and draw parallel lines through these points making a convenient angle different from zero or ^ with the X-axis, the strips bounded by these lines may be taken as period-strips. As 318 SINGLE-VALUED FUNCTIONS [CHAP. VII. a matter of fact, the boundary lines of the regions of periodicity need not even be straight lines; for, all that is essential is that the plane be divided into congruent strips such that for any point z in any strip there corresponds a point z + nw in each of the other strips for which equation (1) holds. It is of importance also to note that a given function may repeat its values in a period-strip; for, as we have seen in the case of w = cos z, the period-strips are not identical with the fundamental regions of the function. Functions like the exponential function and the trigonometric func- tions are, as we have seen, simply periodic. Single-valued analytic functions may, however, have two independent periods, where we understand two periods of a function to be independent if they are not integral multiples bf the same primitive period. For example, a function is said to be doubly periodic if it has two periods 2 coi, 2 w 3 , which are independent of each other and of z, such that a translation of the plane by either of the relations z' = z + 2 wi, (2) z" = z + 2w 3 (3) leaves the function unchanged. We have then the two relations /(* + 2 =/(), (3) /(z + 2 ,)=/(). (4) As in the case of simply periodic functions, any translation of the complex plane by means of the relations z' = z + 2 miwi, mi = 1, 2, . . . , (5) z" = z + 2 maws, m a = 1, 2, . . . , (6) also leaves the function unchanged. It follows at once that any combination of the translations (5), (6) leaves the given function invariant, since any such combination may be regarded as a suc- cession of translations by means of (2), (3). We may then write f(z + 2 m*i + 2 m 3 w 3 ) = f(z), (7) which shows that Q = 2 m^i + 2 maws (8) is likewise a period of the given function. If all of the periods of the given function can be written in the form (8), that is if every such period can be expressed as the sum of integral multiples of these two periods, then 2 wj, 2 w 3 are called a primitive period pair. A = m/ m 3 mi" m 3 " ABT. 58.] PERIODIC FUNCTIONS 319 In order that any two pairs other than 2 coi, 2 o> 3 , for example 2 mi'coi + 2 m 3 'co 3 , 2 m/'coi + 2 m 3 "co 3 , may be taken as a primitive pair, it is sufficient that we have = 1- (9) For, let 2 co/, 2 co 3 ' be any two independent periods of f(z) other than 2 coi, 2 co 3 . Putting 2 m/ coi + 2 m/co 3 = 2 co/, 2 mi" coi + 2 m 3 "co 3 = 2 cos', and solving for 2 coi, 2 co 3 , we have 2m 3 "co/-2m 3 W -2m/V + 2m 1 V , im 2coi = - - , 2 co 3 = - T (10) If A = 1, then 2 coi, 2 co 3 are each a sum of multiples of 2 co/, 2 co 3 ' and consequently any period of /(z) can be written in the form fi = 2 nico/ + 2 w 3 co 3 ' ; hence 2 co/, 2 co 3 ' are a primitive period pair. THEOREM I. Let f(z) be a single-valued doubly periodic analytic function with the independent periods 2 coi, 2 co 3 . Then if f(z) is not a constant, the ratio can not be real. Let us assume first of all that the ratio - - is real and commen- COl surable, say cos = p where p, q are integers prime to each other. We shall show that this assumption leads to a conclusion which is contrary to the given hypothesis. Since 2 coi, 2 co 3 are periods, it follows that 2 micoi + 2 m 3 co 3 is also a period. But from the assumed relation, we may write P\ m 3 - = ) = 2a? 3 (mi- + m 3 ) / \ P / m 3 p) 320 SINGLE-VALUED FUNCTIONS [CHAP. VII. Since p, q are relatively prime, Wi and ma can be so chosen that * Wig + m 3 p = 1. Consequently, we obtain from (11) . 2coi + 2 ra 3 a>3 = -- = -- = 2a>. 5 P where 2 w is a period. Hence, we have 2 o>i = 2 gco, 2 co 3 = 2 pco; that is, the periods 2 i, 2 w 3 are each multiples of a common period 2 u and hence are not independent. Likewise the assumption that the ratio is real and incommen- Wi surable leads to a contradiction. For, let be converted into a con- i timied fraction. Then -- must lie between any two consecutive OJl convergents,f say Pk Hence, the value of differs from either of these convergents by less < 1 than We may then write p k whence | g*w 3 Pk&\ \ < - - Qk+l But since qk+i can be taken as large as we please, it follows that may be made as small as we please. From the foregoing relation, 3 be a primitive period pair of the single-valued doubly periodic analytic function /(z). Since the ratio can not be real, Wl the straight line joining the origin with 2 o>i makes an angle different from zero or TT with the line joining the origin with 2 s. Conse- quently, the set of complex values 12 = 2 micoi + 2 w 3 3 is represented by a net of points covering the complex plane. More- over, any other primitive period pair of /(z) must lead to the same net. If ZQ is any point in the region of existence of /(z), then in the parallelogram ZO, ZQ + 2 i, Zo + 2 W 3 , Zo + 2 W! + 2 <0 3 , the given function takes all of its values. This parallelogram is called a primitive period-parallelogram. If we put 2 (til -f- 2 O>3 = 2 W2, we have the relation 2 coi + 2 2 + 2 o> 3 = 0. By drawing parallel straight lines through the points of the net Zo + ft as shown in Fig. 102, we have a set of congruent period- FIG. 102. parallelograms covering the entire complex plane. If Zo is a singular point, then each point ZQ + fl is likewise a singular point. It is often convenient to choose Zo so that no singular point of /(z) lies upon the boundary of the period-parallelograms. This choice is always possible if the number of singular points of /(z) in a period- parallelogram is finite. For example, if the singular points of /(z) are restricted to poles, then there are but a finite number of poles in the initial period-parallelogram, and hence by the proper choice of ZG none of these poles will lie upon the boundary of any period-parallel- ogram. 322 SINGLE-VALUED FUNCTIONS [CHAP. VII. We shall now set up the convention that the points on one pair of adjacent boundary lines of a primitive period-parallelogram belong to the parallelogram and that the points on the other two boundary lines do not. For example, in the parallelogram PI, Fig. 102, the points on the boundary joining ZQ, 20 + 2 o>i and ZQ, ZQ + 2 co 3 , the points Zo + 2 o>i, 2o + 2 0)3 excepted, are considered as belonging to the period-parallelogram but the points on the other two boundary lines do not. It is sufficient to study the behavior of a doubly peri- odic function for values of 2 in any period-parallelogram, just as in the case of simply periodic functions it is sufficient to examine the function for values of z in any one of the various period-strips. This fact simplifies the discussion of periodic functions. We shall use the term period-region to mean either a period-strip or a period- parallelogram according as the given function is simply or doubly periodic. We shall have no occasion to discuss in this connection functions having more than two independent periods, for it may be shown that a single-valued analytic function can at most be doubly periodic.* As an illustration of a doubly periodic function, let us consider the function p'(z) of Weierstrass, which is defined by the relation (Art. 56) As we have seen, this function is holomorphic in the finite region except for the doubly infinite set of Si-points. Replacing z by z 2 o>i, we have from (12) V 1 \z- (Q=F2i) j From an examination of Fig. 102, it will be seen that the set of points z (2 T 2 o>i) is the same as the set of points z ft, the points being taken in another order. The order of the terms in the series defining p'(z) is immaterial since the series converges absolutely. Consequently, we have * Foreyth, Theory of Functions, 2<* Ed., p. 230. ART. 58.] PERIODIC FUNCTIONS 323 Similarly, it may be shown that Hence, it follows that g/(z) remains invariant by the translation z' = z + 2 wio>i + 2 ra 3 a> 3 , = z + ft; that is, p'(z + 0) = p'Cs), and the given function is therefore doubly periodic. THEOREM II. If a single-valued periodic analytic function f(z) is holomorphic in any given period-region, it is a constant. As we have seen, a periodic function takes the same value at the corresponding points of the various period-regions. If the given function is holomorphic in any period-region, then in that region it is continuous and bounded. But if it is bounded in any period-region, it is bounded over the entire complex plane. Hence f(z) has no singular point and is therefore a constant by Theorem XII, Art. 51. We have also the following theorem. THEOREM III. A single-valued periodic analytic function w = f(z) which is not a constant is necessarily a transcendental function of z. By Theorem II each period-region must contain at least one singu- lar point. In the case of doubly periodic functions these singular points have the point z = oo as a limiting point, and consequently that point is an essential singular point. The same result follows in the case of simply-periodic functions if singular points appear in the finite portion of the various period-strips. However, the point at infinity is a point in each period-strip and this point may be the only singular point. In this case the point z = oo must also be an essential singular point. For, if ZQ is any finite point in one of the period-strips, the function takes then the same finite value f(z ) at each of the corresponding points ZQ + &o>, where o> is the primitive period of the function and k has the values 1, 2, 3, .... The points have the limiting point z = oo, but the limit of the function as k becomes infinite is the finite value /(ZG). Since we obtain by a particular approach to the point z = oo a finite limiting value of the function, that point can not be a pole. Since it is a singular point, it must then be an essential singular point. The given func- tion can not be a rational function; for, in that case the point z = oo 324 SINGLE-VALUED FUNCTIONS [CHAP. VII. can not be an essential singular point. Since all single-valued alge- braic functions are rational, and hence can have no other singular points than poles, the given function must be transcendental. THEOREM IV. Let f(z) be a single-valued doubly periodic analytic function having only a finite number of singular points in each period- parallelogram. The integral I f(z) dz taken around the contour of a period-parallelogram is zero. Denote the contour of a period-parallelogram by C. If ZQ is one of the corner points of the parallelogram, then the other points may be taken as indicated in Fig. 103, ZQ being so chosen that no singular FIG. 103. points lie on the contour of the period-parallelogram. The integral taken in a positive direction around the contour is (* nzv+2ai /o+2,+2, f(z)dz= / /(*)& + / f(z)dz /C t/o 1/ZO+2W, 2, /z<, f(z)dz+ f(z)dz. We can combine the first and third integrals in the right-hand mem- ber, and likewise the second and fourth, thus obtaining Ul * &>3 But as f(z) is doubly periodic having the periods 2 wi, 2 w 3 , we have /(z + 2 0,3) =/(*), /(z + 2 Wl ) =/(*), ABT. 58.] PERIODIC FUNCTIONS 325 from which it follows that both of the integrals in the second member of the foregoing equation vanish. Hence, we have uc f(*)dz = as the theorem requires. By aid of this theorem we can now establish the following theorem. THEOREM V. The sum of the residues in a period-parallelogram of a single-valued doubly periodic analytic function having in any period- parallelogram only a finite number of singular points is zero. The residue of a function at an isolated singular point was defined as the value of the integral where C is a closed path inclosing no other singular point. It has also been shown that this integral taken around the contour of a region containing a finite number of singular points is the sum of the residues of the function at these points. It follows then from The- orem IV that if a function f(z) satisfies the stated conditions, the sum of its residues at the singular points in a period-parallelogram must be zero. THEOREM VI. The number of zero-points in any period-parallel- ogram of a single-valued doubly periodic analytic function, which is not a constant and has no singular points other than poles, is equal to the number of poles in this period-parallelogram, each zero point and each pole being taken a number of times equal to its multiplicity. Let f(z) be the given function. Since it is analytic, its zero points are isolated, and hence in any finite region there can only be a finite f(z) number. By hypothesis f(z) is periodic. Hence, f'(z) and j^- are /w likewise both periodic. Moreover, since /(z) has but a finite number of zero points hi any finite region, JT-^- can have but a finite number j\ z ) f(z) of poles in such a region. Consequently, the quotient -^^- has but J( z ) a finite number of poles in any period-parallelogram. By Theorem V we have 326 SINGLE-VALUED FUNCTIONS [CHAP. VII. where y denotes the contour of any period-parallelogram so chosen f'(z) that no poles of ^T-V or of /(z) lie upon 7. But by Theorem X, J( z ) Art. 54, it follows that the foregoing integral is equal to the number of zero points of /(z) in the period-parallelogram bounded by 7, minus the number of poles of /(z) in the same parallelogram. Since this difference is zero, we have the given theorem. The foregoing theorem may be generalized as follows: THEOREM VII. In any period-parallelogram of a single-valued doubly periodic analytic function f(z) , which is not a constant and has in the finite region no other singular points than poles, takes any arbi- trary value C a number of times equal to the number of its poles, each pole being taken a number of times equal to its multiplicity. To prove this theorem consider the function F(z)=f(z)-C. The conditions of Theorem VI are satisfied by F(z), and hence the number of its zero points is equal to the number of its poles in any period-parallelogram. But the poles of F(z) are at the same time poles of /(z). Moreover, at the zero points of F(z), /(z) takes the value C. Hence, the given function /(z) takes the arbitrary value C in each period-parallelogram a number of times equal to the num- ber of its poles in that parallelogram. If, as in Theorem VI, our attention is confined to functions having no singular points in the finite part of the plane except poles, it follows that there must be at least one pole in each period-parallel- ogram. If there is but one pole, its residue must be zero; conse- quently the order of the pole must be at least two. If only simple poles appear, each period-parallelogram must contain two or more such poles in order that the residue may be zero. Doubly periodic functions having only poles in the finite region form an important class of functions called elliptic functions. We shall, however, not consider the special properties of such functions in the present volume. EXERCISES 1. Show that every rational integral function /(z) is uniquely determined for all complex values of z as soon as/(x) is known for all real values of x from zero to one. 2. Given the function z + 3 /(*) = z(z 2 - z - 2) ART. 58.] EXERCISES 327 Represent this function by an infinite series for values of z; (a) within the unit circle about the origin; (6) within the annular region between the concentric circles about the origin having respectively the radii 1 and 2; (c) exterior to the circle of radius 2. 