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CONTENTS

PREFACE................................... III
PREFACE TO THE ENGLISH EDITION................................... VII

INTRODUCTION. THEORY OF SETS
§ 1. Fundamental definitions................................... 1
§ 2. Denumerable sets................................... 3
§ 3. Abstract topological space................................... 4
§ 4. Closed and open sets................................... 6
§ 5. Connected sets................................... 11
§ 6. Compact sets................................... 13
§ 7. Continuous transformations................................... 14
§ 8. The plane................................... 17
§ 9. Connected sets in the plane................................... 25
§ 10. Square nets in the plane................................... 32
§ 11. Real and complex functions................................... 36
§ 12. Curves................................... 38
§ 13. Cartesian product of sets................................... 40

CHAPTER I. FUNCTIONS OF A COMPLEX VARIABLE
§ 1. Continuous functions................................... 44
§ 2. Uniformly and almost uniformly convergent sequences................................... 46
§ 3. Normal families of functions................................... 49
§ 4. Equi-continuous functions................................... 53
§ 5. The total differential................................... 55
§ 6. The derivative in the complex domain. Cauchy-Riemann equations................................... 57
§ 7. The exponential function................................... 60
§ 8. Trigonometric functions................................... 62
§ 9. Argument................................... 68
§ 10. Logarithm................................... 72
§ 11. Branches of the logarithm, argument and power................................... 74
§ 12. Angle between half-lines................................... 77
§ 13. Tangent to a curve................................... 79
§ 14. Homographic transformations................................... 80
§ 15. Similarity transformations................................... 87
§ 16. Regular curves................................... 91
§ 17. Curvilinear integrals................................... 92
§ 18. Examples................................. 95

CHAPTER II. HOLOMORPHIC FUNCTIONS
§ 1. The derivative in the complex domain................................... 98
§ 2. Primitive function................................... 100
§ 3. Differentiation of an integral with respect to a complex variable................................... 107
§ 4. Cauchy's theorem for a rectangle................................... 112
§ 5. Cauchy's formula for a system of rectangles................................... 112
§ 6. Almost uniformly convergent sequences of holomorphic functions................................... 116
§ 7. Theorem of Stieltjes-Osgood................................... 119
§ 8. Morera's theorem.................................... 120

CHAPTER III. MEROMORPHIC FUNCTIONS
§ 1. Power series in the circle of convergence................................... 125
§ 2. Abel's theorem................................... 128
§ 3. Expansion of Log(1 - z)................................... 134
§ 4. Laurent's series. Annulus of convergence................................... 137
§ 5. Laurent expansion in an annular neighbourhood................................... 140
§ 6. Isolated singular points................................... 143
§ 7. Regular, meromorphic, and rational functions................................... 145
§ 8. Roots of a meromorphic function................................... 150
§ 9. The logarithmic derivative................................... 153
§ 10. Rouché's theorem................................... 155
§ 11. Hurwitz's theorem................................... 158
§ 12. Mappings defined by meromorphic functions................................... 161
§ 13. Holomorphic functions of two variables................................... 165
§ 14. Weierstrass's preparation theorem................................... 167

CHAPTER IV. ELEMENTARY GEOMETRICAL METHODS OF THE THEORY OF FUNCTIONS
§ 1. Translation of poles................................... 171
§ 2. Runge's theorem. Cauchy's theorem for a simply connected region................................... 176
§ 3. Branch of the logarithm................................... 179
§ 4. Jensen's formula................................... 181
§ 5. Increments of the logarithm and argument along a curve................................... 183
§ 6. Index of a point with respect to a curve................................... 186
§ 7. Theorem on residues................................... 189
§ 8. The method of residues in the evaluation of definite integrals................................... 194
§ 9. Cauchy's theorem and formula for an annulus................................... 196
§ 10. Analytical definition of a simply connected region................................... 204
§ 11. Jordan's theorem for a closed polygon................................... 206
§ 12. Analytical definition of the degree of connectivity of a region................................... 209

