PROBLEMS IN COMPLEX VARIABLE THEORY JAN G.KRZYZ I-I PROBLEMS IN COMPLEX VARIABLE THEORY JAN G.KRZYZ MODERN ANALYTIC AND COMPUTATIONAL METHODS IN SCIENCE AND MATHEMATICS Number@@O Richard Bellman Editor Q] tQJ '. EVIER NEW YORK LONDON AMSTERDAM ' .. :,''\, . '-,-" - .... PROBLEMS IN COMPLEX VARIABLE THEORY by JAN G. KRZYZ Maria Curie-Sklodowska University, Lublin (Poland) Institute of Mathematics, Polish Academy of Sciences AMERICAN ELSEVIER PUBLISHING COMPANY, INC. NEW YORK PWN-POLISH SCIENTIFIC PUBLISHERS WARSZAWA 197 1 AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue, New York, N.Y. 10017 ELSEVIER PUBLISHING COMPANY, LTD. Barking, Essex, England ELSEVIER PUBLISHING COMPANY 335 Jan Van Galenstraat, P.O. Box 211 Amsterdam, The Netherlands International Standard Book Number 70-153071 Library of Congress Catalog Card Number 0-444-00098-4 COPYRIGHT 1971 BY PANSTWOWE WYDAWNICTWO NAUKOWE WARSZAWA (POLAND), MIODOWA 10 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue, New York, N.Y. 10017. I'RINTliIl IN (,OI.AND To the memory of M ieczyslaw Biernacki Foreword Notation Contents PROBLEMS 1. Complex Numbers. Linear Transformations 1.1. Sets and Sequences of Complex Numbers . 1.2. Spherical Representation . 1.3. Similarity Transformations 1.4. Linear Transformations . 1.5. Symmetry . . . . . . . 1.6. Conformal Mappings Realized by Linear Transformations 1.7. Invariant Points of Linear Transformations 1.8. Hyperbolic Geometry [3], [21] . . . . . 2. Regularity Conditions. Elementary Functions 2.1. Continuity. Differentiability ..... . 2.2. Harmonic Functions . . . . . . 2.3. Geometrical Interpretation of the Derivative. 2.4. Conformal Mappings Connected with w = Z2 2.5. The Mapping w = +(Z+Z-l) . ...... . 2.6. The Exponential Function and the Logarithm 2.7. The Trigonometric and Hyperbolic Functions 2.8. Inverse Trigonometric and Hyperbolic Functions . 2.9. Conformal Mapping of Circular Wedges. 3. Complex Integration 3.1. Line Integrals. The Index [I], [10] 3.2. Cauchy's Theorem and Cauchy's Integral Formula . 3.3. Isolated Singularities 3.4. Evaluation of Residues. . . . . . . . . 3.5. The Residue Theorem . . . . . . . . . 3.6. Evaluation of Definite Integrals Involving Trigonometric Functions 3.7. Integrals over an Infinite Interval .... . 3.8. Integration of Many-Valued Functions [21] . 3.9. The Argument Principle. Rouche's Theorem. 4. Sequences and Series of Analytic Functions 4.1. Almost Uniform Convergence 4.2. Power Series . . . . . . . . . . Ix xiii xv 3 8 10 11 12 13 14 16 19 21 23 24 25 27 28 30 30 33 38 41 43 45 48 49 52 54 57 58 x 4.3. Taylor Series .......... . 4.4. Boundary Behavior of Power Series .. 4.5. The Laurent Series . . . . . . . . . CONTENTS 60 65 66 4.6. Summation of Series by Means of Contour Integration 69 4.7. Integrals Containing a Complex Parameter. The Gamma Function . . .. 72 4.8. Normal Families [I], [6], [16] . . . . . . . . . . . . . . . . . . . .. 75 5. Meromorphic and Entire Functions 5.1. Mittag-Leffler's Theorem [I] . 5.2. Partial Fractions Expansions of Meromorphic Functions [11] 5.3. Jensen's Formula. Nevanlinna's Characteristic [18] 5.4. Infinite Products [I], [10] . . . . . . . . . . . 5.5. Factorization of an Entire Function [I], [6], [10]. 5.6. Factorization of Elementary Functions [11] 5.7. Order of an Entire Function [6], [10], [13], [18] . 6. The Maximum Principle 6.1. The Maximum Principle for Analytic Functions 6.2. Schwarz's Lemma [16] ......... . . 6.3. Subordination [6], [15], [16], [22] ..... . 6.4. The Maximum Principle for Harmonic Functions 7. Analytic Continuation. Elliptic Functions 7.1. Analytic Continuation [1], [2], [6], [10] 7.2. The Reflection Principle [I] .. . . . 7.3. The Monodromy Theorem [I], [10]. . 7.4. The Schwarz-Christoffel Formulae [I], [10] 7.5. Jacobian Elliptic Functions sn, en, dn [1], [6], [7], [10] 7.6. The Functions cr, 1;, tJ of Weierstrass [1], [6], [7], [10]. 7.7. Conformal Mappings Associated with Elliptic Functions [24] 8. The Dirichlet Problem 8.1. Riemann's Mapping Theorem [I], [6], [10], [16] 8.2. Poisson's Formula [I], [26], [27] . . . . . . . 8.3. The Dirichlet Problem [I], [14], [15], [19], [27] 8.4. Harmonic Measure [I], [15], [16], [19], [25], [26], [27], 8.5. Green's Function [15], [16], [17], [27j 8.6. Bergman Kernel Function [6], [12], [24] 9. Two-Dimensional Vector Fields 9.1. Stationary Two-Dimensional Flow of Incompressible Fluid [4], [9] 9.2. Two -Dimensional Electrostatic Field [4]. . . . . . . . . 77 78 80 82 84 85 88 90 91 92 94 95 96 98 100 105 106 108 Ito III 113 114 116 118 121 124 10. Univalent Functions 10.1. Functions of Positive Real Part [26], [27] . 10.2. Starshaped and Convex Functions [16], [17] 10.3. Univalent Functions [16], [17], [20] .... 10.4. The Inner Radius. Circular and Steiner Symmetrization 10.5. The Method or [nncr Radius Majorization [17] [171. [20]. 127 129 130 133 136 CONTENTS SOLUTIONS 1. Complex Numbers. Linear Transformations . 2. Regularity Conditions. Elementary Functions 3. Complex Integration. . . . . . . . . . . 4. Sequences and Series of Analytic Functions 5. Meromorphic and Entire Functions . . . 6. The Maximum Principle ...... . 7. Analytic Continuation. Elliptic Functions 8. The Dirichlet Problem. . . . . 9. Two-Dimensional Vector Fields 10. Univalent Functions. Bibliography . Subject Index xi 141 155 168 191 212 224 229 245 256 260 277 279 Foreword This collection of exercises in analytic functions is an enlarged and revised English edition of a Polish version first published in 1962. The book is mainly intended for mathematics students who are completing a first course in complex analysis, and its subject matter roughly corresponds to the material covered by Ahlfors's book [1]. Some chapters, for example, evaluation of residues, de-termination of conformal mappings, and applications in the two-dimensional field theory may be, however, of interest to engineering students. Most exercises are just examples illustrating basic concepts and theorems, some are standard theorems contained in most textbooks. However, the 'author does believe that the reconstruction of certain proofs could be instructive and is possible for an average mathematics student. When the subject matter of a particular chap-ter is not covered by standard textbooks, the numbers in parantheses given in the contents indicate a corresponding bibliography position which may be consulted for further information. Some problems are due to the author, and some were adopted by the author from various sources. It was beyond the scope of author's possibility to trace the original sources and therefore the detailed references are omitted. The second part of the book contains solutions of problems. In most cases a complete solution is given; in some cases, where no difficulties could be ex-pected, or when an analogous problem has been already solved in a detailed manner, only a final solution is given. The author is well aware that it was ex-tremely hard to avoid mistakes in a book of this kind. He did his best, how-ever, to reduce their number to a minimum. It is the author's pleasant duty to thank W. K. Hayman, Z. Lewandowski, and Q. I. Rahman, who suggested some problems included in this collection. Thanks are also due to Mrs. J. Zygmunt for her help in preparing the manuscript, as well as to M. Stark for his help and encouragement. Lublin, July 1969 JAN G. KRZYZ XIII Notation 1. Set theory aeA aA BcA AnB A u,B A""-B A ""- a a is an element of the set A a is not an element of A B is a subset of A Intersection of sets A and B Union of sets A and B {a: W(a)} The complement of B with respect to A The set A with the element a removed The set of all a having a property W(a) Closure of the set A A frA r)A o Boundary of the set A The boundary cycle of a domain A taken with positive orientation The empty set 2. Complex numbers I"CZ Imz : 1=1 largz IArgz The real part of a complex number z = x+;y, i.e. the real number x The imaginary part of a complex number z = x+iy, i.e. the real num-ber y The conjugate of z = x+iy, i.e. the complex number x-iy The absolute value of z = x+iy, i.e. yx2+y2 The argument of z =F 0, 00, i.e. any angle 0 satisfying the equations Izl cosO = rez, Izl sinO = imz. There exists a unique value of argz which satisfies -7t < argz ~ 7t. It is called the principal value of argument and is denoted Argz 3. Sets of complex numbers I::, z 2] Closed line segment with end points z 1, Z 2 (Z Z2) Open line segment with end points Zl' Z2 1= Z2 Zn] Polygonal line joining Zl' Z2' , Zn in this order xv xvi NOTATION [Zl' Z2) = [Zl' Z2]""'Z2, (Zl' Z2] = [Zl' Z2]""'Zl [0, + (0), (- 00, 0] Positive and negative real axis [0, +ioo), (-ioo, 0] Positive and negative imaginary axis ( - CXl, + (0) The set of all real numbers K(zo; r) The open disk with center at Zo and radius r K( 00; 1) The set of all z with Izl > 1 C(zo; r) H+ (H-) C (C) (=f; =f) convA The circle with center at Zo and radius r The upper (lower) half-plane The finite (extended) plane The open quadrants of C, e.g. (+; -) = {z: rez > 0, imz < O} The convex hull of a set A dist(a; B) = inf{x: x = la-bl, b E B} dist(A; B) = inf{x: x = la-bl, a E A, bE B} We use the same symbol C(zo; r) for frK(zo; r), as well as- for oK(zo; r); and similarly, [zo, z t1 denotes either a set, or an oriented segment. We hope that this does not cause any misunderstanding. 4. Functions and mappings 1: 1 One-to-one correspondence f(A) The image set of the set A under the mapping f p(a, b) The hyperbolic distance between a, bE K(O; 1), cf. Ex-ercise 1.8.12 0, lei < a; ( ... ) {o z+i 7t } 11l z: a}., lI. > 0; (vi) {z: I ; ~ ~ I < I}; (vii) {z: Izl+rez ~ I}; (viii) {z: Iz2-11 < I}; (ix) {z: re[z(z+i)(z-i)-l] > O}. 