3. Can sin z ever be greater than one? If so at what points? How do the zeros of sin z compare with those of sin x, where x is real? What solutions can the equation cos z = 1 have? 4. Given the infinite series z 2 z 3 z n i+.+g+g+. ..+5+..... From the form of this series, how do we know that it defines a function /(z) that is holomorphic in the entire finite portion of the complex plane? Knowing the expansion of e 1 , how can it be shown that /(z) = e z ? How can this same method be used to obtain the expansion of sin z, cos z from the expansion of sin x, cos x? 6. Given the function * - ?TT Is /(z) an analytic function? What is its region of existence? Find a region S in which the function is holomorphic. Locate the singular points and classify the singularities of the function at these points. Does the function have any zero points? 6. Indicate the general form which the infinite expansion of an analytic function /(z) must have in the neighborhood of ZQ, if f(z) has (a) a pole of order 3 at Zo, (b) a zero of order 3 at Zo, (c) an essential singularity at ZQ. What is the nature of the function F(z) = ^r-r at Zo in each case? J\ z ) 7. Locate the zero points of sin - What is the nature of this function at the Z limiting point of these zeros? From this conclusion what can be said of the sin- gular points of esc z? Why is esc z a transcendental fractional function? 8. Given (z- lz- If this function is expanded in a Taylor series about z = 2, how large can the circle of convergence be? Expand the given function in powers of z and determine the circle of convergence. 9. Giver; the analytic expression I ; - 1^ *<> - L p^j. J M '- - - n = w 1 Z Show that 0(z) is an element of two distinct analytic functions, according as we take | z | < | or | z | > 1. 1 . 10. Show that a doubly-periodic function /(z) which is an integral transcen- dental function is a constant. 11. Given the function f(z} = z*-2z*+z-2' Determine the zero points and poles. Compute the residue at each of the latter. 328 SINGLE-VALUED FUNCTIONS [CHAP. VII. 12. The function f(z) = cos 2 z + sin 2 z has the constant value 1 along the axis of reals. By what property does it follow that it must be equal to one for all finite values of z? What other relations of trigonometric functions can be extended in a similar manner from real to com- plex values of the variable? /z 2 dz taken along a circle C about the origin as center. How large can the radius of C be taken and have the value of the in- tegral zero? WTiat would the integral be if the radius of C is taken | unit larger? 14. Given a function having poles at the points z = on, a 2 , . . . , otk, and an essential singularity at z = . What kind of a function is it? Write an infinite series which will represent such a function. 15. By use of the theory of residues evaluate the integral dx (* + 2) 2 ' 16. Give an illustrative example of a single- valued analytic function hav- ing (a) no singular point, (6) no singular point in the finite region and a pole at infinity, (c) no singular point other than an essential singular point at infinity, (d) a finite number of poles in the finite region and an essential singular point at infinity, (e) an infinite number of poles in finite region dense at infinity, (/) an isolated essential singular point in the finite region. Classify each function. 17. Show that the infinite product converges absolutely. 18. Show without testing the remainder that the expansion of tan z in terms of z must converge for all values of z less in absolute value than - 19. If in the deleted neighborhood of z = ZQ, /(z) is holomorphic and not a constant, and if /(z) takes the same value (3 a,i a. set of points dense at Zo, then z = ZD is an essential singular point of /(z). Apply this result to testing the nature of sin - at z = 0. z HINT: Consider the function ^. - 20. Do the poles of the function ,, . _ 3 sin z + 7 tan z have a limiting point? What is the nature of /(z) at this point? Show that /(z) can be made to approach an arbitrarily chosen number as z approaches this point. Without computing the integral where C is the boundary of a region S having z = oo as an inner point, show that this integral does not vanish. CHAPTER VIII PROPERTIES OF MULTIPLE-VALUED FUNCTIONS 59. Fundamental definitions. In the previous chapters we have frequently had occasion to consider single-valued functions whose inverse functions are multiple-valued, that is, are functions having in general two or more values for the same value of the independent variable. The functions w = Vz, w = log z, w = arc sin z are illustrations of multiple-valued functions. In the present chap- ter we shall consider some of the special properties of such functions. In the consideration of multiple-valued functions we must distin- guish between multiple-valued analytic functions and those multiple- valued expressions that represent two or more single-valued analytic functions. Thus w = Vz 2 is not to be regarded as a multiple-valued analytic function, but rather as two single-valued functions w = z, w = z. These two functions are distinct analytic functions rather than ele- ments of the same analytic function, for neither can be deduced from the other by the process of analytic continuation. In the same way the expression w = Vl -sin 2 z is not a single multiple-valued analytic function, but represents the two single-valued analytic functions w = cos z, w = cos z. The expression w = loge* represents an infinite number of distinct, single-valued analytic functions, namely w = z, z + 2iri, z + 329 330 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. These functions, like the foregoing functions, are distinct because no one of them can be deduced from another by a process of analytic continuation. Likewise, the expression w = a*, where a = a + ib, is not a multiple- valued analytic function but represents an infinite number of single- valued functions which can not be deduced from each other by the process of analytic continuation. For, we have w = a* = which represents the single-valued analytic functions In mapping by means of multiple-valued functions, we have already seen that the whole of one plane may map upon a portion of the other plane. For example, by means of the function w = \^z, the whole of the Z-plane may be mapped upon that half of the TF-plane for which the real part of w = u + iv is positive. Con- versely, by means of the same relation, this half of the TP-plane may be mapped upon the whole of the Z-plane. Likewise, the half of the TT-plane for which the real part of w is negative may be mapped upon the whole of the Z-plane. If the given function is w = Vz, the whole of the Z-plane may be mapped upon a sector of the PP-plane formed by half-rays drawn from the origin and making angles - , -- , respectively, with the positive axis of reals. In the consideration of multiple-valued functions such as w = Vz, w = "V/z, w = log z, we have thus far restricted the discussion to those values of w which correspond to values of z for which we have TT < amp z = TT; that is, we have restricted our consideration to a fundamental region of the inverse function. With this restriction we have been able to consider these functions as though they were single-valued. As we shall see, such an arrangement has the disadvantage that the mapping from at least one of the two planes upon the other is not always continuous. * For the case where a = e, we give to log e only the value 1 since e* must ap- pear as a special case of e*. So also with other logarithms of real numbers. ART. 59.] FUNDAMENTAL DEFINITIONS 331 For example, if we have w = \/z, then by mapping upon the TF-plane the amplitude of z is divided by three; for, putting z = p(cos0 + isind), w = p'(cos 0' + i sin 0'), we have from w = p'(cos B' + i sin 0') = p* (cos 5 + 1 sin ^ J , All points of the Z-plane, when we consider only the principal ampli- tude of those points, map into a region I (Fig. 105), bounded by two half-rays from the origin making an angle of 60 and 60, respec- tively, with the positive half of the axis of reals. The result is as FIG. 104. though the Z-plane were cut along the negative half of the axis of reals and the whole plane contracted by moving each bank in its own one-half plane through an angle of 120 into the positions of O'Q' and O'P'. The mapping from the Z-plane upon the funda- mental region of the TF-plane is single-valued but not necessarily continuous, as may be seen most clearly by considering a continu- ous curve in the Z-plane crossing the negative axis of reals. That portion (a) of the curve above the X-axis maps into the curve (a') to the right of O'Q', while the portion (6) below the X-axis maps into (6') to the right of O'P'. Consequently, what was a continuous curve becomes after mapping a discontinuous curve as shown in the figure. 332 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. Either of the regions II, III might equally well have been selected as the particular region upon which the Z-plane is mapped by plac- ing the proper restrictions upon the variation of the amplitude of z, To any point a in the Z-plane distinct from the origin there corre- spond then three points in the TF-plane when no restriction is placed on the amplitude of z. We may denote these three values by Wi(a), w 2 (oi), W 3 (a). As z takes all values in the Z-plane, subject to the condition that TT < ampz = TT, let wi denote the corresponding functional values represented by the points of region I of the TF-plane. Likewise, let w 2 , wa denote the totality of values represented respectively by the points of the regions II, III. Consequently, w\, 10%, ws are single-valued func- tions of z such that the three taken together give all of the corre- sponding values of w and z included in the functional relation w = \/z. We call Wi, w z , ws the three branches of the given function. Similarly, we may define a branch of any multiple-valued analytic function w = /(z). An assemblage of pair-values (w, z) is said to define a branch Wk of such a function if it possesses the following properties : 1. The aggregate of all z-points of the region of existence which enter into consideration must fill a region Sk exactly once. 2. To each point of S k there corresponds but one value of w. 3. The aggregate of w-points corresponding to points in Sk repre- sents a continuous function of z. Thus, for the function w = \/~z each of the regions Sk, k = 1, 2, 3, of the Z-plane consists of the whole finite portion of the complex plane with the exception of the point z = 0, which is to be thought of as a boundary point of Sk for reasons that will appear later. Tak- ing the negative axis of reals as a portion of the boundary, but nevertheless a part of Sk, it follows that the regions of the TF-plane corresponding to the three branches of the function are definitely determined; that is they are the regions I, II, III in Fig. 105. 7T Had the amplitude varied between other limits, say between ~ Z o and -~ , the values of w corresponding to values of z in Sk would Zt have been different, and the branches of the given function would have been correspondingly changed. ART. 59.] FUNDAMENTAL DEFINITIONS 333 The branches of a multiple-valued analytic function are single- valued functions of the independent variable, and the totality of the value-pairs (w, z), representing all of the branches taken to- gether, is identical with those of the given function. Whenever the inverse function z *= <(io) is single- valued, then the domain of the functional values of the given function w = /(z) breaks up into a finite or an infinite number of non-overlapping regions according as w = f(z) has a finite or an infinite number of branches. Moreover, if in this case the given function has no natural boundary, each of these regions of the TP-plane may be taken as a fundamental region of the inverse function. A point is called a branch-point of the given analytic function if some of the branches interchange as the independent variable de- scribes a closed path about it. The existence of branch-points is a characteristic of multiple-valued analytic functions as distinguished from multiple-valued expressions, such as Vl sin 2 z, representing two or more single-valued analytic functions. The point z = is a branch-point of the function w = \/z. For, let z describe a closed path about the origin, beginning at any point a whose amplitude is 6, and let us consider the change that takes place in the value of the function w = v'z. The initial value of the function is then Wi(a) and is represented by a point in the region I, Fig. 105. After one revolution of z about the origin, the function does not return to its original value, but has changed to a value w 2 (cx), represented by a point in region II, its amplitude having been increased by o After a second revolution of z about the origin the functional value has changed from w 2 (d) to a value w z (a), represented by a point in the region III, and the amplitude of w has changed to 6 + ~^-- o After three revolutions of z about the origin the function again attains its initial value w\(oi). In the discussion of analytic continuation of single-valued func- tions, it was shown that when the continuation is taken along any path between two points ZQ and z\ of a region S in which the function is holomorphic, the same functional value at z\ is obtained irrespec- tive of the path chosen, provided that path