《complex analysis P321》

PART ONE BASIC THEORY3

Chapter ⅠComplex 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 ⅡPower 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

1Analytic 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 ⅣCauchy’s Theorem, Second Part113

1 The winding number113

2 Statement of Cauchy’s theorem117

3 Artin’s proof125

Chapter ⅤApplications 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 ⅥCalculus of Residues151

1 The residue formula151

2 Evaluation of definite integrals167

Fourier transforms,169

Trigonometric integrals172

Mellin transforms174

Chapter ⅦConformal Mappings184

1 Schwarz lemma184

2 Analytic automorphisms of the disc185

3 The upper half plane189

4 Other examples190

Chapter Ⅷ Harmonic Functions197

1Definition197

2 Examples205

3 Construction of harmonic functions212

4 Existence of associated analytic function216

PART TWOVARIOUS ANALYTIC TOPICS221

Chapter ⅨApplications 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 ⅩEntire and Meromorphic Functions241

1 Infinite products241

2 Weierstrass products244

3 Functions of finite order250

4 Meromorphic functions, Mittag-Leffler theorem252

Chapter Ⅺ Elliptic Functions255

1The Liouville theorems255

2 The Weierstrass function258

3 The addition theorem262

4 The sigma and zeta functions265

Chapter ⅫDifferentiating Under an Integral270

1 The differentiation lemma270

2 The gamma function273

Proof of Stirling’s formula277

Chapter ⅩⅢ Analytic Continuation287

1Schwarz reflection287

2 Continuation along a path292

Chapter ⅩⅣThe 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