1.1.22. Show that the set {z: arg(z-a)(z-b) = const} is an arc of an equilat-eral hyperbola whose center is located at 1-(a+b). 1.1.23. Evaluate all R > 0 for which the set {z: Iz2+az+bl < R} i ~ connected. 1.1.24. Explain the geometrical meaning of the set {z: AlzI2_Bz-Bz+C = O}, where A, C are real, A =1= 0, IBI2 > AC. 1.1.25. Find the radius and the center of Apollonius circle Iz-allz-bl-1 = k (k =1= 1, k > 0). 1.1.26. Find the equation of the circle through three not collinear points :I.ZZ,ZJ (cf.Ex. 1.1.24). 6 1. COMPLEX NUMBERS. LINEAR TRANSFORMATIONS 1.1.27. Suppose IZII =/: e, 1 and 0, ZI' Z2 are not collinear. Show that the circle through ZI' Z2' zii has [zl(1+lz212)-z2(1+lzI12)] (ZIZ2-ZIZ2)-1 as center and its radius is IZI-Z2111-ZIZ21IzIZ2-ZIZ21-I. Also show that Zii is situated on this circle. 1.1.28. Suppose mI' m2' m3 are nonnegative, mi +m2-+ m3 = 1, and ZI, Z2' Z3 are not collinear. Show that (i) the point Zo = mizi +m2z2+m3z3 belongs to the closed triangle T with vertices ZI' Z2, Z3; (ii) conversely, for any Zo E T there exists a unique system of nonnegative mI' m2' m3 with mI+m2+m3 = 1 such that Zo = mIzI+m2z2+m3z3' The numbers mJ are called barycentric coordinates of Zo w.r.t. T. 1.1.29. The intersection of all closed and convex sets containing a given set A is called the convex hull of A and is denoted conv A. Show that n mk ~ 0, k = 1, "', n; ~ mk = I}. 1.1.30. Show that conv {ZI' Z2' ... , zn} = U Tk1m , where Tk1m is the closed triangle with vertices Zt, ZI, Zm and the summation ranges over all triples {k, I, m} of positive integers ~ n. n 1.1.31. Prove that the equality L (C - Zk)-I = 0 implies: k=l C Econv {ZI' Z2' ... , zn}. 1.1.32. Prove following theorem (due to Gauss and Lucas): all zeros of the derivative of a polynomial are contained in the convex hull of zeros of the given polynomial. 1.1.33. Show that lim (1+ X+iy)n = eX(cosy+isiny). n .... oo n 1.1.34. Discuss the behavior of the sequence {Zn} , Zn = (1+i) (1+ ~ ) ... (1+ !). 1.1.35. Show that, if {en} and L Ibnl both converge, then the series L Cnb is convergent, too. 1.1. SETS AND SEQUENCES 7 1.1.36. Suppose rez I 0 (n = 1,2, ... ) and both L Zn' L converge. Show that also L IZnlz is convergent. 1.1.37. Prove Toeplitz's theorem: Suppose (a"k) is an infinite matrix of complex numbers (n, k = 1, 2, ... ) which satisfies: 00 (i) L la.kl ::::;; A for n = 1, 2, ... ; k=l (ii) lim ank = 0 for n = 1,2, ... ; k-+oo 00 (iii) lim (L ank) = 1. n--+oo k= 1 00 Then for any positive integer n and any convergent {en} the series L ankCk k=l is convergent. 00 Moreover, if Zn = Lank Cto then lim Zn exists and is equal lim Cn. k=l n-+oo 1138 S P1 +Pz+ ... Pn MOl." .. uppose > lor n = 1,2, ... Ip11+IPzl+ ... +IPnl and lim(!p11+lpzl+ ... +IPnD = +00. Show that for any convergent {zn} 1.1.39. Suppose Zn = UO+U1 + '" +Un-1 +cun is a convergent sequence and rec > -t. Show that also Wn = UO+U1 + ... +Un-1 +un is convergent and has I he same limit. 1.1.40. Suppose {Pn} is a sequence of positive reals monotonically 10 infinity. Show that for any convergent series L Zn with complex terms we have: 1.1.41. Show that f L ,unZn converges and ,un --+ 0 then lim,un(z1+zz+ ... +zn) = O. n-+OO 1.1.42. Suppose {un}, {vn} converge to U and v resp. Show that 1 Wn = - (U1Vn+UZVn_1 + '" +unv1) n rOllverges to 11'1). 8 1. COMPLEX NUMBERS. LINEAR TRANSFORMATIONS 1.2. SPHERICAL REPRESENTATION 1.2.1. Suppose OXlX2X3 is the system of rectangular coordinates whose axes OXl, OX2 coincide with real and imaginary axes Ox, Oy of the complex plane C. Suppose, moreover, that the ray emanating from the north pole N(O; 0; 1) of the unit sphere s: xi + + = 1 and intersecting S at A (Xl; x2; X3) intersects C at the point z. The point z = x+iy is called the stereographic projection of A(xl; X2; x3) whereas A is called the spherical image of z. Show that z+z X 1 - 1+lzI2' and z-z X2 = iO+lzI2)' Xl +iX2 l-x3 IZ12-1 X3 = 1+lz12 ' Hence the points of the sphere S (also called Riemann sphere) can be used for geometrical representation of complex numbers. 1.2.2. Find the spherical images of eilZ, -1 +i, 3-4i. 1.2.3. Describe spherical images of northern and southern hemisphere. 1.2.4. Show that any straight line in C has a circle through N as its spherical image. 1.2.5. Show that the stereo graphic projection of any circle on S not containing N is also a circle. 1.2.6. Show that the spherical images of z, Z-l are points symmetric w.r.t. the plane OXlX2. 1.2.7. Find the relation between the spherical images of following points: (i) z, - z; (ii) z, z; (iii) Z, Z-l. 1.2.8. If cp, () denote the geographical latitude and longitude of A respec-tively, show that the stereographic of A has the representation: z = ei9tan (t7t+tcp). 1.2.9. Show that z, , correspond to antipodal points of S, iff zC = -1. 1.2.10. Prove that the circle AlzI2+Bz+Bz+C = 0 (A, C real) has a great circle on S as its spherical image, iff A + C = O. 1.2.11. Show that C(zo; R) is the stereographic projection of a great circle on S, iff R2 = 1+lzoI2. 1.2.12. Find the stereographic projection of the great circle joining the points ( 3 4 . 12) ( 2 2 I) -] j , - )-3 '-]3' - -f' 3 , f . 1.2. SPHERICAL REPRESENTATION 1.2.13. The distance o"(Zl' Z2) between two points on S whose stereographic projections are Zl, Z2 is called the spherical distance between Zl and Z2' Show that o"(Zl' Z2) = 21z1-z21 [(1 + Iz112)(1+ Iz212)]-1/2. 1.2.14. Suppose dO", ds are lengths of infinitesimal arcs on Sand C resp. cor-responding to each other under stereographic projection. Suppose, moreover, the arc of length ds emanates from the point Z E C. Show that = 2(1 + IzI2)-1. Show, moreover, that the angle between any two regular arcs in C and the angle between their spherical images are equal. 1.2.1S. Suppose the sphere S is rotated by the angle rp round the diameter whose end points have a, _a-1 (cf. 1.2.9) as stereographic projections. Sup-pose, moreover, z, Care stereographic projections of points corresponding to each other under this rotation. Show that C-a z-a --- = el'P---l+aC l+az 1.2.16. Suppose At, A2 E S and at, a2 are stereographic projections of At lind A2, resp. Find the set of all points a E C such that a is the stereographic r>rojection of a point A E S equidistant from At and A2 1.2.17. Find the radius of the circle on S whose stereographic projection is C(a; r). 1.2.18. Suppose r is a regular arc on Sand 'Y is its stereographic projection. Show that the length /(r) of r is equal to ds. y 1.2.19. Find the stereo graphic projection of a rhumb line on S, i.e. of a line 011 S which cuts all meridians at the same angle. 1.2.20. Find the length /(F) of the rhumb line r joining the points whose '\crcographic projection are Zl = r1, Z2 = r2 eif%, 0 < ex < 2". Evaluate /(F) . 1 . 1 . fi lor =1 Z2 =-(3+IJl3). )13 2 1.2.21. Evaluate the area of a spherical domain D being the spherical image III' II regular domain ;1 in C. 10 1. COMPLEX NUMBERS. LINEAR TRANSFORMATIONS 1.2.22. Show that the area I TI of a spherical triangle T with angles (X, {3, y is equal a+{3+Y-7t. 1.2.23. Evaluate the area of the spherical triangle T whose vertices are (0; 2-1/2; 2-1/2), (2-1/2; 2-1/2; 0), (0; I; 0). 1.3. SIMILARITY TRANSFORMATIONS 1.3.1. Show that each similarity transformation w = az+b (a i= 0) can be composed of a translation, a rotation and a homothety with center at the origin. 1.3.2. Prove that a similarity transformation (i) carries circles into circles and (ii) parallel straight lines into parallel straight lines; (iii) leaves the ratio (Z3-Z1)!(Z3-Z2) unchanged; (iv) leaves the angle between two curves unchanged. 1.3.3. Find a similarity transformation mapping the strip {z: 0 < re z < I} onto the strip (w: limwl < +7tl so that (z = -t) +-+ (w = 0). 1.3.4. Find the most general similarity transformation mapping (i) the upper half-plane onto itself; (ii) the upper half-plane onto the lower one; (iii) the upper half-plane onto the right half-plane. 1.3.5. Find the similarity transformation mapping the segment [a, b] onto [A, B] so that a +-+ A, b +-+ B. 1.3.6. Find the similarity transformation mapping the triangle with vertices 0, 1, i onto the triangle with vertices 0, 2, 1 + i. 1.3.7. Find the similarity transformation mapping the strip {x+iy: kx+bl :(; y :(; kx+b2}, bi < b2, onto the strip {IV: 0:(; re w :(; I} so that (z = ibJl) +-+ (II' =-, 0). 1.4. LINEAR TRANSFORMATIONS 11 1.3.8. Show that for any similarity transformation w = az+b (a # 0, 1) there exists a unique invariant point Zo; show that the transformation can be composed of rotation and a homothety center at Zo. . 1.3.9. Find the invariant point zo, the angle of rotation and the ratio of homo-thety for the transformations in (i) Exercise 1.3.6; (ii) Exercise 1.3.7. 1.3.10. Show that for any similarity' transformation w = az+b (a # 0, 1) there exists a family of logarithmic spirals il}variant under the transformation. 1.4. LINEAR TRANSFORMATIONS 1.4.1. Show that any linear transformation w = (az+b)/(cz+d) (a, b, e, d are complex constants, ad-be # 0) is composed of a translation: Zl = z+/X, an inversion: Z2 = llz1 and a similarity transformation: w = Az2+B (some of these transformati R, onto {w: p < Iwl < I}. Show that p = ~ - V ( ~ r - 1 . 1.6.13. Find a linear transformation mapping the bounded domain whose boundary consists of C(O; 2), C(1; 1) onto a strip bounded by two straight lines parallel to the imaginary axis. 1.7. INVARIANT POINTS OF LINEAR TRANSFORMATIONS 1.7.1. If w = (az+b)/(ez+d), ad-be #- 0, and IX = (alX+b)/(elX+d), then IX is called an invariant point of the given linear trans/ormation. Find the general linear transformation with two different and finite invariant points IX, fJ. 1.7.2. Show that the general linear transformation with invariant points 0, ro is a similarity w = Az (A #- 0). 