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图书目录
PART ONE BASIC THEORY 3
Chapter Ⅰ Complex Numbers and Functions 3
1 Definition 3
2 Polar form 8
3 Complex valued functions 12
4 Limits and compact sets 17
5 Complex differentiability 27
6 The Cauchy-Riemann equations 31
Chapter Ⅱ Power Series 35
1 Formal power series 35
2 Convergent power series 45
3 Relations between formal and convergent series 57
Sums and products 57
Quotients 60
Composition of series 62
4 Holomorphic functions 64
5 The inverse and open mapping theorems 67
6 The local maximum modulus principle 73
7 Differentiation of power series 75
Chapter Ⅲ Cauchy’s Theorem, First Part 81
1 Analytic functions on connected sets 81
2 Integrals over paths 88
3 Local primitive for an analytic function 96
4 Another description of the integral along a path 102
5 The homotopy form of Cauchy’s theorem 106
6 Existence of global primitives.Definition of the logarithm 108
Chapter Ⅳ Cauchy’s Theorem, Second Part 113
1 The winding number 113
2 Statement of Cauchy’s theorem 117
3 Artin’s proof 125
Chapter Ⅴ Applications of Cauchy’s Integral Formula 133
1 Cauchy’s integral formula on a disc 133
2 Laurent series 139
3 Isolated singularities 143
4 Dixon’s proof of Cauchy’s theorem 148
Chapter Ⅵ Calculus of Residues 151
1 The residue formula 151
2 Evaluation of definite integrals 167
Fourier transforms, 169
Trigonometric integrals 172
Mellin transforms 174
Chapter Ⅶ Conformal Mappings 184
1 Schwarz lemma 184
2 Analytic automorphisms of the disc 185
3 The upper half plane 189
4 Other examples 190
Chapter Ⅷ Harmonic Functions 197
1 Definition 197
2 Examples 205
3 Construction of harmonic functions 212
4 Existence of associated analytic function 216
PART TWO VARIOUS ANALYTIC TOPICS 221
Chapter Ⅸ Applications of the Maximum Modulus Principle 221
1 The effect of zeros, Jensen-Schwarz lemma 221
2 The effect of small derivatives 226
3 Entire functions with rational values 228
4 Phragmen-Lindelof and Hadamard theorems 234
5 Bounds by the real part, Borel-Caratheodory theorem 238
Chapter Ⅹ Entire and Meromorphic Functions 241
1 Infinite products 241
2 Weierstrass products 244
3 Functions of finite order 250
4 Meromorphic functions, Mittag-Leffler theorem 252
Chapter Ⅺ Elliptic Functions 255
1 The Liouville theorems 255
2 The Weierstrass function 258
3 The addition theorem 262
4 The sigma and zeta functions 265
Chapter Ⅻ Differentiating Under an Integral 270
1 The differentiation lemma 270
2 The gamma function 273
Proof of Stirling’s formula 277
Chapter ⅩⅢ Analytic Continuation 287
1 Schwarz reflection 287
2 Continuation along a path 292
Chapter ⅩⅣ The Riemann Mapping Theorem 299
1 Statement and application to Picard’s theorem 299
2 Compact sets in function spaces 303
3 Proof of the Riemann mapping theorem 306
4 Behavior at the boundary 311
Index 319