lies wholly within S. For multiple-valued functions a different functional value may be obtained at the terminal point z\ if the continuation is taken along different paths. This is always the case for some branches of the 334 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. function if the two paths are so chosen as to inclose a branch-point; for, the two paths taken together then form a closed path about that point, and from the definition of a branch-point it follows that whenever the independent variable z makes but one circuit of this path some of the branches of the function are interchanged and the initial and final values of the function do not coincide. This prop- erty is, by the definition of a branch-point, a characteristic of all paths inclosing such points. The different branches of a function w = /(z) are connected with one another at the w-points corresponding to the branch-points in the Z-plane. For example, it will be observed that all three branches of the function w = \^z come together at the point w = 0, which corresponds to the branch-point 2 = 0. As we shall see later, how- ever, not all of the branches of an analytic function need be con- nected at any particular branch-point. If k + 1 branches coincide at a branch-point, that point is then said to be a branch-point of order A;. In the illustration discussed, the origin is therefore a branch- point of order two. Since the branches of a multiple-valued analytic function are single-valued, such a function may be considered as an aggregate of single-valued functions, so related that their values be- come identical at the branch-points, each being determined from the others by the process of analytic continuation. The point at infinity may of course be a branch-point. To examine its nature we may make use of the usual substitution z = , and examine the nature of the transformed function at the origin. In the neighborhood of a branch-point, the mapping by means of an analytic function ceases to be conformal; for, from the foregoing discussion it follows that angles are not preserved at such a point. In the case of w = Vz, for example, it was seen that at the branch- point z = angles are divided by three in mapping from the Z-plane upon the TF-plane. Suppose we have given a multiple-valued function w = /(z), whose inverse function z = '(w) must either be zero or become in- finite for w = WQ. This fact enables us to formulate a convenient test for finding the branch-points of a function whose inverse func- tion is single-valued. Such a criterion is given in the following theorem. ART. 59.] FUNDAMENTAL DEFINITIONS 335 THEOREM. Given a multiple-valued analytic function w = f(z), whose inverse function z = (w) is single-valued. If the derived function 4>'(w) has a zero point of order k, or a pole of order k + 2, at w , then /() has a branch-point of order k at the corresponding point go- Let us suppose that '(w) has at w a zero point of order k. We may then write 4>'(w) = (w- w^F^w), (1) where k is a positive integer and FI(W) is holomorphic in the neigh- borhood of w and different from zero for w = w . It follows that the factor (w w ) enters into (w} <(w ) to a degree one higher, that is to the degree k + 1, thus giving 0(ti>) - (w Q ) = (w- Wo) k+l F 2 (w), (2) where F%(w) is also holomorphic in the neighborhood of w and different from zero for w w$. Let A be the pj*incij)al_. (& + 1)" root of FZ(WO) and let x( w ) be a function having the point WQ as a regular point such that X(WQ) is equal to A. The function x(w) can be so determined that for values of w in the neighborhood of WQ we have From (2) we now have (w) - 0(tl\>) = \(w - WQ) x(w)$* +1 . As a matter of convenience, we introduce the auxiliary function T = (w - w ) x(w). (3) Let us now suppose the r-plane to be mapped upon the TF-plane by means of this relation. The point T corresponds to the point w = WQ, and the mapping is conformal in the neighborhood of T = 0; because we have 1 "I 1 x(w) + (W - Wo) x'( dw^v, where . r is finite and different from zero, since x(wo) is the prin- cipal (k -j- I) 8 ' root of F Z (WQ), which is different from zero. If we now map the finite portion of the Z-plane upon the r-plane by means of the relation TU_l 1 _ Jt-J-1 _ - (Wo) = V Z -2o, (4) ^|4^ ; ' +^* 336 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. the neighborhood of ZQ maps into the neighborhood of the origin in the r-plane. The two substitutions (3) and (4) are together equiva- lent to mapping at once from the Z-plane to the TF-plane by means of the relation z = (w), or w =/(). From what has been said about mapping by means of the relation w = Vz, it follows at once from (4) that z = z is a branch-point of the order k, which shows that the condition stated in the theorem is valid if w is a zero point of order k of '(w). Let us now consider the case where WQ is a pole of order k + 2 of 4>'(w). The corresponding z-point is the point at infinity. For this case equation (1) takes the form Applying the same method as employed in the foregoing discussion, it follows at once that /(z) has a branch-point of order A; at z = oo .A \ I The details of the proof are left as an exercise for the student. - The difficulties in representing multiple-valued functions in the foregoing manner arise for the most part from the fact that a con- tinuous curve in the one plane does not always correspond to a con- tinuous curve in the other. Such difficulties can be easily avoided by means of a simple device known as a Riemann surface, consisting of n sheets connected with one another in a definite manner depend- ing upon the character of the function. The nature of such a surface can perhaps be most readily made clear by means of an illustrative example. Let us consider the Riemann surface for the function W = V/2. It has already been pointed out (Fig. 105) that the whole of the Z-plane maps by means of this function into any one of the three regions I, II, III of the PF-plane. To each z-point correspond in general three w-points, one in each of the regions I, II, III. Sup- pose we think of the Z-plane as consisting of three sheets (Fig. 106) connected with one another at z = and along the negative axis of reals. As z describes a circuit about the branch-point z = 0, sup- pose it passes from one sheet to another upon crossing the negative axis of reals and that the variable point enters the various sheets ART. 59.] FUNDAMENTAL DEFINITIONS 337 FIG. 106. in the same order as the branches of the function were permuted in the previous discussion when z described a closed path about the same point. Let the points of the region I be placed into corre- spondence with those of the first sheet of the Z-surface, the points of region II with those of the second sheet, and the points of region III with those of the third sheet. Corresponding to the three points Wi(a), w 2 (a), w 3 (a) of the TF-plane we have then a point a in each of the three sheets of the Z-plane, denoted by aW, ( 2 ), a< 3 >, respectively. The branch-point z = is not C[ to be regarded as a point of the region of existence of the given function, but is to be considered as belonging to its boundary. The same is to be said of any branch-point so far as the sheets of the surface af- fected are concerned. Each sheet of the Riemann surface, like the ordinary complex plane is to be regarded as closed at infinity. By aid of the Riemann surface there is thus established a one-to-one corre- spondence between the points of the W-plane and the three-sheeted Z-plane. Let z start from a point a, as before, is an imaginary cube root of unity. The results of successive revolutions about this circuit are shown in the following table Before z' changes, we have Wi, wz', w 3 , W*, w 5 ', WG'; after one revolution, W, w/, w$, w 3 ', w^, Wz', after two revolutions, w%, w 3 , Wi, W&, W, W; after three revolutions, Wi, w$, We after four revolutions, w 3 ', w\, Wz after five revolutions, J 6 ', w*, W* after six revolutions, Wi, Wz', w 3 ', wi, w$, w. It follows that z' = and therefore z = oo is a branch-point where all of the sheets of the Riemann surface are associated with one another. The Riemann surface constituting the Z-plane has then the branch- points z = 0, ZQ, Zi, oo. No two of the finite branch-points can be ', Wz, w 3 ; / ,W,W; / ,to 8 / ,wi / ; 346 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. connected with a branch-cut. At first thought, it may seem possible to connect the two finite branch-points 2=0 and z = Zi with a branch- ^ cut since the same sheets are con- nected in the same way at these O --a 213 . ' * two points. This, however, is not Via 112 possible; for, by successive revolu- tions about each of these points the sheets would be connected along such a branch-cut as shown in Fig. 112. Then, by crossing the branch-cut, to the right of 2 = in the direction of arrow (a) we would pass, for example, from the first sheet to the second sheet, while in passing across the branch-cut to the left of 2 = 2i in the same direction we would pass from the first sheet to the third sheet. Such connection of the sheets along a branch-cut is impossible since the connection must be between the same sheets along its entire length. It is at once apparent that in order to connect any two finite branch-points with a branch-cut, the cyclical interchange of branches must involve the same sheets but in reverse order. We can, however, always draw branch-cuts from any branch-point to the point 2 = co ; that is, the branch-cuts may be taken as lines extending out indefinitely from the branch-points. Drawing these lines in any convenient manner, the six branches of the given function are fully determined by the six definite aggregates of value-pairs (w, z), such that to the values of 2 associated with each sheet of the Riemann surface, there is a definite branch of the func- tion. In this case, however, there are no corresponding fundamental regions in the TF-plane, since the inverse function is not single- valued. The general discussion of the Riemann surface required for the Z- plane is now complete. 62. Riemann surface for w = log z. The logarithm is a func- tion having an infinite number of branches. As we have seen the logarithm of z may be written in the form w = log 2 = log p + 16, (1) where 2 = p(cos0 By means of the relation (1) the whole of the Z-plane maps into any one of the strips (2 k - 1) T < v = (2 k + 1) TT of the JF-plane parallel to the axis of reals. Conversely, any one of this infinite number of strips maps into the whole of the Z-plane. ART. 63.] BRANCH-POINTS, BRANCH-CUTS 347 We may now replace the Z-plane by a Riemann surface having an infinite number of sheets. The point z = is a branch-point of the surface; for, as we see from (1), every revolution of z in a positive direction about a circle having the origin as a center leaves p un- changed but increases 6 by 2 IT, thus changing the value of w. This: change continues indefinitely by successive revolutions. We may take the negative half of the axis of reals as the branch-cut, so that every ^ time the variable point crosses this ' part of the axis it passes from one sheet to the next succeeding or next - -^ preceding sheet, according as 9 is Fl ^ increasing or decreasing. A cross- section of the surface perpendicular to the negative half of the axis of reals, as seen from the origin, is then of the form shown in Fig. 113. In the same way the point z = oo may be shown to be also a branch-point of infinitely high order. The inverse function of w = log z is, as we know, z = e w . The trigonometric functions were defined in terms of the exponential function, so that the general character of the Riemann surfaces con- nected with the inverse trigonometric functions may be easily de- duced by aid of the logarithmic function.* In the discussions to follow we shall have occasion to consider for the most part only Riemann surfaces having a finite number of sheets. 63. Branch-points, branch-cuts. Having discussed some typi- cal illustrations of Riemann surfaces, we shall now consider some of the more important properties of branch-points and branch-cuts and their relation to the Riemann surfaces needed in the representation of multiple-valued functions. The branch-points of a function are always to be found among those points corresponding to the values of the independent variable for which two or more of the values of the function become equal. This common value of the various branches of the function may be finite or infinite. As we have seen, not all of the various sheets of a Riemann surface need be connected at any particular point; for, they may be associated in distinct sets at a branch-point as was the case in the points z = ZQ, z = 0, z = z\ for the function discussed in Art. 61. It is essential, however, that all of the sheets be so con- * Cf. Fouet, Lemons elementaires sur la theorie des fonctions analytiques, 2<* Ed., Tome II, p. 128 et seq. 348 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. nected at the various branch-points that the entire surface forms a connected whole and it is possible to proceed along a continuous path from any point to any other point upon the surface. As already pointed out, the branch-points themselves, while points of the Rie- mann surface, are not to be regarded as points of the region of exist- ence of the given function, but as boundary points of that region. The region of existence is then, as with single-valued analytic func- tions, to be considered an open region, which in the case of multiple- valued functions, however, extends to the several sheets composing the Riemann surface. It is convenient to speak of the branch-points as points of the function in the sense that their character aids in describing the character of the function. Whenever the inverse function is single-valued, we have in the theorem of Art. 59 a convenient test for locating the branch-points and determining their order. This test is not, however, valid when we have as in Art. 61 a function whose inverse is multiple-valued. In that case it is necessary to test each point where some of the values of the function become identical by permitting the indepen- dent variable to traverse an arbitrarily small closed circuit about the point and observe whether the corresponding values of the function return to the initial value. In the illustrative examples discussed, it was observed that the