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图书目录
PART ONE BASIC THEORY3
Chapter Ⅰ Complex Numbers and Functions3
1 Definition3
2 Polar form8
3 Complex valued functions12
4 Limits and compact sets17
5 Complex differentiability27
6 The Cauchy-Riemann equations31
Chapter Ⅱ Power Series35
1 Formal power series35
2 Convergent power series45
3 Relations between formal and convergent series57
Sums and products57
Quotients60
Composition of series62
4 Holomorphic functions64
5 The inverse and open mapping theorems67
6 The local maximum modulus principle73
7 Differentiation of power series75
Chapter Ⅲ Cauchy’s Theorem, First Part81
1 Analytic functions on connected sets81
2 Integrals over paths88
3 Local primitive for an analytic function96
4 Another description of the integral along a path102
5 The homotopy form of Cauchy’s theorem106
6 Existence of global primitives.Definition of the logarithm108
Chapter Ⅳ Cauchy’s Theorem, Second Part113
1 The winding number113
2 Statement of Cauchy’s theorem117
3 Artin’s proof125
Chapter Ⅴ Applications of Cauchy’s Integral Formula133
1 Cauchy’s integral formula on a disc133
2 Laurent series139
3 Isolated singularities143
4 Dixon’s proof of Cauchy’s theorem148
Chapter Ⅵ Calculus of Residues151
1 The residue formula151
2 Evaluation of definite integrals167
Fourier transforms,169
Trigonometric integrals172
Mellin transforms174
Chapter Ⅶ Conformal Mappings184
1 Schwarz lemma184
2 Analytic automorphisms of the disc185
3 The upper half plane189
4 Other examples190
Chapter Ⅷ Harmonic Functions197
1 Definition197
2 Examples205
3 Construction of harmonic functions212
4 Existence of associated analytic function216
PART TWO VARIOUS ANALYTIC TOPICS221
Chapter Ⅸ Applications of the Maximum Modulus Principle221
1 The effect of zeros, Jensen-Schwarz lemma221
2 The effect of small derivatives226
3 Entire functions with rational values228
4 Phragmen-Lindelof and Hadamard theorems234
5 Bounds by the real part, Borel-Caratheodory theorem238
Chapter Ⅹ Entire and Meromorphic Functions241
1 Infinite products241
2 Weierstrass products244
3 Functions of finite order250
4 Meromorphic functions, Mittag-Leffler theorem252
Chapter Ⅺ Elliptic Functions255
1 The Liouville theorems255
2 The Weierstrass function258
3 The addition theorem262
4 The sigma and zeta functions265
Chapter Ⅻ Differentiating Under an Integral270
1 The differentiation lemma270
2 The gamma function273
Proof of Stirling’s formula277
Chapter ⅩⅢ Analytic Continuation287
1 Schwarz reflection287
2 Continuation along a path292
Chapter ⅩⅣ The Riemann Mapping Theorem299
1 Statement and application to Picard’s theorem299
2 Compact sets in function spaces303
3 Proof of the Riemann mapping theorem306
4 Behavior at the boundary311
Index319