1.7.3. A transformation T whose inverse T-1 is identical with T is called an involution. Show that a linear transformation (az+b)/(ez+d) different from identity is an involution, iff a+{[ = O. 1.7.4. Show that an involution different from identity has always two different invariant points. 1.7.5. Prove that any linear transformation with two different invariant points can be written in the standard form: (W-IX)/(W-(J) = A(z-IX)/(z-fJ). 1.7.6. Show that if L1 = (d-a)2+4be and the sign of }/Lf is suitably chosen, then . A -= (a jd+}lIi)/(a j-d--}lt1). 1.7. INVARIANT POINTS 15 1.7.7. Bring the linear transformation w = (z+i)/(z-i) to the standard form of Exercise 1.7.5. 1.7.S. Prove that a linear transformation with only one invariant point 00 is a translation. Also prove that a linear transformation with only one finite invariant point IX (or the parabolic transformation) has the form (W-IX)-l = (z-IX)-l+h (h i= 0). 1.7.9. Find the parabolic transformation mapping ceO; R) onto itself whose only invariant point is z = R. 1.7.10. If IX, {3 are invariant points and A = IAlei9 (cf. Ex. 1.7.5) then the circle Cz with diameter [IX, {3] is carried into a circle Cw with radius R -= +IIX- {3llcosOI-1. 1.7.11. The sequence {zn} is defined by the recurrence formula: Zn+1 =/(zn), wherefis a linear transformation with at most 2 invariant points and Zo is given. Discuss the convergence of {z,,}. 1.7.12. Find the points of accumulation of the sequence {zn}: Zo = 0, 1.7.13. If A in Exercise 1.7.5 is real, the corresponding linear transformation is called hyperbolic, if A = eiq> (with real q;) it is called elliptic. Prove that in both cases there exists a family of circles such that any' circle of the family is mapped onto itself under the transformation. 1.7.14. Prove that for any parabolic transformation with an invariant point rx there exists a family of circles tangent to each other at IX and such that each d rde of the family is mapped onto itself under the given transformation. 1.7.15. Suppose IX, {3 are invariant points of a linear transformation which is not an identity and carries a circle C into itself. Show that either IX, f3 are ,itllated on C, or are symmetric w.r.t. C. 1.7.16. Suppose a linear transformation which is neither elliptic, nor hyper-holic, has two finite and different invariant points. Show that no circle can be IIlIlpped onto itself by this transformation. 1.7.17. Suppose w = (az+b)/(cz+d), ad-bc = 1 and a+d is real. Prove thai the transformation is elliptic if la+dl < 2, hyperbolic if la+dl > 2 and (lilraholic if la+dl = 2. 16 1. COMPLEX NUMBERS. LINEAR TRANSFORMATIONS 1.7.18. Show that the rotations of the Riemann sphere correspond to elliptic transformations in the plane after stereo graphic projection. 1.7.19. Suppose IX, P are invariant points of an elliptic transformation and IIX-PI 2. Prove that this transformation corresponds to a rotation of the Riemann sphere followed by a translation. 1.7.20. Show that the linear transformation w = (az+b)/(-bz+a), lal+lbl > 0, corresponds to a rotation of the Riemann sphere. Evaluate stereographic pro-jections of the end-points of the diameter being the axis of rotation, as well as the angle of rotation in terms of a and b. 1.7.21. Find the linear transformation representing the rotation of the Rie-mann sphere by an angle t'lt round the diameter with end-points (f; f; t), (-f; -f; -t) 1.7.22. Find the general linear transformation representing a rotation of the Riemann sphere by an angle 'It. Show that this is an involution. 1.7.23. Show that for any involution with two finite invariant points IX, P which is different from identity the factor A in Exercise 1.7. 5 -1. Prove that the straight line through IX, fJ is mapped onto itself. 1.7.24. Find all lines remaining invariant under the involution 2wz+i(w+z)-2 = o. 1.8. HYPERBOLIC GEOMETRY In hyperbolic (Lobachevski-Bolyai) geometry the axioms of Euclid are valid except for the parallel axiom: there are at least two different straight lines in the plane through a given point not on the straight line L which do not meet L. There is a very simple and elegant way essentially due to Poincare of satisfying the axioms of non-Euclidean geometry by a suitable choice of con-figurations in Euclidean space. The points in the hyperbolic plane or h-points are the points of the unit disk K(O; 1). The straight lines (hyperbolic straight lines, or h-lines) are the arcs of circles, or straight line segments orthogonal to the unit circle and interior to it. Hyperbolic motions are linear transformations mapping K(O; 1) onto itself. Two sets of h-points are congruent if there exists an h-motion carrying one set intoanother one. I.s. HYPERBOLIC GEOMETRY 17 We can also introduce in a natural way h-distance in the hyperbolic plane which is invariant under h-motion. Complex numbers and linear transformations are very convenient tools in analytic treatment of hyperbolic geometry. 1.8.1. Prove that there exists a unique h-line through any two h-points repre-sented by Zl' Z2 (Zl"# Zz, IZll < 1, IZzl < 1). 1.8.2. Prove that there exists a unique h-line through a given point Zl in a given direction ei!l (i.e. meeting C(O; 1) at ei!l). 1.8.3. The unit circle C(O; 1) is called the h-line at infinity. Two h-lines meeting at infinity (i.e. two circular arcs orthogonal to C(O; 1) intersecting each other at a point on C(O; I) are called h-parallels. Show that there are two h-parallels to a given h-line L through a given h-point Zl not on L, as well as infinitely many h-lines through Zl not meeting L. 1.8.4. Find the general form of an h-rotation (i.e. an h-motion with a unique invariant h-point zo). 1.8.5. Find a general h-translation, i.e. an h-motion with two invariant points 011 the h-line at infinity. 1.8.6. Find a general h-boundary rotation, i.e. an h-motion with a unique invariant point on C(O; 1). 1.8.7. Find a general h-motion. Verify the group property for h-motions. 1.8.8. Write parametric equation of an h-segment [Zl' ZZ]h' i.e. a subarc of h-Iine with end-points Zl' Z2. 1.8.9. Suppose the h-segments [a, Z]h' [b, W]h are congruent in the sense of hyperbolic geometry. Verify that l(z-a)/(I-az)1 = l(w-b)/(l-bw)l 1.8.10. Suppose C and r are two regular curves situated in the unit disk and carried into each other under an h-motion. Show that 1.8.11. Consider all regular curves situated inside K(O; 1) and joining two Ji1lcd points 0, R (0 < R < I) of K(O; 1). Show that 18 f. COMPLEX NUMBERS. LINEAR TRANSFORMATIONS has a minimum for y being the segment [0, R], the minimum being equal 1 I+R Tlog l-R = artanhR. Hint: Verify first that we can restrict ourselves to regular curves with para-metric representation 0 = O(r), where 0, r are polar coordinates. 1.S.12. If C is a regular curve situated inside K(O; 1) then C Idzi J l-lz[2 c is called hyperbolic (or h-) length of C. Show that the h-segment with end points Zl' Z2 is the curve with shortest h-Iength among all regular curves in K(O; 1) joining Zl to Z2' The h-Iength of [Zl' Z2]h is called hyperbolic (or h-) distance P(Zl' Z2) of points Zl' Z2' Also show that P(Zl' Z2) = artanhlzl-z2111-Z1Z21-1. 1.S.13. Find the h-circle with h-radius R, i.e. {z: p(zo, z) = R}. Also find its h-Iength lh' 1.S.14. Verify the usual properties of a metric for P(Zl' Z2)' 1.S.lS. Show that P(Zl, Z2) = tlog (Zl , Z2' ei/l, ei"), where ei", Zl, Z2' ei/l are successive points of a circle orthogonal to C(O; 1). 1.S.16. Suppose a regular domain D, D c K(O; 1), is carried under h-motion into [j. Prove that CC dxdy rc JJ (l-x2_y2)2 =:' JJ . D Q The integral on the left is called hyperbolic (or h-) area of D and will be denoted IDlh' -, 1.S.17. Find l[jlh for Q = {z: Izl < R}. 1.S.lS. Consider an h-triangle T, i.e. a domain bounded by three h-segments with angles rx, (:1, y. Show that ITlh = t[1t-(rx+(:1+y)]. Hint: Take the origin as one of the vertices. 1.8.19. Evaluate the h-area or an h-triunglc with ZI' =2' =.\. CHAPTER 2 Regularity Conditions. Elementary Functions A complex function w = fez) defined on a set of complex numbers A is actually defined ~ y a pair of real-valued functions u(x, y), vex, y) of two real variables x, y (x+iy = z) with a common domain A. In a formally identical manner as in real analysis we can introduce the notions of limit, continuity and differentia-bility. If fez) = u(x, y)+iv(x, y) is differentiable at zo = xo+iyo, then u, v have partial derivatives at zo satisfying Cauchy-Riemann equations at this point: Ux = vy, uy = -Vx' On the other hand, if all the four partials of first order of u, v exist in some neighborhood of zo, are continuous and satisfy Cauchy-Riemann equations at zo, then f = u+ iv is differentiable at Zo, i.e. f(zo+h) = f(zo) + ah +h'fJ (h) , where lim 'fJ(h) = 0; h .... O the constant a is called the derivative off at zo. The most interesting and most important case occurs when f is defined and has a derivative at every point of some domain (or open, connected set) D in the plane. Thenfis called analytic, Iw/omorphic or regular in D. Regularity has far-reaching consequences that go much beyond what one can obtain from differentiability in the real case. The theory of analytic functions has as its central theme just the investigation or these consequences. So, for example, regularity in a domain D implies the existence of derivatives of all orders at all points of D. Since the definitions of Ihe derivative in real and complex domain are formally identical, the usual I"lIlcs of differentiation as the formulas concerning the derivative of a sum, a prod-nct or a quotient, as well as the chain rule, remain the same in complex case. 2.1. CONTINUITY. DIFFERENTIABILITY 2.1. t. Discuss the continuity at z = 0 offunctions defined at z :1= 0 as follows: (i) / ~ ~ Z l ; (ii) z-lrez; (iii) z-2rez2; (iv) z-2(rez2)2 and equal 0 at z = O. 19 20 2. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS 2.1.2. Suppose J is defined and uniformly continuous in K(O; I). Prove that for any sequence {zn}, Zn E K(O; I), convergent to C (lCJ = I) there exists a limit p(C) depending only on C, and not on a particular choice of {zn}. Also prove that F: F(z) = J(z) for z E K(O; I), F(C) = p(C) for C E C(O; I), is continuous in K(O; 1). 2.1.3. Verify that the function J: J(O) = 0, J(z) = Izl-2(I+i)imz2 for z i= 0, satisfies Cauchy-Riemann equations at z = 0. Is J differentiable at z = O? 2.1.4. Verify that J(z) = zrez is differentiable at z = only. 2.1.5. Suppose J(z) = u(x, y)+iv(x, y) and the limit limreh-1 [f(zo+h)-J(zo)] h-+O exists. Show that the partials ux, Vy at Zo both exist and are equal. 2.1.6. Suppose u(x, y), v(x, y) are continuous and have continuous partials of first order at Zo = xo+iyo' If J = u+iv and the limit lim Ihl-1If(zo+h)-J(zo)1 h-+O exists, then either f, or 1 = u-iv has a derivative at Zo. 2.1.7. Verify that the following functions fulfill the Cauchy-Riemann equations in the whole plane: (i)J(z) = Z3; (ii)J(z) = eXcosy+iexsiny. 2.1.8. Verify that J(z) = x(x2+y2)-1_iy(x2+y2rl is analytic in C"-O. 2.1.9. If J = u+iv is analytic and satisfies u2 = v in a domain D, then J is a constant. 2.1.10. Suppose a2u a2u Llu = ax2 + ay2 . If J is analytic and does not vanish in a domain D, then 2.1.11. Prove that for an analytic function J: 2.2. HARMONIC FUNCTIONS 21 2.1.12. Write Cauchy-Riemann equations for fez) = U(r, O)+iV(r, 0), where z = rei8 Express l' in terms of partials of U, V. 2.1.13. Prove thatf(z) = z" (n is a positive integer) satisfies Cauchy-Riemann equations and f'(z) = nz,,-l. 2.2. HARMONIC FUNCTIONS A real-valued function u of two real variables x, y (resp. of one complex varia-ble z = x+ iy) defined in a domain D is said to be harmonic in D, if it has contin-uous partial derivatives of second order that satisfy in D Laplace's equation: Llu = u.u+uyy = O. Notice that continuity of partial derivatives of second order -implies continuity fix and Uy, as well as continuity of u. Two functions u, v harmonic in a domain D and satisfying Cauchy-Riemann equations in D: UX = Vy, Uy = -Vx are called conjugate harmonic functions. Any pair of conjugate harmonic functions u,v determines an analytic function u+iv. 2.2.1. Find all the functions harmonic 10 C"", (- 00, 0] which are cOllstant one the rays argz = const. 2.2.2. Find all the functions harmonic in C"", 0 which are constant on the circles ceO; r). 2.2.3. Verify that the functions u = log Izl, v = argz are conjugate harmonic functions in C"", (-00,0] and Logz = log Izl+iArgz, where Argz is the principal value of argument: -7t < Argz < 7t, is analytic in C"",(- 00, 0]. 2.2.4. Verify that are conjugate harmonic functions in C. Also verify that the analytic function expz = eXcosy+iexsiny flllfills the identity: expLogz = Logexpz = z III C "'-, ( 00 , 0] . 2.2.5. Show that ! Logz = Z_l in C"", (00, 0] and ! expz = expz in C. 2.2.6. Suppose w = fez) is analytic in a domain D and feD) II (- 00,0] = 0. Show that F(z) CO" loglf(z)l+i Argf(z) is analytic in D. Evaluate P'. 22 1. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS 2.2.7. Suppose u is harmonic in a domain D. Verify that! = u,,-iuy is analytic in D. 2.2.8. Suppose V1' V2 are harmonic and conjugate with u in a domain D. Verify that V1-V2 is a constant in D. 2.2.9. Suppose u is harmonic in {z: Izl > O} and homogeneous of degree m, m =/: 0, i.e. for any t > 0, u(tz) = tmu(z). Verify that v = m-1(yux-xUy) is a conjugate harmonic function. 2.2.10. Find a conjugate harmonic function v for u equal: (i) x2_y2+xy; (ii) x3+6x2y-3xy2_2y3; (iii) X(X2+y2)-1; (iv) (X2_y2) X X (X2+y2)-2. Evaluate in each case a corresponding analytic function u+iv as depending on z = x+iy. 2.2.11. Find a conjugate harmonic function v for (1 +x2 +y2) X u(x, y) = 1+2(x2_y2)+(x2+y2)2 . Write u+iv as depending on z = x+iy. 2.2.12. Show that a function u harmonic in a domain D has a conjugate har-monic function v in D, iff! = ux-iuy admits a primitive in D. 2.2.13. Find a conjugate harmonic function v for u(x, y) = eX(xcosy-ysinx). 2.2.14. Write Laplace's equation in polar coordinates r, O. Verify that r"cosnO, r"sinnO are harmonic for any positive integer n. 2.2.15. Discuss the existence of nonconstant harmonic functions having the form: (i) lP(xy); (ii) !p(x+f'x2+J2); (iii) tp(x2+y), where tp is a suitable, real-valued function of one real variable. Fisd a corresponding conjugate harmonic function in case it does exist. 2.2.16. Given a real-valued function F with continuous partial derivatives of second order in a domain D such that F;+F; > 0 in D. Suppose a < F(x, y) < b for x+iy ED and 1p is a real-valued, continuous function of t E (a, b) such that (F",,+Fyy)(F;+F;)-l = 1pO F. Then there exists a real-valued function tp defined in (a, b) and ~ u c h that tp 0 F is harmonic in D. Evaluate tp as depending on 1p. 2.2.17. Find an analogue of Exercise 2.2.16 in case F is given in polar coordi-nates r, O. 2.3. GEOMETRICAL INTERPRETATION 23 2.2.18. Verify the existence of functions u harmonic in C'\, (- 00, 0] and constant on confocal parabolas with foci at the origin and vertices on (0, +(0). Find all these functions. 2.2.19. Find all the functions I analytic in C '\,0 such that III has a constant value on circles x2+y2-ax = O. 2.2.20. Find all the functions I analytic in C'\, (-IX), 0] such that argl has a constant value on circles C(O; r). 2.2.21. Verify the existence of functions u(r, 0) harmonic in C '\, ( - 00, 0] having a constant value on arcs of logarithmic spirals r = ke)./J, where r, 0 are polar coordinates, A. is fixed for all the spirals and k is a parameter determining the individual arcs. 2.2.22. Find all the functions regular in C '" 0 whose absolute value is constant on lemniscates r2 = a2sin20. 2.3. GEOMETRICAL INTERPRETATION OF THE DERIVATIVE If I is analytic in a domain D, Zo ED and J'(zo) =F 0, then I(zo+h) = l(zo)+hJ'(zo)+O(h2) as h -+ O. This means that locally I is a similarity transformation composed of a rotation hy the angle argf'(zo) , a homothety with the ratio 1f'(zo)1 followed by a translation (=0)' The angle between any regular arcs intersecting at Zo and the angle between Ihe image arcs are equal, therefore the mapping realized by an analytic functionl withf'(z) =F 0 is said to be conformal. An analytic function realizing a conformal lind homeomorphic mapping of a domain D is said to be univalent in D. 2.3.1. The linear transformation w = (z+ I)/(z-I) carries the boundary of Ihe upper half-disk of K(O; 1) into two rays emanating from the origin (why?). Iii nd the angle (l( between the image ray of (-I, I) and the positive real axis liN well as the local length distortion A. at z = -I. 2.3.2. Given a linear transformation w = (az+b)/(cz+d) with c =F 0, find I he sets of all z for which (i) the length of infinitesimal segments is preserved; (ii) the direction of infinitesimal segments is preserved. 2.3.3. Suppose z = z(t) is a differentiable, complex-valued function of a real \1Il'inhle t E (a, b) such that z'(t) =F 0 and IV =/(z) is a conformal mapping 24 2. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS defined in a domain D containing all the points z(t), t E Ca, b). Show that if arg[f'(z(t))z'(t)] is constant in (a, b), then the image of the arc z = z(t), a < t< b, is a straight line segment (not necessarily bounded). 2.3.4. Show that for any linear mapping w = (az+b)/(cz+d), ad-be:F 0, c :F 0, some straight line has as its image a parallel straight line. 2.3.5. Find the sets of all z where an infinitesimal segment is expanded, contrac-ted or preserved under the given transformation: (i) w = Z2; (ii) w = z2+2z; (iii) w = Z-l. 2.3.6. Find local magnification and the angle of local rotation at Zo = -3+4 under the mapping w = Z3. -2.3.7: Show that the Jacobian of the mapping f= u+iv, where f is analytic in a domain D, is equal 11'12. Give a geometrical interpretation. 2.3.8. Verify that if f = u+iv is analytic and !'(zo) :F 0, then the lines u = const, v = const, intersect at Zo at the right angle. 2.3.9. Find the lines u = const, v = const for the mappings (i) w = Z2; (ii) w = Logz. 2.3.10. Find the length of the image arc under the univalent mappingf: D - C of the arc given by the equation z = z(t), a: t : b. Also find the area of the image domain of Q, QeD. 2.3.11. Show that under the mapping w = Z2, the image curve of C(1; 1) is the cardioid w( If!) = 2(1 + cos If!) eiq>. Find its length and the area enclosed. 2.3.12. Evaluate the integral where D is a domain situated D in {z: rez > 0, imz > O} whose boundary consists of the segment [1,2] and three arcs of hyperbolas x2 - y2 = 1, x2 - y2 = 4, xy 1. Hint: Cf. Ex. 2.3.9 (i). 2.3.13. Evaluate the length of the image arc of the segment [0, i] under the mapping w = z(1_Z)-2. 2.4. CONFORMAL MAPPINGS CONNECTED WITH w = z'-2.4.1. Evaluate the maxiq1al error in K(i; +0) if the mapping w = Z2 of this disk is replaced by its differential at z = i. 2.4.2. Find the image domain of the square: 0 < x < 1, 0 < y < 1 and the length of the boundary of the image domain under the mapping w ",c Z2. Z -=, x+ iy. 1.S. THE MAPPING w = t(z+z-l) 25 2.4.3. Find the univalent mapping of the domain {z: rez > 0, imz > O} onto K(O; 1) such that Zo = l+i corresponds to the center. 2.4.4. Find the univalent mapping of K(O; 1) onto the inside of w(O) = 2(1+cosO)eif1, 0::::;; 0::::;; 2". 2.4.5. Find the univalent mapping of the domain situated on the right-hand side branch of the hyperbola x2_y2 = a2 onto K(O; 1) which carries the focus of the hyperbola into w = 0 and the vertex into w = -1. 