branches of a function associated with each other at a branch-point were cyclically interchanged by successive revolutions about the point. As a matter of fact, it is always true that all of the branches affected at a branch-point can be so arranged that by successive circuits of the independent variable about the point these branches are cyclically interchanged. The cycle may include all or only a portion of the branches affected. In the latter case there may be two or more sets each of which undergoes a cyclical interchange as the independent variable traverses a circuit about the branch-point a suitable number of times. To show that the branches of the func- tion are interchanged as stated, suppose we let the branches affected be wi, 102, ws, . . . . Then as z describes a small circuit about the branch-point, w\ must change into some other branch, say w z . By a second revolution about the point, w z can not remain unchanged; for, in that case, tracing the circuit in a reverse order would not restore the initial value Wi, as it should do. It is possible that this second revolution changes the branch ~w z back into wi, in which case 101, W* constitute a complete cycle. If this is not the case then tt? 2 ART. 63.] BRANCH-POINTS, BRANCH-CUTS 349 must change into some one of the remaining branches, say w$. As before, when z describes the circuit about the branch-point a third time w 3 can not remain unchanged and can either change back into wi or into some one of the remaining branches, say W*. In the first case, the three branches wi, w*, w 3 constitute by themselves a set which cyclically interchange. In the other case, w t changes in a similar way into w$, say, by another revolution about the branch- point. This method of interchange may involve all of the branches in one set, or in several such sets. There may be other sheets that are not affected at this particular branch-point, the connection with one or more of the remaining sheets being made at another branch- point. The same sheets may be affected at two branch-points. If the order of the cyclic permutation of the sheets in the one case is the reverse of what it is in the other, then it is always possible to connect the two points with a branch-cut; for, by crossing this cut at any point, the variable passes from one sheet into another particular sheet of the cycle. Instead of connecting the branch-points with each other it is always possible to connect the various branch-points with the point at infinity with branch-cuts. Thus far in the dis- cussion the various branch-cuts have been taken as straight lines. This restriction is unnecessary, however, as the choice of a particular curve for the branch-cut is a purely arbitrary convention. A branch-cut is, however, always to be taken so that it has no double- points, that is, the cut must not intersect itself. While" the partic- ular aggregate of value-pairs (w, z) constituting the various branches are changed by varying the branch-cut, the connection of the branches with each other at the branch-points remains unchanged. The following illustration will make this statement clear. Ex. 1. Discuss the branch-cuts of the Riemann surface required for the func- ,. . [i z tion w = y . The Z-plane is a double-sheeted Riemann surface and the points z = i and z = i are simple branch-points. The corresponding functional values are re- spectively w = and w = oo . The result of mapping the Z-plane upon the TF-plane is exhibited in Figs. 114, 115. Any curve in the Z-plane joining the points i and i may be taken as a branch-cut and will map into a curve in the TF-plane joining the points w = and w = oo . For example, if we select that portion of the axis of imaginaries joining the two branch-points, then the cor- responding curve in the TF-plane is the positive half of the axis of reals. To every point in the Z-plane, however, correspond two points in the TF-plane and 4 , 350 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. this same portion of the axis of imaginaries also maps into the negative half of the axis of reals in the TF-plane. Consequently, in this case the two branches wi, ioj of the function corresponding to the values of z in the first and second sheets of the Z-plane are represented by points in the upper and the lower half of the TF-plane respectively. If we take any circle through i and i and select that portion (2) lying to the right of the line x = as the branch-cut, then Z-plane W-plane FIG. 114. FIG. 115. the corresponding curve in the TF-plane is a half-ray (2') proceeding from the origin. The same curve (2) also maps into the half-ray making an angle of 180 with (2'). The two half -rays taken together divide the TF-plane into two regions and the points in these two regions represent the values of w corre- sponding to the two branches of the given function. If instead of a circle through t and i any ordinary curve had been taken as a branch-cut, it would map into a curve joining the points w = and w = and again into a congruent curve, which may be obtained by rotating the first curve through an angle of 180. These two curves taken together constitute a continuous curve dividing the TF-plane into two regions whose points give the values of the two branches 101, Wz of the function corresponding to the two sheets of the Riemann surface constituting the Z-plane. Again we may select as the branch-cuts of the Rie- mann surface any curves joining the two branch-points i and i with the point 2 = oo, for example those portions of the axis of imaginaries exterior to the circle of unit radius about the origin. This selection likewise leads to a division of the TF-plane into two regions and corresponding branches of the function. From this discussion, it will be seen that the branch-cuts can be selected in a variety of ways and may or may not be straight lines. By the selection of the branch-cuts particular values of the function constituting the various branches are determined. The number and the association of such branches are determined by the character of the function itself. While the selection of the branch-cut is arbi- trary, there is often an advantage in selecting it in a particular manner. For example, in the discussion of the Riemann surface for w = ~ ur V" ART. 63.] BRANCH-POINTS, BRANCH-CUTS 351 we chose the negative half of the axis of reals as the branch-cut in order that one branch of the function should correspond to the prin- cipal value of the amplitude of z. Again in the logarithmic function the negative half axis was chosen for the same reason. In discussing the inverse of a periodic function, it is likewise an advantage to select the branch-cuts so that the previously determined fundamental regions shall correspond to single sheets of the Riemann surface rather than conversely. The following theorems concerning branch-points and branch-cuts give additional information that will be useful in the discussion of special Riemann surfaces. THEOREM I. A free end of a branch-cut is a branch-point. Let a be a free end of a branch-cut C. Suppose that as z crosses this branch-cut it passes from the ki th sheet into the kz th . Then as z makes a complete circuit about a starting from an initial position Zo (fcl) in the ki th sheet it does not return at the end of the first revolution to that initial position, but it ends in a point Zo (il) in the k^ ih sheet. Hence, from the definition of a branch-point, a is such a point. / \ / \ ktfo a a \kt C k l FIG. 117. It is to be noted that in case a branch-cut ends in a point on the boundary of the region of existence it is not necessarily a branch- point. THEOREM II. // but one branch-cut passes through a branch-point, the connection of the sheets on the two sides of the branch-point is not the same. Let a be a branch-point through which but one branch-cut passes. The connection between the sheets can not be the same on the one side of a as on the other; for, in that case as the variable z describes a circuit about a it returns after each revolution to its initial posi- tion. For suppose the ki th and k 2 th sheets are connected along the given branch-cut C, as indicated in Fig. 117. If z has the initial 352 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. value 2o (tl) and describes a circuit about a, say in the positive direc- tion, then as 2 crosses C at a it passes from the ki th sheet to the k^ th . Upon crossing the branch-cut again at 6, z passes from the k 2 tfl sheet back to the ki th sheet returning to its initial value Zo (tl) . THEOREM III. // two branch-cuts, having different sequences of interchange of sheets associated with them, meet in a point, that point is either a branch-point or is an extremity of at least one other branch-cut. Let a be the common point of the two branch-cuts, /, //, Fig. 118. Along the branch-cut 7, let the k^ sheet be associated with the k z th and along 77, since the sequence of inter- change of sheets can not be the same, suppose the k z th to be connected with a third sheet ks th . If the variable starts from an initial point 2o (il) in the ki th sheet, then by passing around a in a positive z (fcl) direction it passes into the k 2 th sheet, FIG. 118. r TT TT [ upon crossing 7. Upon crossing 77 it passes from the k 2 th sheet to the k 3 th sheet and, if it crosses no fur- ther branch-cut, then instead of returning to the initial position 2o (tl) at the close of the first circuit it ends in a point z$ (k *> in the k 3 th sheet and a is a branch-point. Hence either a is a branch-point or there must be at least one more branch-cut like 777 ending in a by which the k 3 lh sheet is connected with the ki th sheet. THEOREM IV. If a change of sequence in the branches of a function occurs at any point of a branch-cut, then that point is a branch-point or it lies also on some other branch-cut. Since by hypothesis a change of sequence occurs at some point, say a, then as 2 describes a circuit about a it does not return to the sheet from which it started but passes into another sheet of the sur- face. Hence, the point a must either be a branch-point or another branch-cut must terminate at a. THEOREM V. A branch-cut passing through but one branch-point can not be a closed curve. If a branch-cut having but one branch-point is a closed curve, then by encircling that point along an arbitrarily small circuit, the variable returns to the same sheet; for, the connection between the sheets must be the same on both sides of the branch-point, since the portions of ART. 63.] BRANCH-POINTS, BRANCH-CUTS 353 the cut on the two sides of the point belong to the same branch-cut. It is, however, impossible for the variable to encircle a branch-point and not change sheets. Hence under the conditions set forth in the theorem, the branch-cut can not be a closed curve. In general we have considered only paths which encircle a single branch-point. In Art. 60 we considered certain paths encircling two branch-points. We shall now consider the general effect of a path encircling two or more branch-points. We have the following theorem. THEOREM VI. The effect of describing a dosed circuit about several branch-points is the same as though the variable point had described a closed path about each of the branch-points in succession.* We shall prove the theorem for the case of a circuit about three branch-points. The same argument holds for any finite number of such points. It is at once evident that a path can be deformed in any manner without affecting the result, provided in such a defor- mation a branch-point is not encountered. For by such a deforma- tion no additional branch-cuts need be crossed an odd number of tunes. If crossed an even number of times, the final position of the variable point is in the same sheet as the initial point. Consequently the closed path C about the three points Zo, 2i, 22 can be de- formed without affecting the final result into the succession of paths \o, AI, X 2 about the three points ZQ, Zi, Z2, respectively. Each of these paths begins and ends at z 3 , as shown in Fig. 119, and consists of a small circle about the branch-point and a path connecting that circle with z 3 . This connecting path, however, is traversed twice, once in each direction, so that any branch-cut crossed in going in one direction will be crossed again in an opposite direction when the path is traversed in the opposite direction. The effect of this por- * The group of permutations which the function values undergo as the inde- pendent variable describes a closed path is often called a monodromic group. Cf. Encyklopadie der Math. Wiss., Bd. II 2 , p. 121. 354 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. tion of the path crossing a branch-cut can therefore be neglected. The total effect of traversing the closed path C is then the same as traversing in succession the closed circuits about the separate branch- points, as stated in the theorem. The foregoing theorem also gives us a convenient way for testing the point z = oo for a branch-point. Consider a closed path inclos- ing all of the finite branch-points. If this path be traversed in a counter-clockwise direction the result is easily obtained by the theorem. However, since no finite branch-point lies exterior to this path, it follows that traversing the path in a clockwise direction is equivalent to encircling the point at infinity. Traversing the path in a clockwise direction gives the same interchange of sheets but in opposite order as traversing it in the opposite direction. Therefore, if the sheets are interchanged by traversing such a path in either direction the point z = oo is a branch-point. Ex.2. Given the function w = (z z )(z zi). Determine whether the point z = oo is a branch-point. The given function has a branch-point of the second order at z = Zo and at z = ZL At Zo and zi the branches interchange by successive clockwise revolutions about the point as follows : Before z changes, . wi, Wi, after one revolution, 103, tcj, after two revolutions, w 3 , w\, w*; after three revolutions, wi, w 2 , w*. As z describes clockwise a closed circuit inclosing both z fl , z\, we have: Before z changes, Wi, w 2 , w; after one revolution, u> 3 , wi, u> 2 ; after two revolutions, w 2 , 103, w\.; after three revolutions, w\, w 2 , w*. The point z = oo is then a branch-point of order two. It is to be observed that had the cyclic arrangement of the sheets at Zo, z\ been such that these points might have been connected by a branch-cut, then there would have been no interchange of branches as z described a closed circuit about the two branch-points and consequently the point z = oo would not have been a branch-point. In order that the two points z and zi might have been so con- nected the cyclic arrangement of the branches at this point would necessarily have involved the same sheets taken in opposite order. Such would be the case, for example, with the function ZQ Zi 64. Stereographic projection of a Riemann surface. As with single- valued functions, stereographic projection upon a sphere may ART. 65.] PROPERTIES OF RIEMANN SURFACES 355 often be employed with advantage in the discussion of multiple- valued functions. The multiple-sheeted Riemann surface projects into a multiple-sheeted Riemann sphere whose sheets are associated at the branch-points. The branch-cuts in the plane go over into curves upon the sphere along which the variable point passes from one sheet to another. In the case of the inverse of the exponential and trigonometric functions, namely the logarithmic and inverse trigonometric func- tions, the projection of the PF-plane upon the sphere is also of inter- est. The infinite number of strips congruent with the fundamental strip map into similar regions having a common point and bounded by curves having a common tangent at the north pole. This result exhibits the fact that the exponential function e w , in terms of which the other functions are defined, is a function which takes in the neigh- borhood of the essential singular point w = every complex value except zero and infinity. The branch-points may be regarded as boundary points of the region of existence on the sphere just as they are regarded upon the Riemann surface. 