复分析

Foreword

Introductlon

Chapter 1．Preliminaries to Complex Analysis

l Complex numbers and the eompicx plane

1.1 Basic properties

l.2 Convergence

1.3 Sets in tim complex plane

2 Functions on the complex plane

2.l Conltinuous fnetions

2.2 Holomorphic fimctions

2.3 P0weI series

3 Integration along crvcs

4 Exorcises

Chapter 2 Cauchy’s Theorem and Its Applications

1 Goursat’s theorem

2 Local existencc of primitives and Cauchy s theorem in a

disc

3 EvaIuatlon of some integrals

4 Cauchy’s integral formulas

5 1lrther applications

5.1 Morera’s tImorem

5.2 Sequences of holomorphic functions

5.3 Holomorphic functions defined in terms of integrals

5.4 Schwarz reflection principle

5.5 Runge’s approxlnlatlon theorem

6 Exereises

7 Problems

Chapter 3．Meromorphic Functions and the Logarithm

1 Zeros and polcs

2 The residue formuia

2.l Examples

3 Singularities and meromorphic functions

4 The argmuent principle and applications

5 Homotopies and simply connected domains

6 The complex logarithm

7 Fourier series and harmonic functions

8 Exercises

9 Problenis

Chapter 4. The Fourier Transforin

1 The class

2 Action of the Fourier transform on

3 Palev-wiener tbeorem

4 Exercises

5 Problems

Chapter 5. Entire Functions

1 Jensens formUla

2 Functions of finite order

3 Infinite products

3.1 Generalities

3.2 Example: the product foemula for the sine function

4 Weierstrass infinite products

5 Hadamards factorozatoon theorem

6 Exercises

7 Problems

Chapter 6. The Gamma and Zeta Functions

1 The gamma function

1.1 Analytic continuation

1.2 Furtiicr properties of F

2 The zeta function

2.1 Functional equation and analytic continuation

3 Exercises

4 Problems

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

1 Zeros of tile zeta function

hl Esthnates for 1/C(S)

2 Reduction to the functions

2.1 Proof of the asymptotics for

Note on interchanging double sums

3 Exercises

4 problems

Chapter 8. Conformal Mappings

Chapter 9. An Introduction to Elliptic Functions

Chapter 10. Applications of Theta Functions

Appendix A: Asymptotics

Appendix B: Simple Connectivity and Jordan Curve Theorem

Notes and References

Bibliography

Symbol Glossary

Index

本书由在国际上享有盛誉的普林斯顿大学教授Stein等撰写而成, 是一部为数学及相关专业大学二年级和三年级学生编写的教材，理论与实践并重。为了便于非数学专业的学生学习，全书内容简明、易懂，读者只需掌握微积分和线性代数知识。关于本书的详细介绍，请见“影印版前言”。
　　本书已被哈佛大学和加利福尼亚理工学院选为教材。与本书相配套的教材《傅立叶分析导论》和《实分析》也已影印出版。

《复分析》已被哈佛大学和加利福尼亚理工学院选为教材。与《复分析》相配套的教材《傅立叶分析导论》和《实分析》也已影印出版。