2.4.6. Find the univalent conformal mapping of the domain {z = x+iy: -00 < y < +00, 2px < y2}, p > 0 onto the unit disk such that the points z = -t and z = 0 correspond to w = 0 and w = 1 respectively. 2.4.7. Find the univalent conformal mapping of the domain bounded by the branch of hyperbola: x2_y2 = 1, X > 0, and the rays argz = 1=+" onto the strip {w: limzl < I}. 2.4.8. Show that the mapping z = ay2w(1+w2)-1/2 carries the unit disk {w: Iwl < I} into the domain bounded by the branches of the hyperbola x2_yZ = a2 2.4.9. Map conformally the inside of the right half of the lemniscate 11I'2-a21 = p2, 0 < p ::::;; a, onto the unit disk. 2.4.10. Map conformally the inside of lemniscate Iw2-a21 = p2, P > a, onlo the unit disk. 2.4.11. Map conformally the strip domain between the patabolas: I,l 4(x+l), y2 = 8(x+2) onto the strip {w: lim wi < I}. 2.5. THE MAPPING w = t(Z+Z-l) 2.5.1. Suppose C is an arbitrary circle through -1, 1 and Zl' Z2 do not lie tin C and satisfy ZlZ2 = 1. Show that one of these points lies inside C and an-t II hl'r one outside C. 2,5.2. Show that the mapping w = t(Z+Z-l) carries both the inside and the oiliside of any circle C through the points z = 1=1 in a 1: 1 manner onto the domain in the w-plane. Find the image domain. /lint: Show that (w-l)j(w+ 1) = [(z-l)j(z+ lW. 2.S.3. Show that the image domain of the upper half-plane under the mapping II' is C,,",-{(-oo, -1] u [1, +oo)}. 26 2. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS 2.5.4. Show that the image domain of the unit disk under the mapping W= t(Z+Z-l) is C",[-I, 1]. 2.5.5. Suppose C is a circle through z = =f 1 and r is a circle having a com-mon tangent with C at z = 1 and situated in the outside of C. Describe the image curve of r under the mapping w = t(Z+Z-l). 2.5.6. Find the image curves of: (i) circles C(O; R); (ii) rays argz = (1 under the mapping w = t(Z+Z-l). 2.5.7. Map conformally the ellipse {w: Iw-21+lw+21 < l00/7} slit along [-2,2] onto the annulus {z: 1 < Izl < R}. Evaluate R. 2.5.8. Map conformally the outside of the unit disk onto the outside of the ellipse: {w: Iw-cl+lw+cl > 2a} (c2 = a2_b2, a, b, c > 0). 2.5.9. Map conformally the domain C",{K(O; 1) u [-a, -1] u [1, b]} (a > 1, b > 1) onto the outside of the unit disk. 2.5.10. Map conformally the outside of the unit disk slit along (- 00, -1) onto: (i) w-plane slit along the negative real axis; (ii) the right half-plane. 2.5.11. Map conformally the domain whose boundary consists of three rays: (-00, -1], [1, +00), [2i, +ioo) and of the upper half of C(O; 1) onto a half-plane. 2.5.12. Map conformally the domain bounded by two confocal ellipses: {w: 2V5 < I w ~ 2 1 + l w + 2 1 < 6} onto an annulus {z: Rl < Izl < R2}. Evaluate Rl I R2 2.5.13. Map conformally the domain bounded by the right-hand branches of the hyperbolas u2coS-21X-v2sin-21X = 1, u2cos-2(J-v2sin-2(J = 1 (0 < IX < (J < +1t) onto the angle {z: IX < argz < (J}. Hint: Cf. Excercise 2.5.6 (ii). 2.5.14. Show that the image W of the point w under the mapping W = w3 - 3 w describes three times an ellipse with foci -2,2, when w describes once a COIl-focal ellipse. 2.6. THE EXPONENTIAL FUNCnON 27 2.5.15. Show that under the mapping W = w3-3w the quadrant {w: rew > 0, im w < O} is carried 1: 1 onto the complementary domain of the set {W: re W ;;;::, 0, im W ;;;::, O} u [-2, 0]. 2.5.16. Map conformally the part of the z-plane to the left of the right-hand branch of the hyperbola x2 - y2 = 1 on a half-plane. Hint: Map the upper half of the given domain by the mapping W = Z2. 2.6. THE EXPONENTIAL FUNCTION AND THE LOGARITHM 2.6.1. Use the identity: to verify that (i) le%1 = eX; expz = e% = eXcosy+iexsiny, z = x+iy, (ii) exp(z+2rci) = expz; "-(iii) exp(zl+Z2) = (expzl)(expz2)' 2.6.2. Show that for any complex w =f. 0 and any real ct the equation e% = w has exactly one solution z satisfying ct < imz < ct+2re. 2.6.3. Find the image domain of the strip -re < imz < re under the mapping II' = e%. Also find the images of segments (xo-rei, xo+rei) and of straight lines y =Yo 2.6.4. Find the image of the straight line y = mx+n under the mapping II' .. '" e% (m =f. 0). 2.6.5. Find the image domain of the strip mx-re < y < mx+re under the llIapping w = eZ 2.6.6. For which z is the exponential function (i) real; (ii) purely imaginary? Evaluate the real and imaginary parts of exp(2+i) up to 4 decimals. 2.6.7. Find the image domain of the square Ix-al < e, Iyl < e under the mapping w = ~ (a, e are real, 0 < e < re, z = x+iy). Evaluate the limit of I he ratio of the areas of both domains as e ~ O. 2.6.8. Show that the principal branch of the logarithm maps conformally "d I ; r), 0 < r < 1, onto a convex domain symmetric W.r. t. the straight lines: illlll' . ~ 0, rew = ilog(1-r2). 2.6.9. Show that the function w = Log[(z-ct)j(z-P)], where Log denotes I he principal branch of logarithm corresponding to largzl < re, is univalent in (' '.,'[ct,/J]. Find the image domain of C",[ct, 1'1] and also the images of: 28 1. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS (i) circular arcs with end-points IX, (3; (ii) Apollonius circles for the points IX, (3; (iii) the point z = 00. 2.6.10. Show that the function W = (3-oc){LOg[(Z-IX)/(Z-(3))}_l is univalent in C"'-[IX, (J]. Find the image domain. 2.6.11. Suppose f is analytic in a domain D and does not take real, non-positive values in D. Show that If(z) I = If(X)tp(y), x+iy = zED, implies: fez) = exp(az2+bz+c) where a is a real constant and b, c are complex constants. 2.6.12. Find a conformal mapping of the sphere into the w-plane such that the straight lines v = const are image lines of parallels and the straight lines u = const are the image lines of meridians (u+iv = w). Express u, v in terms of geographical coordinates 0, If on the sphere. Hint: If z is the stereographic projection of a point on the sphere, then w = fez) is anaiytic. 1.7. THE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 2.7.1. Starting from the definition of eZ (Ex. 2.6.1)-verify that are analytic in C and coincide with cosz and sinz resp. on the real axis. This defines the functions sin and cos for complex z. 2.7.2. Write cosz, sinz, tanz = s i ~ z / c o s z in the form u(x, y)+iv(x, y) where u, v are real-valued functions of real variables x, y (x+iy = z). Verify that u, v are conjugate harmonic functions. 2.7.3. Show that Isinzl2 = sin2x+sinh2y, Icoszl2 = cos2x+sinh2y. Find all zeros of sine and cosine. 2.7.4. Verify that Isinzl ~ 1 on the boundary of any square with vertices 7t(m+t)(=Fn=i), m = 0,1,2, ... 2.7.5. Verify that Icoszl ~ 1 on the boundary of any square with vertices 7tm(=Fl =j=i), m =-= 1,2, ... 2.7. TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 2.7.6. Show that Isinzl ~ coshR, Icoszl ~ coshR for any z E .R(O; R). 2.7.7. Show that for any z with limzl ~ t5 > 0 Itanzl ~ [1+ (sinh 0)-2]1/2, [cotzl ~ [I+(sinho)-2]1/2. 2.7.8. Verify the identity sin2z+cos2z = 1 for all complex z. 29 2.7.9. Verify the identity sinz = sinz and its analogues for cos, tan, cot. 2.7.10. Find all z for which sinz, cosz, tanz are (i) real; (ii) purely imaginary. 2.7.11. Evaluate cos(5 - i), sin(1- 5i) up to 4 decimals. Show that if Zo = fTC+ilog(4+ ]115) then sinzo = 4. 2.7.12. Verify for complex Zl' Z2 the addition formulas: COS(Zl+Z2) = COSZ1COsz2-sinz1 sinz2' sin(zl+z2) = sinz1cosz2+Cosz1sinz2' 2.7.13. Write coshz = t(ez+e-Z), sinhz = t(eZ-e-z)- in the form u(x, y)+ +iv(x, y), x+iy = z. 2.7.14. Express [sinhz[2, [coshz[2 as functions of x, y. 2.7.15. Find the relation between corresponding trigonometric and hyper-holic functions and give a geometric interpretation. 2.7.16. Verify the identity: (1 .) ( 'R) (1 .) ( 'R) 2 sin2IX+sinh2f3 +1 cot IX+If' + -I cot IX-If' = h2 2R . cos IX-COS f' 2.7.17. Find the image domain of the strip [rezJ < tTC under the mapping II' sinz. Find the image arcs of segments (-tTC+iyo, tTC+iyO) and of straight lincs x = xo and verify the univalence of sine in the strip considered. 2.7.18. Find the image domain of the rectangle: 0 < rez < tTC, 0 < imz < a, IInder the mapping w = sinz. Are the angles at all vertices preserved? 2.7.19. Show that cosine is a univalent function in the strip 0 < rez < fTC, t hc image domain being the right half-plane with they ray [1, + (0) removed. 2.7.20. Map 1: 1 conform ally the strip 0 < rez < fTC onto the unit disk slit IIlong a radius. 2.7.21. Map 1: 1 conformally the domain D to the left of the parabola 1'/ 4(I-x) onto the unit disk. /lint: Consider the image domain of D''-.., (- 00,0] under the mapping t = liz. 30 1. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS 1.8. INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 2.8.1. Find all solutions of the equation z = cos w in terms of the logarithm. 2.8.2. Show that w = Arccosz maps 1: 1 conformally the z-plane slit along the rays (-00, -1], [1, +(0) onto the strip 0 < rew < 7t. 2.8.3. Find Arccos+(3+i). 2.8.4. Show that w = Arctanz maps 1: 1 conformally the z-plane slit along the rays [i, +ioo), [-i, -ioo) onto the strip Irewl < t7t. Also show that .. 1 l+iz w = -2' Log-1-- 1 -IZ 2.8.5. Discuss the image arcs of Apollonius circles with limit points -I, 1 and the image arcs of circles through these points under the mapping w = Arctanz. 2.8.6. Evaluate (i) Arctan(1+2i); (ii) Arctaneif1, -t7t < (J < t7t. 2.8.7. Find the image domain of the unit disk under the mapping w = Arctanz. 2.8.8. Find the principal branch Artanh of the inverse of tanh (i.e. the branch whose restriction to the real axis coincides with the real function artanh) in terms of Log. Find the image domain of the unit disk under Artanh. 