65. General properties of Riemann surfaces. Thus far we have considered only special cases of Riemann surfaces. In general to con- struct such a surface for a given function, we may proceed as follows. 1. Determine the number of branches. This number is equal to the largest number of distinct values which the function has for each value of the independent variable. In case the function is algebraic, the number of branches is equal to the degree of the algebraic equa- tion defining the function, when that equation is freed from radicals. 2. Locate the branch-points. If the inverse of the given function is single-valued, the branch-points may be found by the theorem of Art. 59. In any case, they are to be found among the points of the | complex plane representing values of the independent variable for I which two or more values of the function are equal, and they may be 1 either all at finite points or one of them may be the point at infinity. From among these points those that are branch-points may be found by permitting the independent variable to describe a closed path about each point and observing which of these paths leads to the initial value of the function after each circuit. 3. Find the connection of the branches of the function at the various branch-points. Having determined the number of branches and located the branch-points, the branches themselves are not as yet uniquely determined. This, however, is not necessary in order 356 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. that we may determine the connection of the branches. To show this connection find the cyclic permutation of the branches at each branch-point as the independent variable describes a closed path in the ordinary complex plane about that point. The number of branches affected at any branch-point is one greater than the order of the branch-point. 4. Draw the branch-cuts. The branch-cuts may be inserted in a variety of ways. They should be so chosen as suits best the pur- poses of the discussion in hand. When the cyclic permutation of the branches permits, the branch-cuts may connect the various branch-points, or when more convenient they may be drawn from the various branch-points to the point at infinity. As we have seen, a branch-cut need not be a straight line and may in fact be any ordinary curve that does not intersect itself. When once the branch- cuts are drawn, the various branches of the function are definitely determined aggregates of value-pairs (w, z), and with each sheet of the Riemann surface there is associated a definite branch of the function. The various sheets of the surface should be so connected along the branch-cuts that as the variable passes over one of these cuts the proper branches of the function interchange. While the branches of a function are single-valued, it may happen that both w and z are multiple-valued functions of the other. In that case it is often convenient to introduce a third variable T so related to w and z that the Riemann surfaces constituting the TF-plane and the Z-plane, respectively, map continuously upon the single- sheeted T-plane. Each branch of the given function w = /(z) asso- ciated with a sheet of the Z-plane maps into a fundamental region of the T-plane, which we may designate as a (z, T) fundamental region. Likewise each sheet of the Riemann surface constituting the TF-plane is associated with a branch of the inverse function 2 = (w) and maps upon a fundamental region of the T-plane, which we may designate as a (w, T) fundamental region. The (z, T) regions do not coincide with the (w, T) regions. As a result, that portion of the Riemann surface constituting the TF-plane which corresponds to a sheet of the Z-plane, and hence to a branch of the function w = f(z)> does not coincide exactly with the whole of one or more sheets of the TF-Riemann surface. It may be less or it may be more than one sheet, depending upon the nature of the functional relation between w and z. We shall refer to that portion of the TF-Riemann surface which corresponds to the whole of a sheet of the Z-plane as a ART. 65.] PROPERTIES OF RIEMANN SURFACES 357 fundamental region on the Riemann surface. When the branch- cuts are inserted, the correspondence between the points of this region and the particular sheet of the Z-plane is definitely determined; that is, this branch of the function w = f(z) is fully determined. Since to a single sheet of the Z-plane there may correspond more than one sheet of the PF-plane, it follows that some of the w-values may be repeated, although the branches of the given function re- main single-valued. An illustrative example will aid to make clear the foregoing discussion. Ex. 1. Consider the function We introduce the auxiliary variable r by putting The two-sheeted Z-Riemann surface required for this function maps upon the whole of the r-plane, Fig. 120. If we take the negative half of the axis of reals as the branch-cut, then the upper sheet maps into the half of the r-plane to the right of the axis of imaginaries, while the second sheet maps into the half of the plane to the left of the same axis. On the other hand, the three-sheeted W-plane maps r-plane U FIG. 120. FIG. 121. into the whole of the r-plane as follows, where again we take the negative half of the axis of reals as the branch-cut. The first sheet maps into the region / bounded by the lines OP\ and OP 2 making angles of = and --, respectively, with the o o positive axis of reals. The other two sheets map likewise into regions // and ///, respectively. By direct comparison of the Z-surface and the TF-surface, it will be seen that the region S of the TF-surface corresponding to the first sheet of the Z-plane consists of the first sheet of the TF-surface together with the second quadrant of the third sheet and the third quadrant of the second sheet, Fig. 121. 358 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. All of the values of S lying to the left of the axis of imaginaries are repeated, as will be seen from the figure; for, that portion of the region *S lies in two sheets of the TF-surface and the one directly over the other. However, no two of these values of w correspond to the same value of z; that is, while some of the values of the particular branch of the given function are repeated in S, nevertheless the branch is a single-valued function of z. 66. Singular points of multiple-valued functions. Since each branch of a multiple-valued analytic function is single-valued, if we exclude the branch-points from consideration we may, as we have already seen, regard such a function as an aggregate of single-valued functions, each of which may be holomorphic for those values of the independent variable which belong to the particular sheet of the Riemann surface with which the corresponding branch is associated. Aside from the branch-points, each branch of the given function may have such other singular points as any single-valued function. The singular points may affect one sheet or more than one sheet and consequently may be singular points of one or of more than one branch of the given function. For example, a branch of a multiple-valued function may have a singular point which does not affect any other branch. As an illustration, consider the function w = log log 2. (1) Let 2 = 1. For this value of z,w = log z takes any one of the values 2 kiw, k = 0, 1, 2 These values of w correspond to the value z = 1 in the various sheets of the Riemann surface constituting the Z-plane. For the sheet corresponding to k = 0, w becomes infinite for z = 1 and the function (1) ceases to be regular. For k j 0, however, the function can be expanded in powers of (z 1) and therefore z = 1 is a regular point for these sheets. Consequently, the given function has a singularity at 2 = 1, which, however, affects only one branch. In the neighborhood of a point z which is not a branch-point, a multiple-valued analytic function can be expanded in a series in- volving only integral powers of (z 2 ). Such a point is a pole or an essential singular point according as the expansion contains a finite or an infinite number of terms having negative exponents. If there are no negative exponents, then ZQ is a regular point. Let us now examine the situation when ZQ is a branch-point. Suppose that at ZD a finite number of branches, say k, of the given analytic function w = /(z) are cyclically connected. Take a small region about z bounded by a curve C closed upon the Riemann surface, such that it incloses no singular point nor branch-point other than ZQ. Such AKT. 66.] SINGULAR POINTS 359 a curve must make k circuits about ZQ before it can be said to be closed. Let any convenient line extending out indefinitely from ZQ be taken as a branch-cut. We shall speak of the fc-sheeted open region thus obtained on the Riemann surface, bounded by ZQ and C, as the region R. Denote the function defined in R by the k branches of f(z) connected at z by ^(2). By means of the substitution Z - 2o = T*, the region R is mapped in a single-valued and continuous manner upon a region S of the one-sheeted r-plane. Corresponding to the point ZD, we have the point T = 0, and except at the point ZQ itself, the mapping is conformal. The transformed function 4>(r) corre- sponding to F(z) is single-valued, and with at most the exception of the point T = it is holomorphic in S. At T = 0, the function <(T) may have a pole or an essential singularity. In either case, the limit of <(T) as T approaches zero does not exist. On the other hand the function <(T) may approach a definite limiting value as T ap- proaches zero. If in the latter case we assign this limiting value as the value of <(T) at T = 0, then, by virtue of Theorem I, Art. 51, the origin is a regular point of (r). In any case <(T) can be expanded in the neighborhood of r = by means of Laurent's expansion, thus obtaining *(T) = 2) nr, (2) n=m which holds for values of T exterior to an arbitrarily small circle about the origin and interior to any concentric circle within S. If 0(-r) becomes infinite as T approaches zero, then the origin is a pole and there are a finite number of negative terms in (2) equal in number to the order of the pole. If the pole is of order X, then m = X. If on the other hand r = is an essential singular point, m becomes - oo . If it is a regular point, m is equal to or greater than zero and the expansion reduces to a Taylor series. The character of (T) in the neighborhood of T = enables us to determine the nature of F(z), and hence of /(z) for those branches affected at ZQ. The character of those branches of /(z) not connected at ZQ is quite independent of the existence of a branch-point at ZQ for the branches already considered. For those sheets not affected, this point may be a regular point or a pole or an essential singular point. It may also be a branch-point at which two or more of the remain- ing branches are associated with each other. The expansion of F(z) 360 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. in the deleted neighborhood of ZQ is obtained by replacing r in (2) by i (z zo)* giving **(*) = n(z - z/ (3) n = m which holds for values of z in R, that is in all of the sheets affected. As we know from Art. 8, there are k, k th roots of (z ZQ), namely: 111 i (z - ZQ)*, co(z - zo)*, " 2 (z - Zo)*~, . . . , a k ~ l (z - Zo)*, i where (z zo) fc is now restricted to the principal value of the root and co is one of the imaginary k th roots of unity. To get the expansion of the individual branches of /(z) composing F(z), all we need to do is to replace n in (3) by a n , a n u> n , . . . , a n (co fc - 1 ) n , respectively. In case r = is a regular point of $(r), then from (2) we have L 0(r) = A, T=0 where A is different from zero or equal to zero according as we have m = or m > 0; hence, since z approaches ZQ as r approaches zero, we have L F(z) = A. z=zo Assigning this value as the value of F(z) at Zo, then the branch-point ZQ is called a point of continuity of F(z). The expansion (3) is in this case a series of increasing positive fractional powers of (z ZQ), and we have m = 0. If m is greater than zero, say equal to r, we say that F(z) and hence /(z) has a zero point of order r at ZQ. The function w defined by w z = z has a point of continuity at the branch- point z = 0. The expansion consists of one term, namely z 5 , and the given function has a zero point of order one at this point. If T = is a pole of (r), m is negative and finite. Let the order of the pole at T be r. Then the expansion (2) and therefore (3) has r terms with negative exponents. We say that F(z) has a pole of order r at ZQ. The function w defined by the relation w 2 = - z furnishes an illustration. The point z = is a branch-point, and at the same time it is a pole of order one, the expansion consisting of a single term having a negative exponent, namely z~*. AKT. 66.] SINGULAR POINTS 361 If T = is an essential singular point, then the expansion (2) contains an infinite number of terms with negative exponents; that is, we have m = QO . In this case, the transformed series (3) has also an infinite number of terms having negative fractional expo- nents. The point ZQ is called an essential singular point of F(z). j_ The function w = e^l has such a point at z = 0. This point is a simple branch-point. The form of the expansion in the deleted neighborhood of the origin is The origin is therefore an essential singular point as well as a branch- point. Since the single-valued function <(T) can not have other singular points than poles and essential singular points, it follows that the foregoing discussion exhausts the possibilities as regards the singu- larities of a function at a branch-point when a finite number of sheets are connected. That is, a branch-point may also be at the same time a point of continuity, a pole, or an essential singularity. In any case we shall class a branch-point among the singular points of a function, since the expansion of the function in the deleted neigh- borhood of the point involves fractional powers of the variable, that is, the function does not permit a proper power series develop- ment. Consequently, a branch-point is not to be included in the region of existence of the given function, and, moreover, we can not reach such a point by the process of analytic continuation from any regular point in the Riemann surface. If an infinite number of sheets are connected at a branch-point, then the singularity at that point may be of a transcendental char- acter.