CHAPTER V. CONFORMAL TRANSFORMATIONS
§ 1. Definition................................... 214
§ 2. Homographic transformations................................... 216
§ 3. Symmetry with respect to a circumference................................... 217
§ 4. Blaschke's factors................................... 220
§ 5. Schwarz's lemma................................... 222
§ 6. Riemann's theorem................................... 225
§ 7. Radó's theorem................................... 231
§ 8. The Schwarz-Christoffel formulae................................... 233

CHAPTER VI. ANALYTIC FUNCTION
§ 1. Introductory remarks................................... 238
§ 2. Analytic element................................... 239
§ 3. Analytic continuation along a curve................................... 246
§ 4. Analytic functions................................... 247
§ 5. Inverse of an analytic function................................... 254
§ 6. Analytic functions arbitrarily continuable in a region................................... 255
§ 7. Theorem of Poincaré-Volterra................................... 258
§ 8. An analytic function as an abstract space................................... 259
§ 9. Analytic functions in an annular neighbourhood of a point................................... 261
§ 10. Analytic functions in an annular neighbourhood as an abstract space................................... 264
§ 11. Critical points................................... 265
§ 12. Algebraic critical points................................... 267
§ 13. Auxiliary theorems of algebra................................... 268
§ 14. Functions with algebraic critical points................................... 271
§ 15. Algebraic functions................................... 275
§ 16. Riemann surfaces................................... 277

CHAPTER VII. ENTIRE FUNCTIONS AND FUNCTIONS MEROMORPHIC IN THE ENTIRE OPEN PLANE
§ 1. Infinite products................................... 286
§ 2. Weierstrass's theorem on the decomposition of entire functions into products................................... 295
§ 3. Mittag-Leffler's theorem on the decomposition of meromorphic functions into simple fractions................................... 301
§ 4. Cauchy's method of decomposing meromorphic functions into simple fractions................................... 305
§ 5. Examples of expansions of entire and meromorphic functions................................... 309
§ 6. Order of an entire function................................... 319
§ 7. Dependence of the order of an entire function on the coefficients of its Taylor series expansion................................... 324
§ 8. The exponent of convergence of the roots of an entire function................................... 327
§ 9. Canonical product................................... 329
§ 10. Hadamard's theorem................................... 332
§ 11. Borel's theorem on the roots of entire functions................................... 338
§ 12. The small theorem of Picard................................... 341
§ 13. Schottky's theorem. Montel's theorem. Picard's great theorem................................... 346
§ 14. Landau's theorem................................... 354

CHAPTER VIII. ELLIPTIC FUNCTIONS
§ 1. General remarks about periodic functions................................... 356
§ 2. Expansion of a periodic function in a Fourier series................................... 360
§ 3. General theorems on elliptic functions................................... 363
§ 4. The function p(z)................................... 368
§ 5. Differential equation of the function p(z)................................... 371
§ 6. The function ζ(z) and σ(z)................................... 375
§ 7. Construction of elliptic functions by means of the function σ(z)................................... 378
§ 8. Expression of elliptic functions in terms of the functions ζ(z) and σ(z)................................... 380
§ 9. Algebraic addition theorem for the function p(z)................................... 384
§ 10. Algebraic relations between elliptic functions................................... 386
§ 11. The modular function J(τ)................................... 387
§ 12. Further properties of the function J(τ)................................... 392
§ 13.Solution of the system of equations $g_2(ω,ω')=a$, $g_3(ω,ω')=b$................................... 403
§ 14. Elliptic integrals................................... 404

CHAPTER IX. THE FUNCTIONS Γ(s) AND ζ(s) DIRICHLET SERIES
§ 1. The function Γ(s)................................... 411
§ 2. The function B(p,q)................................... 416
§ 3. Hankel's formulae for the function Γ(s)................................... 418
§ 4. Stirling's formula................................... 420
§ 5. The function ζ(s) of Riemann................................... 424
§ 6. Functional equation of the function ζ(s)................................... 428
§ 7. Roots of the function ζ(s)................................... 429
§ 8. Dirichlet series................................... 432

INDEX................................... 441
ERRATA................................... 446