2.8.9. Find the image domain of the quadrant rez > 0, imz > 0 under w = Arsinhz. 2.8.10. Show that w = arcsinez maps the z-plane slit along the rays y = k7t, - 00 < x ~ 0 (k = 0, =f 1, =f2, ... ) onto the upper half-plane. 1.9. CONFORMAL MAPPING OF CIRCULAR WEDGES Elementary functions such as logarithm, exponential function, linear functions, power w = z" = exp(cdogz), carry some families of rays, circular arcs, or straight lines, into a similar family, e.g., Logz carries the family of rays argz = const into the family of parallel straight lines. A suitable superposition of such transformations enables us to map any circular wedge, i.e. a simply connected do-main whose boundary consists of two circular arcs (not necessarily different, either arc can be replaced by a straight line segment), onto a disk, or half-plane. In particular the mapping w = z" for Ot: suitably chosen carries an angular sector with vertex at the origin into a half-plane. On the other hand the exponential 1.9. CONFORMAL MAPPING OF CIRCULAR WEDGES 31 function transforms zero angle into a half-plane. It may even happen that a par-ticular circular triangle with two right angles can be mapped onto a circular wedge and hence onto a half-plane. 2.9.1. Map 1: 1 conformally a circular wedge in the z-plane whose sides in-tersect at a, b and make an angle IX orito a half-plane. Hint: Consider first a linear mapping carrying a, b into 0, 00 respectivel.y. 2.9.2. Map 1: 1 conformally the angle D = {z: IX < argz < {3}, 0 < {3-1X < 27t, onto the right half-plane. 2.9.3. Map 1: 1 conformally the upper half of the unit disk onto the upper half-plane. 2.9.4. Map 1: 1 conformally the circular sector: 0 < argz < t7t, 0 < Izi < 1, onto the unit disk. 2.9.5. Map the circular wedge {;: Izl < I} n {z: Iz+iv31 > 2} onto the upper half-plane. 2.9.6. Map the wedge K(-l; Y2) ("'I K(I; y2) onto K(O; 1) so that 0 +-+ O. 2.9.7. Map the strip {z: limzl < t} onto K(O; 1) so that 0 +-+ O. Find the image arcs of segments rez = Xo and straight lines imz = Yo. 2.9.8. Map the domain K(O; 1) n (C",-K(t; t)) onto K(O; 1). 2.9.9. Map the complementary domain of the set K(i; 1) n K( -i; 1) onto the outside of K(O; 1) so that 00 +-+ 00. 2.9.10. Map the circular triangle Oal a2 : K(1; 1) n [C",,-:K(I-i; y2)] n {z: Iz/(z-2)1 < 2} onto the upper half-plane. 2.9.11. Map the w-plane slit along the ray (- 00, -{-] onto the unit disk so that 0 +-+ O. 2.9.12. Map the w-plane slit along the rays (-00, -t)], [ - ~ - +00) onto the lInit disk so that 0 +-+ o. 2.9.13. Map the upper half-plane with the points of the upper half of the unit disk removed onto the upper half-plane. 2.9.14. Map the upper half-plane slit along the segment (0, ih], h > 0, onto I he upper half-plane. 32 2. REGULARITY CONDITIONS. ELEMENTARY FUNCTIONS 2.9.15. Map the upper half-plane slit along the ray [ih, +ioo), h > 0, onto the upper half-plane. 2.9.16. Show that the mapping (l_p)2 z w = p . (l_Z)2' 0 < p < 1, carries the unit disk Izi < 1 slit along the radius (-1, 0] onto the w-plane slit along the negative real axis. 2.9.17. Map the unit disk Izl < 1 onto the w-plane slit along the negative real axis so that (z = 0) - (w = 1). 2.9.18. Map the unit disk Izl < 1 slit along the radius (-1, 0] onto the unit disk It I < 1 so that the points t = 0, z = p (0 < p < 1) correspond to each other. 2.9.19. Map the domain K(O; 1) "" (-1, -p], 0 < p < 1, onto the unit disk 1 t 1 < 1 so that 0 - O. ' 2.9.20. Assume that lal < 1, Ibl < 1, a i= b, and y is the circle through a, b orthogonal to C(O; 1). Let D(a, b) be the unit disk slit along the arc of y joining b to C(O; 1), situated inside C(O; 1) and not containing a. Map conformally D(a, b) onto the unit disk It I < 1 so that (C = a) - (t = 0). 2.9.2i. Evaluate (ddC) for the mapping of Exercise 2.9.20. t t=O 2.9.22. Show that the mapping w = pz(z+p)(l +pz)-t, p > 1, carries' 1: 1 the outside of the unit disk in the z-p1ane into the outside of a circular arc on the circle C(O; p) symmetric w.r.t. the real axis. Verify that the angle subtended by the arc is equal to 4arcsinp-l. 2.9.23. Find the 1:1 conformal mapping of the domain C""K(O; 1) onto the w-plane slit along the arc: Iwl = 1; largwl ~ +7t s,o that the points at 00 correspond to each other. 2.9.24. Find the 1: 1 conformal mapping of the domain C""K(O; 1) onto the w-plane slit along the arc: Iwl = R, largwl ~ lI. so that the points at 00 cor-respond to each other. CHAPTER 3 Complex Integration 3.1. LINE INTEGRALS. THE INDEX Suppose F = u+iv is a complex-valued function of a real variable t E [a, b]; F is said to be integrable (e.g. in Riemann's sense), if both real-valued functions b b b U, v are integrable; then the integraliF(t)dt is defined as a a a Most of the properties of the real integral also hold for the integral of a complex-valued function. The lihe integrals of complex-valued functions which are a very important tool in complex analysis, can be reduced to the integrals over an interval. Suppose I' is a regular curve, i.e. a curve which is represented by the equation z = z(t), a t b, with continuous z(t) having a piecewise continuous non-vanishing derivative z'(t). If J is a complex-valued function defined for all z b on 1', then the line integral J(z)dz may be defined as G(t)dt, where G(t) 1 a ,J(z(t))z'(t) = J(z(t))[x' (t)+ iy'(t)] . The line integral does not depend on the choice of parameter; if -I' is the arc with the opposite orientation, i.e. the arc defined by the equationz = Z(-T), -b T -a, then = - V(z)dz. -]I " Ir ris defined in a domain D and has a primitive Fin D (i.e. F is analytic in D and F' = f) then for any regular arc I' contained in D with end points Zl' Z2 we have = F(Z2}-F(Zl}' In particular, for any closed curve C in D we ]I have J(z)dz = O. Conversely, if the line integral of J over any regular, closed c curve in D vanishes, then J has a primitive in D. The notion of line integral can be extended on chains r, which are linear forms n k",I'"" where I'm are regular curves and km are integers, the integral being III-I r n defined as L km . If all curves I'm are closed, the corresponding chain is m-l 1m .. tid to be a cycle. 33 34 3. COMPLEX INTEGRATION Similarly as in real analysis, we also consider unoriented line integrals. If f = u+iv is defined for all points of a regular curve 'Y and s is the arc-length on 'Y, then uds+i vds is called unoriented line integral and is denoted fds, y y y or l' 3.1.1. Evaluate the line integral re z dz for: (i) 'Y = [0, 1 + i]; (ii) 'Y = ceO; r). l' 3.1.2. Show that, for a regular curve 'Y with the equation z = z(t), a R}, then 1(1/z) has an isolated singularity at 0 and the character of singularity of/at 00 is defined to be the same as that of/(1fz) at z = O. 3.3.1. Suppose I is analytic in K(a; R)",a and lim (z-a)/(z) = O. Prove z_ that for any closed, regular curve in K(a; R)""a we have = O. ,. 3.3.2. Show that under the assumptions of Excercise 3.3.1 we have 1m = _1_. r I(z) dz, 2m J z-C C(a;r) where O O. o Hint: Show that sinO> 20/rr: for 0 E (0, -trr:). 3.7.3. Use Exercise 3.7.2 and the equality e'x = cosx+isinx to show that + 0) over a suit-able contour show that 3.7.9. By integrating fez) = eOZ(1 +ez)-l (0 < a < 1) round the rectangle with vertices =FR, =fR+27ti show that +00 --dx=-- ~ eax 7t 1 +ex sina7t . -00 Also prove that +00 ~ ta-1 7t (i) d l+t t = sina7t; o (ii) +00 . ~ X"'-l 7t --dx- (0 < m 0, prove that Hint: Integrate [(z2+a2) LogZ]-l round the annulus {z: r < Izl < R} slit along [-R, -r]. 3.8.7. By integrating log(z-a) (0 < a < 1) round the cycle consisting of ('(0; 1), C(a; r) and [a+r, 1], a+r < 1, show that 2", ~ loglei8-aldO = O. o 3.8.8. By integrating (Z2_1)-1Iogz round the boundary of {z: r < Izl < R}n(+; +), S4 3. COMPLEX INTEGRATION show that 3.8.9. Prove that 1 (i) o 1 (ii) o dx 3 X2n 1.4 .... . (3n-2) Vx(I-x2) }fJ 3.6 ... 3n 3.8.10. By integrating z1-'(I-z)'(I+z)3 round the boundary of K(O; R)". {K(O; r) u K(1; r) u [0, In, 0 < r t, show that 1 = 2P-37tp(1-p)(sinp7t)-1 (-1 I, show that the equation z + e - ~ = A has one solution with positive real part. 3.9.14. Prove that the polynomial I+z+az", n ~ 2, has for any complex a at least one root in R(O; 2). 3.9.15. Suppose 0 < lal < 1 and p is a positive integer. Show that the equation (;=-I)P = ae-Z has exactly p simple roots with positive real part and all of these arc located inside K(I; I). 3.9.16. Prove that all roots of the equation tanz-z = 0 are real and each ill tcrval n - -}) 'It, (n + ~ - ) 'It), n =1= 0, contains exactly one root All' S6 3.9.17. If laml < 1 (m = 1,2, ... , n), Ibl < 1 and n F(z) = IT z-am , in=1 l-amz 3. COMPLEX INTEGRATION show that the equation F(z) = b has exactly n roots in the unit disk. 3.9.18. If F is analytic in K(O; a) and continuous in .K(O; a), IF(z) I > m on C(O; a) and IF(O) I < m, show that F has at least one zero in K(O; a). 3.9.19. If I is analytic in the annulus A = {z: r < Izi < R}, I(z) =1= a for all z E A and Yt = I( C(O; t)), show that for any t E (r, R) the index n(y" a) has the same value. 3.9.20. If I is analytic in a domain D except for one simple pole Zo and con-tinuous in l5"",zo, I/(z) I = 1 on D",D, show that I takes in D exactly once every value a with lal > 1. CHAPTER 4 Sequences and Series of .Analytic Functions 4.1. ALMOST UNIFORM CONVERGENCE Following Saks and Zygmund [10] we shall call a sequence of functions {In} ddi'iied in an open set G almost uniformly (a.u.) convergent on G to alunction!, if {In} tends to I uniformly on each compact subset of G. In what follows w ~ use the notation: In =l GI. If all functions!.. are analytic in a domain D and!.. =l'DI, then also lis analytic in D. Moreover, for any fixed, positive integer k, JC:) =l DJCk). We can also consider a.u. convergent series of functions. The so-called M-test of Weierstrass yields a quite convenient, sufficient condition for a.u. convergence of functional series. Let L Un be a series of functions defined on an open set G. If for each compact subset F of G there exists a convergent series L Mn with positive terms such that I un(z) I ~ Mn for all z E F, then L Un is a.u. convergent on G. 4.1.1. Show that the sequence {zexp (_-}n2z2)} is uniformly convergent on the real axis and at the same time it is not a.u. convergent in any disk K(O; r). 