* For example, in the case of the logarithmic function, the origin is a singular point where log z becomes infinitely great by every possible approach of z to the origin. It is not, however, a pole because the order of the singularity is not finite. It is in this case sometimes spoken of as a logarithmic singularity. It is not within the scope of this volume to discuss the character of the various transcendental singularities that may occur at branch-points, further than to point out the illustration already cited. We have excluded from the region of existence of an analytic * Cf. Zoretti, Lemons sur le prolongement analytique, p. 61. 362 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. function the poles and essential singular points in the case of single- valued functions, and in the case of multiple-valued functions we have excluded also branch-points. It is often convenient to con- sider the region of existence as thus used, together with those singu- lar points where w and z have a definite one-to-one correspondence. Thus in single-valued functions we may include in our consideration the poles of a function, if we associate with the pole z = ZQ the functional value w = GO. Likewise in multiple-valued functions, we may include those branch-points at which the function has a point of continuity or a pole, by associating with the branch-point z = Zo as the corresponding functional values, w = L f(z) and z=o to = oo, respectively. When these points of the Riemann surface and their corresponding functional values are added to the region of existence of an analytic function, we shall speak of the result- ing aggregate of pair-values (w, z) of the given function as an analytic configuration. The essential singular points are not in- cluded hi the notion of an analytic configuration; for, corresponding to such a singular point no particular w-point can be associated. Just as in the consideration of single-valued analytic functions, the singular points are boundary points of the region of existence. In case of multiple-valued analytic functions, these singular points include the branch-points as well as the poles and essential singular points. When these boundary points constitute a closed curve upon the Riemann surface, then the region of existence of the given function has a natural boundary; that is, a boundary beyond which it is impossible to proceed by analytic continuations. The region of existence may be different in different sheets of the Riemann sur- face for the function. 67. Functions defined on a Riemann surface. Physical appli- cations. The chief advantage of a Riemann surface is that it en- ables us not only to establish a one-to-one correspondence between the points of the Z-plane and those of the TF-plane, but also to state that any continuous path in the one plane corresponds to a continu- ous path in the other. Regarding the branch-points as boundary points of the region of existence on the Riemann surface and using the term region as in the case of single-valued functions to mean an aggregate of inner points, unless otherwise stated, we can by means of Riemann surfaces extend to multiple-valued functions the general properties already discussed for single-valued functions. When use is made of Riemann surfaces, the mapping by means of multiple- ART. 67.] FUNCTIONS ON RIEMANN SURFACES 363 valued analytic functions is conformal in any region which contains no branch-points or other singular points of the function, provided that the derivative of the function is different from zero; that is, in any region in which the branches of the function are holomorphic and their derivatives do not vanish. In the neighborhood of a branch-point ZQ of finite order, the ex- pansion of a multiple-valued function requires an infinite series whose terms involve fractional powers of (z z ). In the neighbor- hood of any isolated singular point other than a branch-point, the expansion of any branch takes the form of a Laurent series. In the neighborhood of a regular point, the branches of the function can each be expanded in a Taylor series. The circle of convergence may, however, lie partly in one sheet and partly in another depend- ing upon the position of the point in whose neighborhood the func- tion is expanded. In no case can the brandi-point lie within the region of convergence; for, otherwise there would exist a regular power series development in the neighborhood of a branch-point, which we have seen is impossible. As a consequence it will be seen that by the process of analytic continuation a branch-point can not be included in the region of existence of an analytic function. It is for this reason, as already stated, that we have regarded branch- points as belonging to the boundary of the region of existence. Having thus excluded the branch-points, analytic continuation upon the Riemann surface can take place along any ordinary curve just as in the case of single-valued functions. The path of analytic con- tinuation may lie wholly in one sheet or may pass from one sheet to another as the conditions require. Along any closed path of ana- lytic continuation the terminal value of the function is identical with the initial value. If the path incloses a branch-point, then it must make as many circuits around that point as there are sheets connected at the point before the path can be said to be closed. Similarly the process of integration can be extended to multiple- valued functions, the path of integration being any continuous ordinary curve upon the Riemann surface. In case the path does not inclose a branch-point, there is nothing new in the process. If, however, the path incloses a branch-point, say of order k, then it must pass k times around that point before the path is closed. For exajuple, suppose it to be required to integrate the function w = Vz along a closed path C about the origin. The closed path C 364 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. can be deformed into a double circle about the origin without affect- ing the value of the integral. We have then f* /* ' /M r ' " / z*dz = I " P *e~ 2 iped0 = ip* / e~*d0 Jc Jo Jo = IP* r*cos^<# - P * r*wM* = o. Jo & Jo & The residue of a multiple-valued function at an isolated singular point is defined, as in the case of single-valued functions, to be where C is a closed curve about the given singular point and inclosing no other singular point. In case the point ZQ is a branch-point at which k sheets are connected, the form of the expansion of the function is = 2, where m = X or ra = according as ZQ is a pole or an essential singular point. Since this series converges uniformly, it can be integrated term by term, and since C can be taken as a circle about ZQ, traversed k times, we have as the residue at z where m = k. When we have k < m < 0, the residue is zero. To find the residue when there is an isolated singular point at z = QO, we have as the expansion of the function and hence obtain It is often necessary to employ multiple-valued functions in map- ping from one complex plane to another. In case the inverse function is single-valued, we can do as was done in Chapter IV, namely: we can map the whole of one plane upon a portion of the other plane, or we can now introduce a Riemann surface in place of ART. 67.] FUNCTIONS ON RIEMANN SURFACES 365 the one plane, thus making it possible to map in a continuous and single-valued manner the whole of either plane upon the whole of the other. As an illustration, consider the function w = z 2 . The W-plane is a double-sheeted Riemann surface having branch-points at w = and w = oo. Let us choose the negative half of the axis of reals as the branch-cut. By comparing Figs. 122 and 123, it will be seen that (6) X (a) U FIG. 122. FIG. 123. the half of the Z-plane lying to the right of the axis of imaginaries maps into the first sheet of the TF-plane while the half of that plane to the left of this axis maps into the lower sheet of the Riemann sur- face constituting the PF-plane. Any line (a) lying in the right-hand half of the Z-plane and parallel to the F-axis maps into the parabola (.A) lying wholly in the first sheet of the Riemann surface. Likewise any line parallel to the F-axis and lying in the left-hand half plane maps into a parabola lying wholly in the second sheet. The line (6) parallel to the X-axis maps into a parabola (B) situated symmetrically with respect to the C/-axis and lying partly in the first sheet and partly in the second sheet as indicated, the dotted portion of the curve indicating that part which lies in the second sheet. A circle about the origin in the Z-plane maps into a double circle in the TF-plane, one in each sheet. The 366 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. two circles thus obtained constitute a closed path as shown in Fig. 123. The line (C) situated in the first sheet and parallel to the F-axis maps into the branch (c) of an equilateral hyperbola lying in the first quadrant of the Z-plane. Similarly, the straight line (C') lying in the second sheet directly underneath (C) maps into the branch (c') of the same hyperbola lying in the third quadrant. Lines parallel to the t/-axis map into hyperbolas cutting the first hyperbola at right angles. It is of interest hi this connection to point out some of the uses of Riemann surfaces in the discussion of physical problems. A solu- tion of Laplace's differential equation gives a potential function. The solution may give a multiple-valued function u and the corre- sponding analytic function w = u + iv = /(z) to which it gives rise may have singular points other than the branch-points. In the corresponding physical problem, however, the potential must be single-valued when the boundary conditions are given.* Similarly, if we arrive at the potential function through a process of integration, the function may be a cyclic function; that is, it may be multiple-valued because the path of integration encircles a singular point of the region under consideration. If such points be excluded from the region by arbitrarily small circles being drawn about them, the resulting region is multiply connected. It may be made simply connected by the introduction of barriers or cross- cuts, in which case the potential function obtained in this manner is single- valued. If the given region lies upon a Riemann surface, it will be seen that in order that the potential shall be single-valued in that region it is necessary to restrict the region to one sheet of the Riemann surface by excluding from the region all branch-points and branch-cuts. Thus in Fig. 123, the transformation w = & enables us to consider the potential in an electrostatic field bounded by the parabola A and lying exterior to this curve. This region corresponds to the region in the Z-plane lying to the right of the line o parallel to the F-axis, Fig. 122. This transformation does not, however, enable us to consider a field upon the Riemann surface lying interior to the parabola, since this region contains the branch-point w = and the negative (/-axis, which we have here taken as a branch-cut. * See Jean, Electricity and Magnetism, Art. 330, also Lamb, Hydrodynamics, 3d Ed., Art. 62. AOT. 68.] FUNCTION OF A FUNCTION 367 As another illustration, consider the function 2 - 1 w log We have already discussed the mapping of this functipn where the amplitude z 1 of T s is restricted to its chief amplitude. If we *ow allow 6 to take all z ~"r~ \. values we have a multiple-valued function and the corresponding Z-plane must consist of an infinite number of sheets. The branch-points of the function are located at 2 = 1,2 = 1. All of the sheets of the surface are connected at these two points, but the cyclic arrangement is in the one case the reverse of the other. The two points can then be connected by a branch-cut. Having excluded the branch-points by arbitrarily small circles extending in each sheet in succession, the resulting surface is triply connected. If we now take the branch-cut as a barrier or cross-cut and draw an additional cross-cut from some point on this inner boundary of the surface to the point at infinity, we shall make the region simply-connected. Any region in any sheet exterior to a curve inclosing both of the two branch-points may therefore be taken as a suitable region for the con- sideration of a potential. 