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Chapter 1．Preliminaries to Complex Analysis1

1 Complex numbers and the complex plane1

1.1 Basic properties1

1.2 Convergence5

1.3 Sets in the complex plane5

2 Functions on the complex plane8

2.1 Continuous functions8

2.2 Holomorphic functions8

2.3 Power series14

3 Integration along curves18

4 Exercises24

Chapter 2．Cauchy's Theorem and Its Applications32

1 Goursat's theorem34

2 Local existence of primitives and Cauchy's theorem in a disc37

3 Evaluation of some integrals41

4 Cauchy's integral formulas45

5 Further applications53

5.1 Morera's theorem53

5.2 Sequences of holomorphic functions53

5.3 Holomorphic functions defined in terms of integrals55

5.4 Schwarz reflection principle57

5.5 Runge's approximation theorem60

6 Exercises64

7 Problems67

Chapter 3．Meromorphic Functions and the Logarithm71

1 Zeros and poles72

2 The residue formula76

2.1 Examples77

3 Singularities and meromorphic functions83

4 The argument principle and applications89

5 Homotopies and simply connected domains93

6 The complex logarithm97

7 Fourier series and harmonic functions101

8 Exercises103

9 Problems108

Chapter 4．The Fourier Transform111

1 The class ？113

2 Action of the Fourier transform on ？114

3 Paley-Wiener theorem121

4 Exercises127

5 Problems131

Chapter 5．Entire Functions134

1 Jensen's formula135

2 Functions of finite order138

3 Infinite products140

3.1 Generalities140

3.2 Example:the product formula for the sine function142

4 Weierstrass infinite products145

5 Hadamard's factorization theorem147

6 Exercises153

7 Problems156

Chapter 6．The Gamma and Zeta Functions159

1 The gamma function160

1.1 Analytic continuation161

1.2 Further properties of г163

2 The zeta function168

2.1 Functional equation and analytic continuation168

3 Exercises174

4 Problems179

Chapter 7．The Zeta Function and Prime Number The-orem181

1 Zeros of the zeta function182

1.1 Estimates for 1/ζ(s)187

2 Reduction to the functionsψandψ1188

2.1 Proof of the asymptotics forψ1194

Note on interchanging double sums197

3 Exercises199

4 Problems203

Chapter 8．Conformal Mappings205

1 Conformal equivalence and examples206

1.1 The disc and upper half-plane208

1.2 Further examples209

1.3 The Dirichlet problem in a strip212

2 The Schwarz lemma;automorphisms of the disc and upper half-plane218

2.1 Automorphisms of the disc219

2.2 Automorphisms of the upper half-plane221

3 The Riemann mapping theorem224

3.1 Necessary conditions and statement of the theorem224

3.2 Montel's theorem225

3.3 Proof of the Riemann mapping theorem228

4 Conformal mappings onto polygons231

4.1 Some examples231

4.2 The Schwarz-Christoffel integral235

4.3 Boundary behavior238

4.4 The mapping formula241

4.5 Return to elliptic integrals245

5 Exercises248

6 Problems254

Chapter 9．An Introduction to Elliptic Functions261

1 Elliptic functions262

1.1 Liouville's theorems264

1.2 The Weierstrass ？ function266

2 The modular character of elliptic functions and Eisenstein series273

2.1 Eisenstein series273

2.2 Eisenstein series and divisor functions276

3 Exercises278

4 Problems281

Chapter 10．Applications of Theta Functions283

1 Product formula for the Jacobi theta function284

1.1 Further transformation laws289

2 Generating functions293

3 The theorems about sums of squares296

3.1 The two-squares theorem297

3.2 The four-squares theorem304

4 Exercises309

5 Problems314

Appendix A:Asymptotics318

1 Bessel functions319

2 Laplace's method;Stirling's formula323

3 The Airy function328

4 The partition function334

5 Problems341

Appendix B:Simple Connectivity and Jordan Curve Theorem344

1 Equivalent formulations of simple connectivity345

2 The Jordan curve theorem351

2.1 Proof of a general form of Cauchy's theorem361

Notes and References365

Bibliography369

Symbol Glossary373

Index375