4.1.2. Prove that the series co 00 l>n(Z) = I: zII[(1-zII)(1- zn+1)]-1 n=1 n=1 is a.u. convergent in K(O; 1) to z(1-Z)-2 and also a.u. convergent in C,,-K(O; 1) to (1-z)-2. co 4.1.3. Prove that the ~ e r i e s l: 3-nsinnz is a.u. convergent in the strip n=O lil11zl < log3, its sum I being analytic in this domain. Evaluate /,(0). co 4.1.4. If I is analytic in K(O; 1) and 1(0) = 0, show that the series l:/(z") is !l.U. convergent in K(O; 1) and its sum is analytic. n=1 57 58 4. SEQUENCES AND SERIES 4.1.5. Prove that in any closed disk K(zo; r) leaving outside all negative in-00 tegers the series L (_1)n+1(z+n)-1 is uniformly convergent and its sum is ana-n=1 lytic in C""{-l; -2; -3; ... }. 4.1.6. Suppose y is a closed, regular curve not meeting any negative integer and I is the analytic function of Exercise 4.1.5. Evaluate ~ I(z) dz. l' 4.1.7. If Izl < 1 and .,;(n) is the number of divisors of the integer n, show that 00 00 I zn(1-z")-1 = L .,;(n)z". n=1 n=1 Prove the a.u. convergence of both series in K(O; 1). 4.1.8. Prove the identity 1+ (ktl)z+ (kt2)Z2+ ... = (l-z)-k-l, k = 0, 1, 2, ... , Izl < 1. 4.1.9. Find the set of points of convergence of the functional series L unCz) in case un(z) equals to: (i) zn(1 +z2n)-1; (ii) n-2 cosnz; (iii) (z+n)-2; (iv) (q"z+q-nz-1-2)-1 (0 < q < 1). 4.1.10 .. Prove that the series 00 1+ ,-" z2(z2+12) ... (Z2+n2) ~ [(n+1)!j2 .n=O S convergent for any z, its sum being analytic in the finite plane. 4.1.11. By using the identity n=-oo 00 z 1= 0, =F1, =f2, ... (cf. e.g. Ex. 4.5.15) find the sum L (z-n)-3. "=-00 4.1.12. If the sequence of functions {In} analytic and univalent in a domain G is a.u. convergent in G and 1= lim/n, show that I is univalent, unless it is a constant. 4.2: POWER SERIES 00 A series of functions of the form L an(z-a)n is called a power series. The n=O least upper bound R of nonnegative r for which the sequence {Ianl rn} is bounded 4.2. POWER SERIES 59 is called the radius of convergence of the given power series and the disk K(a; R) is called the disk of convergence. In case R -> 0 the power series is a.u. convergent in its disk of convergence, its sum being analytic in this disk. Outside its disk of convergence, i.e. in R), the power series is divergent. The radius of convergence can be evaluated by means of the Cauchy-Hadamard formula: R = (lim VianO-I. 4.2.1. If an:l= 0 (n = 1, 2, ... ) and the limit lim an/an+! = q exists, show 00 that the radius of convergence of the power series 2: anz" is equal to Iql. n=O 4.2.2. Evaluate the radius of convergence of: 00 (2n)1 (1) L (nl)2 z"; n=O 00 I . (ii) L z"; n=1 00 00 (iii) 2: 2-nz2n; n=O (iV) L (n+an)z". n=O 4.2.3. If the radii of convergence of L anzn, L bnzn are equal R1, R2 resp., show that: (i) the radius of convergence R of L anbnz" satisfies R RI R2; (ii) the radius of convergence R' of ," ban z" (bn :1= 0, n = 0, 1, 2, ... ) L n satisfies R' R1/R2; (iii) the radius of cohvergence Ro of 2: (anbO+an_l bl+ ... -t-aobn)z" satisfies Ro min(RI' R2). 4.2.4. If the sum of the power series L an z" is real in some interval (- " > 0, show that all coefficients an are real. 00 4.2.5. Suppose R > 0 is the radius of convergence of the power series 2: anz" ILnd fez) is its sum. Prove ParsevaI's identity: n=O 21< 00 If(reifl)l2dO = L lanl2r2n, 0 < r < R. o n=O Hint: 1/12 =fJ. 60 01. SEQUENCES AND SERIES 4.2.6. (cont.) If 1 is bounded: I/(z) I ~ M for all z E K(O; R), show that 00 L lanl2R2n ~ M2. n=O 4.2.7. (cont.) If r < R and M(r) = sup I/(reif1) I , 0 ~ () ~ 27t, show that lanl ~ ,-nM(r), n = 0, 1, ... 4.2.8. If the sum of a power series L anz" is defined and bounded in the unit disk, show that an = 0(1). 00 00 4.2.9. If I(z) = L anz' in K(O; r) .and lall ~ L nlanlrn-l, show that 1 is n=O n=2 univalent in K(O; r) (i.e. Zl' Z2 E K(O; r), Zl =/: Z2 implies I(Zl) =/: I(Z2). 00 4.2.10. By using Exercise 4.2.9 find a disk of univalence of )-, z"1 . "--J n n=O 4.2.11. If L cnz" has a positive radius of convergence R, show that the power series L ~ i z" has an infinite radius of convergence and the sum 1 of the latter series satisfies I/(z)1 ~ M(()exp(lzll()R), w ~ e r e 0 < () < 1. 00 4.2.12. If I(z) = L anr' is univalent in K(O; 1) and A(r) is the area of the n=O image domain of K(O; r), 0 < r < 1, show that 4.3. TAYLOR SERIES If 1 is analytic in a domain D and a is an arbitrary point of D, there exists a sequence of complex numbers {an} such that for any K(a; r) c::: D the p o ~ e r 00 series L an(z-a)n is a.u. convergent in K(a; r) its sum being equal/(z). The power n=O 00 series L an(z-a)n is called Taylor series 011 center at a and the Taylor's coeffi-n=O cients an can be expressed either by successive derivatives of 1 at the point a: I(n)(a) ao=/(a), an= I (n=I,2, ... ), n. 4.3. TAYLOR SERIES 61 or by Cauchy's coefficient formula: an = dC (n = 0,1,2, ... ) C(a:r) where r is such that [((a; r) c D. The radius of convergence R of Taylor series center a is not less than the distance between c",-.n and the point a. On the other hand, the circle of con-00 vergence C(a; R) of the power series I an(z-a)n contains at least one singular n=O point. Each point on C(a; R) which is not regular, is called singular. A point . 00 bE CCa; R) is called regular if there exists a power series L bn(z-b)n conver-n=O 00 00 gent in some disk K(b; > 0, such that both series L an(z-a)n, L bn(z-b)n n=O n=O have identical sums in K(a; R) n K(b; 4.3.1. Evaluate four initial, non-vanishing coefficients of Taylor series with center at the origin for the following functions and find the corresponding radius of convergence R: (i) z Log(1+z) (ii) z Arctanz' (iii) ]I cosz (take the branch corresponding to the value 1 at the origin); (.) 1 IV cosz' (v) Log (1 +e); (vi) expe. 4.3.2. By applying Weierstrass theorem on term by term differentiation of II.U. convergent series of analytic functions prove the following theorem: 00 Suppose L unCz) is a.u. convergent in K(O; R) and fez) is its sum; suppose, lIloreover, ulI(z) =, anO+alllz+ ... +anlczk+ ... in K(O; R) for n = 0,1,2, ... 62 4. SEQUENCES AND SERIES co Then all the series Lank are convergent and their sums are equal to correspond-n=O ing Taylor coeffiCients of f at the origin. 4.3.3. Evaluate four initial, non-vanishing Taylor coefficients at the ongm for the following functions and find the corresponding radius of convergence R:--(.) z (")' Z 1 exp -1--; II sm -1--' -z -z 4.3.4. Evaluate nth coefficient of Taylor series at the origin and its radius of convergence for following functions: . 1 ( 1)2 (1) 2" Log l-z ; (ii) (Arctanz)2; (iii) (Arctanz) Log(l+z2): (iv) cos2z; (v) Log(1+Yl+z2). Hint: Differentiate the given 'function; (vi) Log( Yl+z+ Yl-z). [Take in (v), (vi) these branches of the square root which are equal 1 at the origin.] . 4.3.5. Find the radius of convergence for Taylor series with center at the origin for sin 7tZ2 jsin 7tZ . 4.3.6. Find Taylor series with center at the origin for these branches of (i) (1+ J/l+Z)1/2; (ii) (1+ yl+Z)-1/2 which are equal 21/2, 2-1/2 at the origin. Hint: Cf. Exercise 1.1.5. 4.3.7. Show that the coefficients Cn of Taylor series of (1- z-Z2)-1 with center at the origin satisfy: Co = C1 = 1, cn+2 = cn+1 +cn (n ~ 0). Find the explicit formula for Cn by representing the given function as a sum of partial fractions. co Also find the radius of convergence of L cnz". The sequence {cn} is so called n=O Fibonacci sequence. 4.3.8. Prove that fez) = ,_1_ - ~ + ~ has a removable singularity at the eZ-l z 2 origin and is an odd function. Hence 4.3. TAYLOR SERIES co _1 __ ~ ~ = ~ (_I)k-1 ~ 2k-1 e%-1 z+2 L..J (2k)!z. k=1 63 Evaluate B1, B2, ... , Bs. Find the radius of convergence of the power series on the right and show that lim V IBn I = + 00. The constants Bk are called Ber-noulli numbers. 4.3.9. Show that co (1') 1 hi - 1 + ~ ( l)k-1 Bk 2k-1 Tzcot TZ - L..J - (2k)! z , k= 1 co 22k (1'1') 1 "'-, Bk 2k I I zcotz = - ~ (2k)! z , z < 7t. k=1 4.3.10. Express by means of Bernoulli numbers the nth Taylor coefficient at the origin and the radius of convergence of the corresponding Taylor series for the following functions: (.) L sinz 1 og--: z (ii) Logcosz; (iii) tanz; (iv) (COSZ)-2; (v) tan2z; ( .) L tanz VI og--; z ( .. ) Z VB -.-. smz 4.3.11. By using the power series expansions of Log(1 + z), sinz near the origin evaluate three initial, non-vanishing terms of the power series expansion of Log sinz and compare the obtained result with Exercise 4.3.10 (i). ;> 4.3.12. Verify that q;(z) = -. / z Arctan'" / z V l-z V l-z is analytic in some neighborhood of z = O. Evaluate the radius of convergence of its Taylor series at the origin. 64 4. SEQUENCES AND SERIES 4.3.13. (cont.) Verify that rp satisfies the differential equation 2z(l-z)rp'(z) = rp(z)+z with the initial condition rp(O) = o. Prove the identity: _ 2 3 24 5 246 7 rp(z) - z+3z +TIz +T5.'7z + ... , Izl < 1. 4.3.14. Prove that . 2 _ 2 1 2 4 1 2.4 6 (Arcsmz) - z +"23z +3TIz + ... , Izl < 1. 4.3.15. If f is analytic in K(O; R) and Rn is the remainder of Taylor series center at the origin, i.e. z" Rn(z) = f(z)- f(O)-z!, (0)- ... - -, j b. 4.5.6. Find the Laurent development of (1 + Z2)-1 (2+ Z2)-1 for (i) 1 < Izl < V2; (ii) Izl > V2. 4.5.7. Find the Laurent expansion of Log [Z2 j(z2-l)] for Izl > 1. 