68. Function of a function. We have the following theorem. THEOREM. An analytic function of an analytic function is an analytic function. We shall demonstrate this theorem for the case of single-valued functions. From the discussion in this chapter it follows that the theorem may be extended to multiple-valued analytic functions. Suppose F(w) to be a single-valued analytic function of w, and let w = /(z) likewise be a single-valued analytic function of z. We are to show that F\f(z)\ = <(z) is also an analytic function of z. Let z be any regular point of /(z) such that Wo = /(ZQ) is a regular point of the function F(w). Then F(w] can be expanded as a power series in (w w ) ; that is, we may write F(w)=ao-\-ai(w Wo)-\-az(w itfo) 2 + +ak(w w>o) fc + . (1) Let the circle of convergence of (1) have a radius equal to p. Since w = f(z) is analytic, it follows that ak(w Wo) k is also analytic for all finite values of k and hence may be expanded as a power series in (z Zo). The circle of convergence of these power series may be denoted by r, since they have the same radius of convergence for all values of k. We may write <*o = 0o,o ai(w w ) = 0i,o + 0i,i(z Zo) + 0i,2(z Z ) 2 + , az(w Wo) 2 = 02,o ~\~ 02,i(z Zo) -f- 02, 2 (z Zo) 2 -(- i a 3 (w - Wo) 3 = 03,o + 03,i(z - ZQ) + 3 , 2 (z - Zo) 2 + , if ' 368 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. Substituting these values in (1), we have the double series 4>(z) = 0o.o + 0i.o + 0u(z - Zo) + ll2 (z - zo) 2 + + 02,0 + 02,l(z - Zo) + 2,2(2 - Zo) 2 + + 03,0 + j8 8 .i(2 - Zo) + 03. 2 (Z - 2o) 2 + . . (2) The rows of this series converge absolutely for all of those values of z for which | z ZQ \ = r' < r, since each row is a power series with the radius of convergence r. Summing by rows, the resulting series each term of which is the sum of a row in (2) is none other than (1), which, as we know, also converges absolutely for all values of w for which | w W Q | = R < p. Since the series formed by taking the absolute values of the terms of (2) converges by rows, it follows that the double series of these absolute values also converges *; that is, the given series converges absolutely. Consequently, by the theorem of Art. 44, we may sum it by columns as well as by rows. We have then GO 00 OQ ) + 2 0*,2 (Z - Zo) 2 1 where we may put 00 00 V0*, = 0o, ^0t.n = 0n, n= 1, 2, 3, .... 1 The function (z) is therefore represented by a power series in the neighborhood of 2o; hence, 2 is a regular point of (z). But z is any regular point of w = f(z) for which the corresponding point w is a regular point of F(w). Hence, within the region for which z lies in the region of existence of F(w], we may regard $(2) as identical with F\f(z)\. It is possible, however, that the region of existence of the analytic function $(2) thus defined may extend beyond the region of existence of w = f(z). On the other hand, it is possible that the values of w given by the relation w = f(z) may not lie within the region of ! existence of F(w), in which case F\f(z)\ has no meaning. 69. Algebraic functions. In the preceding chapter, we dis- cussed a special kind of algebraic functions, namely rational func- tions. In the present chapter we have had occasion to consider several particular algebraic functions. We shall now consider the * See Bromwich, Theory of Infinite Series, Art. 31. ART. 69.] ALGEBRAIC FUNCTIONS 369 general case where w = /(z) is defined by an irreducible equation of the form F(w, z) = w* + / : (z) w^+Mz) w"- 2 + + f n (z) = 0, n > 0, (1) where /i (z), fz(z), . . . , f n (z) are rational functions.* In some dis- cussions it is convenient to write the foregoing equation in the follow- ing form poGO w" + PiO) W- 1 + + p k (z}w n - k + + Pn(z)=0, (2) where Pk(z), &= 0, 1, 2, . . . , w, is a rational integral function of z. For each value of z these equations have n roots and there are then in general n distinct values of w. We shall denote these values by Wi, Wz, . . . , w n . The function w = /(z) thus defined is then a multi- ple-valued function, and w\, w 2 , . . . , w n are all functions of z. In fact the functions Wi, w^, . . . , w n become the n branches of the given function when once the branch-cuts are properly chosen. THEOREM I. Every value of ZQ for which all of the n branches of an algebraic function remain finite and distinct is a regular point of each branch of the function. Corresponding to a circle C about ZQ as a center there may be drawn a circle C* in the W-plane about each of the distinct points W ,k (k = 1, 2, . . . , n) as a center such that for all values of z in the region S bounded by C each of these circles C k shall inclose values of w belonging to one and only one branch of the given function. Conse- quently, each branch of the function is single-valued for values of z in/S. We shall now show that ZQ is a regular point of each branch of the given algebraic function. To do this, it is sufficient to show that each of the functions w k (k = I, 2, . . . , n) admits of a derivative for values of z in S. Denote by Aw h the increment of w k corresponding to the increment' Az of z. If the given function is defined by the equation F(w, z) = 0, we have AF = F(w + Aw, z + Az) - F(w, z) F(w + Aty, z -f Az) F(w, z + Az) Aw F(w, z + Az) - F(v>, z) ~~ A Aw * For a somewhat different definition of an algebraic function, see Forsyth, Theory of Functions, 2d Ed., Art. 95. Compare also Encyklopddie d. Math. Wiss., Vol. II, B 2 , Art. 1. 370 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. ftp 1 ftp 1 But since the derivatives , both exist, it follows that for dw dz w = Wk the foregoing equation can be written in the form ( dF ) (dF where ei, 2 approach zero with Ate*, Az, respectively. We have from the foregoing relation dF dz Az ~ ~ dF As Az approaches zero, Awk also approaches zero and hence we have in the limit the same law as holds for the differentiation of implicit functions of real variables, namely dF_ dwie _ dz , Q v ~dz~" ~~d^' . dw k dF The value of - is different from zero for all values of z in S: for, dw h otherwise F(w, z) = would have a multiple root * for some value of z in S, which is contrary to the hypothesis. The value of -^ is given by dz (3) for any Wk, k = 1, 2, . . . , n. Consequently, the point ZQ is a regular point for each of the functions Wk (k = 1, 2, . . . , n). THEOREM II. The number of points at which two or more of the branches of an algebraic function may become equal or infinite is finite. The finite values of z for which two or more of the values Wi, Wz, . . . , w n become equal are those values of z that cause the dis- criminant of F(w, z) = to vanish. Consequently, forming the resultant f R of the two polynomials dF(w, z) F(W ' Z} ' dw^> the desired values of z are the roots of the equation obtained by equating R to zero. There can be at most a finite number of roots of this equation. The finite values of z for which two or more of the values w\, Wz, * See Forsyth, Theory of Functions, 2d Ed., Art. 94. t See B6cher, Introduction to Higher Algebra, Art. 86. ART. 69.] ALGEBRAIC FUNCTIONS 371 . . . , w n become infinite are those for which the coefficient p (z) of (2) vanishes. Since p (z) is the least common multiple of the denominators of fi (z),fa(z), . . . , f n (z) and therefore of finite degree in z, there can be only a finite number of roots of the equation p (z) = 0. The only remaining z-point at which two or more values of Wi, Wz, . . . , w n can become equal or infinite is the point z = oo . Con- sequently, the total number of points at which the branches of an algebraic function can be infinite is finite in number, and hence the theorem. From Theorem I it follows that any one of the branches Wk can be expanded in a Taylor series W k = 00,* + ai, fc (z - Zb) + a z ,k( z - Zo) 2 + ' ' ' , (4) which holds at least for all values of 2 in the region bounded by the circle of convergence C. The expansions for the various branches are of course different and in general the radius of convergence is not the same for all branches. We now see that there are only a finite number of points at which the branches of an algebraic func- tion may become infinite or two or more of them be finite and equal. Since all other points must be regular points, it follows that every branch of an algebraic function is holomorphic except at a finite number of points, and hence we have the following theorem. THEOREM III. Every algebraic function is analytic and has only a finite number of singular points. The expansion (4) of any branch Wk in the neighborhood of a point where that branch is finite and distinct defines an element of the function, and from this element the algebraic function is completely and uniquely determined. It is also of interest to note that it fol- lows from Theorem I that the singularities of an algebraic function can occur only at points where two or more of the branches have the same finite value or where one or more of the branches become infinite. We shall use the expression infinity of a function, or more briefly an infinity, to mean a singular point of a multiple-valued function at which the function becomes infinite by at least one approach of the independent variable to the critical point. Infinities include both poles and essential singular points but exclude branch-points, unless those points are at the same time poles or essential singular points. The order of an infinity may therefore be either finite or 372 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. infinite. By the following theorem we shall show that the infinities of an algebraic function are necessarily poles. THEOREM IV. The infinities of an algebraic function are singular points of the coefficients and conversely. Moreover, an algebraic func- tion can have only polar infinities. Let the given function w = f(z) be defined by the algebraic equa- tion F(w, z) = w n + /i(z)w- 1 + + / n (z) = 0. (5) We shall first show that if ZQ is an infinity of any branch Wk, it is at the same time a singular point of at least one of the coefficients /i> /2> , fn of (5) and conversely. We shall also show that the order of the singularity of this coefficient is equal to or greater than the order of the infinity of w k . From (5) we have by aid of the relation between the roots and coefficients of an algebraic equation /i (2) = - (wi + w z + + w k + + w n }, /a (2) = Wiw-i + tPitOs + + w k w 2 + w k w 3 + + w n -iW n ; /n(2) = ( l) n WiUW . . . W k . . . W n . Let the branches of the given function, other than Wk, be determined by an equation of the form W"- 1 + 0l(2) W"- 2 + n - 2 (z) W + 0-i(2) = 0. (6) Then we may write (7) /-GO = - From the last of equations (7) it follows that z must be a singular point of f n (z) unless n -i(z) has a zero point of as high order as the infinity of w k . The singularity of / n (z) may, however, be of higher order than the infinity of Wk in case z is also an infinity of < n _i(z). It may be of lower order provided n -i(z) has a zero point at z of order less than the order of the infinity of w k . Consequently, / n (z) has a singular point at ZQ of order equal to or higher than the infinity of Wk unless ZQ is at the same time a zero point of n _i(z). In the latter case, it follows in a similar manner from the next to the last of these equations that/ n -i(z) has a singularity at ZQ of order equal ART. 69.] ALGEBRAIC FUNCTIONS 373 to or higher than the infinity of Wh unless < n _ 2 (2) has a zero point at ZQ. Continuing in this manner, it follows that either some one of the coefficients fa(z), . . . , f n (z) has an infinity at ZQ of at least as high an order as that of Wk, or #1(2) has a zero point at ZQ. In the latter case, it follows from the first equation that fi(z) has an infinity at ZQ of the same order as w^ We can conclude that at least one of the coefficients fi(z), fz(z), . . . , f n (z) has a singularity at z of equal or higher order than the infinity of Wk. However, the coeffi- cients fi(z), fi(z), - - , f n (z) are all rational functions of z and there- fore the function itself can have at most polar infinities. Conversely, if some one of the coefficients /i(z), fo(z), . . . , f n (z) has a singular point at ZQ, then it must be a pole. From (7) it fol- lows that if ZQ is not a pole of Wk, it must be a pole of at least one of the coefficients i(z), 2 ) n , . . . , a n (w r - l ) n , where w is one of the r imaginary r th roots of unity, as explained in Art. 66. The remaining m r branches may remain distinct or form by themselves one or more cycles. ART. 69.] ALGEBRAIC FUNCTIONS 375 Any symmetric polynomial of the m branches w\, w?, . . . , w m can be expressed in terms of the sums of equal powers of the Wk's; that is, in terms of functions of the type * S a = Wf + W 2 a + ' + W m a . In adding equal powers of Wk's, however, the coefficients of terms having fractional exponents vanish by virtue of the relation 1 + u + co 2 + + w 1 - 1 = 0. Hence, in this case also, the expansion of any symmetric polynomial of Wi, Wz, . . . , w m involves only positive integral powers of (z ZQ), and consequently z is a regular point of the given function /(*) Finally, let us suppose 2 is a pole of the given function. In this point then one or more values of Wk become infinite as z approaches ZQ. In the latter case the point z may be a branch-point. The expansion of Wk in the neighborhood of such a point involves a finite number of terms with negative exponents, which may be either frac- tional or integral. In forming a symmetric function of w\, w%, , . , , w m the exponents all become integral even in case ZQ is a branch-point and hence the resulting function has a pole at z . As we now see, any symmetric function of w\, Wz, . . . , w m can have in the region S only polar singularities. There can be but a finite number of poles in S, for otherwise there must be at least one essential singular point. Consequently, any symmetric func- tion of Wi, w 2 , . . . , w m must be meromorphic in the given region S as the theorem requires. We shall now consider the following theorem. THEOREM VI. Every analytic function w = f(z) having n values for each value of z and having in the entire complex plane no other singu- larities than poles and branch-points can be expressed as a root of an algebraic equation of degree n in w, the coefficients of which are rational functions of z, and consequently w = f(z) is an algebraic function. Corresponding to any point z of the complex plane, it follows from the hypothesis set forth in the theorem that w has n values, which as before we denote by w\, w%, . . . , w n . These values are in general distinct. * See B6cher, Introduction to Higher Algebra, p. 241. 376 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. We shall now consider the following symmetric polynomials of tfli, 10j, . . . , W n . /l(z) = -(Wi + V*+ ' ' + Wn), -f WiWa + ' ' ' + W n -iW n , . . W n . From Theorem V, it follows that the functions fi(z), fa(z), . . . , f n (z) are meromorphic for all values of z, since the region S may now include the whole complex plane. These functions therefore have only polar singularities and must be rational functions. To complete the proof, we have only to equate to zero the product n JJ (w Wk), namely t=i (w Wi) (w ? 2 ) . (w w n ) = 0. (10) By the principles of elementary algebra, this equation can be written in the form w" + /i(z) w n ~> + f 2 (z)w n - 2 + + f n (z) = 0. The function w = f(z) is a root of this equation, and as we have seen the coefficients /i(0),/ 2 (2), . . . , f n (z) are rational functions. Hence the theorem. The Riemann surface for an algebraic function can now be de- termined. If the equation F(w, z) = defining the function is of the degree n in w then the Z-plane must be an n-sheeted surface. The branch-points and the connection of the sheets at each can be determined by the methods given. The branch-cuts can be drawn between branch-points where the same sheets are affected but their cyclic arrangements are of opposite order, or they may be drawn to the point z = oo . All of the sheets must form a connected sur- face since by definition F(w, z) = is an irreducible equation. In case this equation is reducible, that is, can be separated into two or more factors involving w and z, then it defines not one algebraic function but two or more such functions and the corresponding Riemann surface separates into two or more distinct surfaces. In fact, a given relation between w and z may be said to represent one multiple-valued function or more than one function according as the corresponding Riemann surface consists of one connected part or of two distinct parts. ABT. 69.] EXERCISES 377 EXERCISES 1. Show that z, where a = a -j- ib, is a multiple-valued analytic function with a branch-point of an infinitely high order at the origin. HINT: Put z a = e alogz . O 2. Given the analytic function w = . Locate the branch-points V2 + Z 2 and determine whether they are at the same time poles. 3. Evaluate the integral I , where C is a curve closed on the J C Vz 2 + 1 Riemann surface about the point z = i. f.' 4. Knowing that dz arc tan z r* = I ? J 1 . , f + z* expand in an infinite series the function /(z) = arc tan z. How large is the circle of convergence of this series? What singular points restrict the size of the circle of convergence? Are these singular points also branch-points? 6. Determine the branch-points of the function w = arc sin z. 6. By aid of the definitions of circular functions, show that 1 , 1 + iz w = arc tan z = log - - - /r * & % 1 zz Locate the branch-points of w and determine a region on the Riemann surface in which each of the infinite number of branches is holomorphic. Show whether this region is simply or multiply connected. 7. Show that the function /(z) = log (sin z) is analytic. _ i 8. Discuss the Riemann surface for the function w = v (z a) (z /3) 2 . What physical phenomena does this functional relation represent? 9: Construct a model showing the connection of the sheets of the Riemann surface required for the function discussed in Art. 61. 10. Discuss the Riemann surface for the function w 2 1 = z 3 . Determine a fundamental region on the TF-Riemann surface. 11. By mapping the Z-plane upon the TP-plane by means of the function w = e z , show that w takes every value, except zero and infinity, in the neighbor- hood of z = oo . 12. Given the algebraic function, tc 3 18 w 35 z = 0. Determine the character of the Riemann surface suited to this function by the method of Art. 60. L* 13. Expand the function w = Vz in an infinite series for values of z in the ,f neighborhood of z = a, where a = 2 e 3 . cL^ 14. Given the function w = Vl z 3 . Examine the character of this func- tion in the neighborhood of z = oo . o 16. Indicate the form of the expansion in an infinite series which the analytic function w = /(z) takes for values of z in the neighborhood of z = i, (a) if z = i is a point of continuity and a branch-point of order 2; (6) if z = i is a pole of order k but not a branch-point; (c) if z = i is a branch-point where 3 branches | come together and at the same time a pole of order 4; (d) if z = i is a branch- point of order 6 and an essential singular point; (e) if z = i is a zero-point of order 3 and a branch-point where 2 branches become equal. 378 MULTIPLE-VALUED FUNCTIONS [CHAP. VIII. 16. Distinguish between an algebraic function and a transcendental function (a) as to the number of possible zero points, (b) as to the number of poles and essential singular points that may occur, (c) as to the number and order of the branch-points that the function may have. Illustrate in each case by a particu- lar function. 17. Does the expression arc sin (cos z) represent a multiple-valued analytic function or a number of single- valued analytic functions? 18. Discuss the Riemann surfaces for the following functions: (a) w = Vz i + Vz 2, (6) .vrri + ^=L=' (c) w* 4 w = z. INDEX INDEX (Numbers refer to pages.) Absolute convergence of infinite prod- Cauchy-Riemann differential equations, ucts, 311. Absolute convergence of series, 201. theorems concerning, 201, 203, 204, 206, 208, 212, 214, 227, 228, 236. Addition of complex numbers, 8. of infinite series, 206. Algebraic functions, 23, 368. theorems concerning, 369, 370, 371, 372, 375. Amplitude, definition of, 6. Analytic configuration, 362. Analytic curve, 253. Analytic continuation, 245, 249. by power series, 251. by Schwarz's method, 252. theorems concerning, 250, 255. Analytic function, definition of, 45, 257. of an analytic function, 367. Anharmonic ratio, 178. Applications to physics, 96, 103, 104, 107, 112, 113, 117, 118, 119, 120, 132, 137, 138, 139, 140, 141, 152, 190, 196, 362, 366. Boundary point, 20, 348. Bounded sequence, 28. Branch of a function, 332, 355. theorems concerning, 369, 370, 374, Branch-cuts, 338, 347, 356. theorems concerning, 351, 352. Branch-point, 333, 347, 355. order of, 334. theorems concerning, 335, 351, 352, 353, 374, 375. Cauchy-Goursat theorem, 66, 68. extension of, 71. Cauchy's integral formula, 75. 83. Change of variable, 64, 89. Chief amplitude of a complex number, 7. Circle of convergence, definition of, 230. theorems concerning, 235, 236, 237, 274. Class of a function, 316. Closed region, 20. Complex number, modulus of, 6. amplitude of, 6. chief amplitude of, 7. Complex numbers, definition of, 5. addition and subtraction of, 8. comparison of, 8. division of, 16. geometric representation of, 6. multiplication of, 11. Complex plane, 6. Conditional convergence of series, defi- nition of, 204. theorems concerning, 205, 237. Conditional convergence of infinite products, 311. Conformal mapping, definitions of, 107. theorems concerning, 107, 175, 189. Conjugate functions, definition of, 101. Conjugate points, definition of, 170. theorems concerning, 172, 173, 174. Continuity, definition of, 33. theorems concerning, 34, 35, 38, 39, 40. point of, 360. Continuity of f'(z), 77. Convergence of infinite series, defini- tion of, 198, 213. theorems concerning, 199, 200, 201, 207, 208, 214. Cross-cut, 53. 381 3S2 INDEX Degree of a function, 22. Deleted neighborhood, 20. DeMoivre's theorem, 12. Derivative, definition of, 43. continuity of, 77. Derived function /'(z), properties of, 77. Differentiation of series, 222, 225, 237. under integral sign, 52. Difference of two series, 208. Division of complex numbers, 16. Double series, 213. Doubly periodic functions, 318, 319, 324, 325, 326. Element of an analytic function, 250. Electric potential, 96. Elliptic motion, 194. Equianharmonic points, 182. Equipotential lines, 104. Equipotential surface, 103. Essential singular point, 263, 273. theorems concerning, 270, 271, 292. Expansion, 160. modulus of, 160. Expansion of functions in series, 238. Function, definition of, 21. analytic function, 45, 257. Functions, definition of elementary, exponential, 122. hyperbolic, 150. logarithmic, 133. trigonometric, 144. Function of a function, 367. Fundamental region, 116, 357. Fundamental theorem of Algebra, 291. Geometric inversion, 166. of straight line, 168. of circle, 168. effect on angles, 169. theorems concerning, 171, 172, 173. Green's theorem, 54. Harmonic points, 181. Higher derivatives of / (z), 79. Holomorphic in a region, 45. theorems concerning, 45, 66, 68, 71, 74, 75, 77, 80, 82, 83, 85, 92, 93, 224, 235, 238, 247, 248, 250, 263, 265, 268, 273, 278, 284, 285, 296, 304, 323. Hyperbolic functions, 150. Hyperbolic motion, 193. Indefinite integrals, 90. Infinite product, definition of, 308. absolute convergence of, 311. theorems concerning, 309, 311. Infinity, point at, 157. Infinity of a function, 371. theorem concerning, 372. Inner point, 20. Integral, positive direction of, 52, 63. Integral of / (z), definition of, 60. theorems concerning, 62, 63, 64, 72, 73, 74, 80. Integral of / (z), when independent of path, 73. Integral, function / (z) defined by, 80. Integration of series, 222, 224, 237. Inverse functions, property of, 85. Irrational function, 23. Irrational numbers, 2. Isogonal mapping, 107. theorem concerning, 107. Isolated singular point, 263. Isolated line of singularities, 265. Lacunary space, 258. Laplace's equation, 92. consequence of, 93. Level lines, 103. Laurent's series, 278. function uniquely determined by, 280. principal part of, 279. Limits, 23. Limit of a function, 26. theorem concerning, 27. Limit of a sequence, 24. theorems concerning, 24, 29, 30, 31. Linear automorphic functions, 114. INDEX 383 Linear fractional transformations, 156. classifications of, 190. general properties of, 173. Line-integral, definition of, 46. theorems concerning, 51, 52, 54, 56, 57, 58. Lines of force, 104. of slope, 103. of level, 103. Liouville's theorem, 274. Logarithm, definition of, 133. principal value of, 135. Logarithmic derivative, definition of, 296. properties of, 296, 297. Logarithmic potential, 96. Logarithmic spiral motion, 161. Loxodromic motion, 194. Lower limit, 28. Maclaurin's series, 239. Map of w = 2 2 , 105. w = z n , 114. z = w + e w , 131. = log , 138. 1) (2-1), 139. (z I) 2 -^+I)' 141 ' w = cos z, 147. w = coshz, 151. the anharmonic ratios, 181. Mapping in neighborhood of a regular point, 107. Maximum, 28. M creator's projection, 137. Meromorphic in a region, definition of, 293. theorem concerning, 374. Minimum, 28. Mittag-Leffler's theorem, 303. Modulus, definition of, 6. Moduli, relation between, 9, 10. Modulus of expansion, 160. Monogenic analytic function, 258. Morera's theorem, 80. Multiplication of complex numbers, 11. Multiply connected regions, 53. integral over boundary of, 74. Multiple-valued function, definition of, 22. applications of, 366. illustrations of, 329. Natural boundary, 258, 362. Neighborhood, 20. Net of points, 321. Non-essential singular point, 262. Norm of a complex number, 7. Open region, 20. Order of a pole, 262. a zero point, 266. Ordinary curve, definition of, 47. Parabolic motion, 191. Painlev^'s theorem, 250. Partition, definition of, 3. Path of integration, definition of, 47. deformation of, 73. Periodic function, 125, 317. theorems concerning, 319, 323, 324, 325, 326. Period-parallelogram, 321. Period region, 322, 323, 326. Plane doublet, 118. Point of continuity, 360. Point of equilibrium, 141, 143. Pole, definition of, 262, 273, 360. order of, 262, 360. Poles, theorems concerning, 267, 268, 270, 284, 285, 286, 290, 292, 294, 295, 296, 297, 298, 325, 326, 335, 372, 374, 375. Potential, definition of, 96, 98. Power series, 226. theorems concerning, 227, 228, 230, 232, 235, 236, 237, 238, 274. with negative exponents, 275. Primitive element of an analytic func- tion, 258. Primitive period, 126. Primitive period pair, 318. Product of series, 208. Products, infinite, 308. INDEX Quotient of two series, 212. Radius of convergence, 230, 232. Ratio of magnification, 109. Rational numbers, 1. Rational functions, 290. theorems concerning, 292, 294, 295, 298, 299. Rational integral function, 22. theorems concerning, 290, 291. Rational fractional function, 22. Real numbers, system of, 4. Reciprocation, 167. Reflection, 167, 254. Region, definition of, 20. of convergence of a series, 218. of existence, 258, 362. of periodicity, 125. Regular point, definition of, 45. theorems concerning, 263, 265, 369. Residue, definition of, 284, 364. theorems concerning, 284, 285, 286, 295, 296, 297, 325. Riemann's theorem, 263. Riemann surface for w = \/z, 336. for to* 3 w 2 z =0, 338. f or w = V V,2 -=--, Z Zi 343. for w = log z, 346. Riemann surfaces, properties of, 355. functions defined on, 362. mapping on, 364. Roots of a complex number, 13-16. Root, principal value of, 14. Sequence, limit of, 24. theorems concerning, 24, 31. Sequence of circles, limit of, 29. Sequence of rectangles, limit of, 30. Series, convergence of, 198, 199, 200, 201. differentiation of, 222. double, 213. integration of, 222. operations with, 206. power, 226. uniform convergence of, 217. Simple pole, 263. Simply connected region, definition of, 52. Simply periodic functions, 126. Single-valued function, 22, 245. Singular point, 45, 262, 358. Sink, 118. Source, 118. Stereographic projection, 184, 354. Subtraction of complex numbers, 10, Sum of two series, 206. Taylor's series, 239. Trigonometric functions, 144. Transcendental function, 23, 300. Transcendental functions, theorems concerning, 301, 313, 317, 323. Transcendental integral function, 300, 301. Transcendental fractional function, 302, 317. Translation, 159. Transformation, linear fractional, de- fined, 156. w = z + 0, 159. w = az, 159. w = az + /3, 162. w = -, 166. Upper limit, 28, 39, 40. Uniform continuity, 35. Uniform convergence of function along a boundary, 37. consequence of, 38, 71, 75, 255. Uniform convergence of series, 217. condition for, 220. consequence of, 221, 223, 225, 235, 236. Velocity-potential, 98. Zero point, definition of, 266, 360. order of, 266, 360. Zero points, theorems concerning, 267, 269, 294, 296, 298, 301, 313, 325, 335. V $ 33 - -J N / / / -7 -^' ' f-C fc-r, /3s < to " ?V *;*/. 2.3 y tie. R~J: ^ * * Y 331 THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. 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