68 4. SEQUENCES AND SERIES 4.5.8. Express the Laurent coefficients of exp(z+z-l) at the origin in terms of (i) integrals involving trigonometric functions; (ii) sums of infinite series by using the identity exp(z+z-l) = eZel/z 4.5.9. The Bessel function In(z) is defined as the nth coefficient of Laurent expansion at the origin (n 0): . (-1)" __ 1t_ v2 v2 v2 v2 ..:.......J. n4+a4 - a3y2 . . 2 1ta + .nh2 1ta "=-00 sm y2 SI y2 4.6. SUMMATION OF SERIES 71 00 (iv) L (-1)"n2 7t n4+a4 = ay2 . 7ta. tta . tta 7ta cos 172 smh l72 sm l72 cosh y2 2 tta . h2 7ta sm y'2+sm y2 n=-oo =f1 =fi a #- 2 n (n = 0, 1, 2, ... ) in (iii) and (iv). 4 6 9 B . 7t sin az y mtegratmg 3. , Z sm7tZ where -7t < a < 7t, round the boundary i)QN of the square QN with corners (N+}) (=f1=fi), verify the formulas: 4.6.10. If x is a complex number different from an integer, -7t < a < 7t and QN is the square with corners (N +-t) (=f 1 =fi), show that I = r cos az dz -+ 0 as N -+ + 00 . N J (X2_Z2) sin 7tZ iJQN Verify the formula: 00 7tcosax = 2x (-1)" . sin7tx x x -n 11=1 4.6.11. If a#-O is real and RN are rectangles with corners (N +-t ) (=f 1 , show that the integrals a IN = sin 7tZ 7taz dz -+ 0 iJRN By using this show that 00 00 as N -+ +00. (1) L (-l)"n _- - .1 ___ 1 L (-l)"n - for real a#-O; sinh 7tan 27ta a2 sinh( 7tn/a) "=1 "=1 1 (.1.1) 1 2 3 eTt_e-Tt - e27t _e- 27t + e37t -e 37t = s;-. 4.6.12. Verify that the equality of Exercise 4.6.11 (i) also holds for complex " outside the imaginary axis. 72 4. SEQUENCES AND SERIES 4.6.13. If x is complex and not an integer, -7t < a < 7t, and QN is the square with comers (N +i-)(=f 1 =fi), show that the integrals zsinaz d 0 I - N - (x2-z2)sin7tz ilQN as +00 and verify the formula co 4.6.14. Show that co = (_I)n nsman . 2 sm 7tX L x2-n2 ,,=1 . 7t _I(-I)n(2n+l) 1 3 5 (1) --- ( + 1 )2 2' Z 1= =fT' =fT' =fT' ... , COS7tZ n T -Z n=O co (oo) 7t I (-I)n(2n+l) l' 3' 5' 11 h = 2+ ( 1)2' Z 1= =fTI, =fTI, =fTI, ... cos 7tZ Z n+T n=O 4.6.15. By integrating fez) = 7tZ-7cot7tzcoth 7tZ round a suitable contour show that coth7t coth27t coth37t 197t7 37 + oo. = 56700 4.7. INTEGRALS CONTAINING A COMPLEX PARAMETER THE GAMMA FUNCTION Suppose W(z, C) is a complex-valued function of two complex variables z, C defined and continuous on Gx {r}, where G is a domain and {r} is the set of points of a regular curve r. If W;(?, C) exists and is continuous on G X {r}, then H(z) = W(z, C)dC is analytic in G; moreover, H'(z) = W;(z, OdC. r r In particular r can be a segment [a, b] of the real axis. If the limit +co lim W(z, t)dt exists, it will denoted W(z, t)dt. b->co [a,b] a +co 4.7.1. The integral W(z, t)dt is said to be almost uniformly (a.u.) con-a vergent in a domain G, if for any compact subset F c any B > 0 there exists A such that for any b, B with A < b < B we have 4.7. THE GAMMA FUNCTION 73 B W(Z, t)dt! < e for all z in F. b +00 Ifboth W(z, t), W;(z, t) are continuous on Gx [a, +(0) and H(z) = i W(z, t)dt a is a.u. convergent in G, show that H is analytic in G. Hint: Use Weierstrass theorem on a.u. convergent series of analytic functions. +00 4.7.2. Prove that e-Iltdt is a.u. convergent m the right half-plane o + 00 1 {z:rez>O}and e-ztdt--=Ofor all z with rez>O. By separating o z real and imaginary parts in both terms find the values of two real integrals. +00 4.7.3. Prove that H(z) = e-'t,,-ldt is analytic in the open plane and 1 +00 H'(O) = t-1e-:,}dt. 1 4.7.4. If is a complex-valued function of a real variable x E (- 00, + (0) +00 and l < +00, show that -00 +00 J(z) = -00 is analytic in the upper half-plane. 4.7.5. The gamma function as .defined by the equality +00 r(z)= (Z-l e-'dt, rez>O, o ill analytic in the right half-plane and satisfies zr(z) = r(z + 1). Show that this functional equation defines a function meromorphic in C whose only singulari-lies are simple poles at z=O,-I,-2, ... with res(-n;r(z) = (-I)R/n! 4.7.6. Prove that a function G meromorphic in the finite plane C satisfies Ihe functional equation G(z+l) = zG(z), iff G(z) = r(z)p(z), where P is mero-IIwrphic in C and has the period 1. 4.7.7. Prove that +00 exp( -x")4x = oc-1r(oc-1), oc > o. o 4.7.8. By integrating e-"zS-l round the boundary of R) = {z: < Izl < R, 0 < argz < n/2}, 74 4. SEQUENCES AND SERIES show that +00 ~ y,-le-iYdy = F(s)exp(-i-7tis), 0 < res < 1. o 4.7.9. If 0 < oc < 1, show that +00 (i) ~ x-a.cos x dx = r(1-oc)sini-oc7t; o +00 (ii) ~ x-a.sinxdx = r(l-oc)cos-}OC7t. o Also verify the left-hand side continuity of the integral (ii) at oc = 1. 4.7.10. Verify that "'/2 ~ 1 r(i-p)F(i-q) sinP-10cosq-10dO = -2 . - - - - . o ~ _ ; _ ~ ~ o r(i-(p+ q)) 4.7.11. Show that "'/2 (i) ~ (tanO)"'dO = i-7t(cos-}7toc)-l, -1 < oc < 1; o ' "'/2 (ii) ~ ysinOdO = (27t)3/2[F(i-)r2. o Hint: Use the formula F(z)F(I-z) = 7t/sin7tz. 4.7.12. Express the following elliptic integrals by means of gamma function: "'/2 (i) ~ (l-tsin20)-1/2dO; o "'/0 (ii) ~ (1-i-sin20)1/2dO. o Hint: Put sinO = V 1- Y u. 4.7.13. Prove that r{! )F{!} ... r( n:l ) = n-1/2(27t)(n-1)/2. n-1 ( 2k .) Hint: Show that n = IT 1-exp ~ and k=l n hence deduce the formula n-1 k I1 sin ~ = n . 21-n k=1 n 4.8. NORMAL FAMILIES 75 4.7.14. Prove that 1 logr(t)dt = tlog27t. o 4.7.15. Evaluate the integral a+ 1 /(a)= logr(t)dt, a>O. a 4.8. NORMAL FAMILIES A family ff of functions I analytic in a domain D is said to be normal, if every sequence {J,,} of functions In E ff contains a subsequence {Ink} which either converges a.u. in D, or diverges to 00 a.u. in D. The limit function I = lim Ink , is analytic, unless it reduces to the 00. If any sequence {[,.} of functions fn E ff contains an a.u. convergent subsequence, then ff is said to be compact. A necessary and sufficient condition for compactness of the family ff is the existence of a common, finite upper bound of the absolute values of all IE ff on each compact subset of D (Stieltjes-Osgood theorem, also called Montel's compactness condition). Clearly each compact family is normal. The real-valued !'unction p(z,f) = 2 If'(z) 1 (1+I/(zW)-l is called the spherical derivative of I and has an obvious geometrical meaning (cf. Ex. 4.8.1). Now, a family ff of functions I analytic in a domain D is normal, if there exists in every compact of D a common finite upper bound for the spherical rlerivative of all !'unctions of the family. This criterion is due to F. Marty. Another sufficient condition for normality is due to Monte! and its proof is based on the properties of the modular function: if all functions I of a family :iF are analytic in a domain D and every IE ff does not take in D two fixed, lillite values ct, {3 (ct =f. {3), then I is normal. The concept of normal family is very important in the existence questions for solutions of extremal problems. 4.8.1. Explain the geometrical meaning of the spherical derivative. 4.8.2. Verify that the family of all similarity transformations az+b is not II normal family in the finite plane. 4.8.3. If ff is normal in a domain D and there exists a point Zo E D and a real I"llllstant Mo < + 00 such that I/(zo)1 Mo for all I E ff, show that ff is a com-plld family. .. 76 4. SEQUENCES AND SERIES 4.8.4. If ff is a compact family of functions analytic in a domain D, show that also ffl = {g: g = 1', IE ff} is compact. By considering the sequence {n(z2 _n2)} verify that the derivatives of functions of a normal family not neces-sarily form a normal family. 4.8.5. Suppose F(w) is an entire function (i.e. a function analytic in the finite plane C) and ff is a compact family of functions analytic in a domain D. Verify that the functions F 0 I, IE ff also form a compact family in D. 4.8.6. Suppose ff is the family of all functions I(z) = az, where a is a complex constant and F(w) = eWsinw. Verify that ff is normal in C""K(O; 1), whereas the functions Fol do not form a normal family. 4.8.7. Prove Hurwitz's theorem: If {In} is an a.u. convergent sequence in a domain D and/n(z) 1= 0 for all ZED and all In, then the limiting function I is either identically 0 in D, or does not vanish in D. Hint: Verify that p(z,J) == p(z, Iff). Also prove that the terms of an a. u. sequence form a normal family. 4.8.8. Suppose {In} is an a.u. convergent sequence of univalent functions in a domain D and g = limfn. Show that either g is univalent in D, or is a constant. Hint: Consider I(z)-/(a) in D ~ a . 4.8.9. Show that the family of all functions I analytic in the unit disk and such that 1(0) = 0,1'(0) = 1, is not a normal family. 4.8.10. Let To be the family of all functions I analytic in the unit disk and such that 1(0) = 0,1'(0) = 1, I(z) 1= 0 for z 1= o. Show that To is compact. Hint: Consider log(f(z)/z). 4.8.11. (cont.) Show that there exists a constant ct > 0 such that any I E To takes in the unit disk any value Wo E K(O; ct). Hint: If In does not take the value ctn and ctn ~ 0, consider the sequence gn(z) = log (1-/n(z)/ctn). 4.8.12. Show that the family So of ail functions analytic and univalent in the domain D omitting one fixed value ct is normal. 4.8.13. Suppose G(M) is the family of functions analytic in a domain D and such that ~ ~ If(z)12dxdy ~ M. Show that G(M) is compact in D. D Hint: If K(zo; R) cD, verify that 2", If(zoW ~ 217': ~ If(zo+rei6WdO, 0 < r ~ R, o and deduce that 7':R21/(zo)12 ~ M. CHAPTER 5 Meromorphic and Entire Functions 5.1. MITTAG-LEFFLER'S THEOREM Let {a"} be an arbitrary sequence of complex numbers such that ao = 0 < lall < la21 < ... and lima" = 00 and let {G"(w)} be an arbitrary sequence of polynomials with vanishing constant terms. There exists a meromorphic function F which is analytic in the finite plane except for the poles ao, a1' a2, ... and has G"(I/(z-a")) as singular parts at a". 00 Let L Un be an arbitrary, convergent series with