复分析.pdf

复分析.pdf
 

书籍描述

编辑推荐
《复分析(英文版)》由在国际上享有盛誉普林斯大林顿大学教授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。

作者简介
作者:(美国)斯坦恩(Elias M.Stein) (美国)Rami Shakarchi

目录
Foreword
Introduction
Chapter 1.Preliminaries to Complex Analysis
1 Complex numbers and the complex plane
1.1 Basic properties
1.2 Convergence
1.3 Sets in the complex plane
2 Functions on the complex plane
2.1 Continuous functions
2.2 Holomorphic functions
2.3 Power series
3 Integration along curves
4 Exercises
Chapter 2.Cauchy's Theorem and Its Applications
1 Goursat's theorem
2 Local existence of primitives and Cauchy's theorem in a disc
3 Evaluation of some integrals
4 Cauchy's integral formulas
5 Further applications
5.1 Morera's theorem
5.2 Sequences of holomorphic functions
5.3 Holomorphic functions defined in terms of integrals
5.4 Schwarz reflection principle
5.5 Runge's approximation theorem
6 Exercises
7 Problems
Chapter 3.Meromorphic Functions and the Logarithm
1 Zeros and poles
2 The residue formula
2.1 Examples
3 Singularities and meromorphic functions
4 The argument principle and applications
5 Homotopies and simply connected domains
6 The complex logarithm
7 Fourier series and harmonic functions
8 Exercises
9 Problems
Chapter 4.The Fourier Transform
1 The class ξ
2 Action of the Fourier transform on ξ
3 Paley—Wiener theorem
4 Exercises
5 Problems
Chapter 5.Entire Functions
1 Jensen's formula
2 Functions of finite order
3 Infinite products
3.1 Generalities
3.2 Example: the product formula for the sine function
4 Weierstrass infinite products
5 Hadamard's factorization theorem
6 Exercises
7 Problems
Chapter 6.The Gamma and Zeta Functions
1 The gamma function
1.1 Analytic continuation
1.2 Further properties of г
2 The zeta function
2.1 Functional equation and analytic continuation
3 Exercises
4 Problems
Chapter 7.The Zeta Function and Prime Number Theorem
1 Zeros of the zeta function
1.1 Estimates for 1/ζ(s)
2 Reduction to the functions ψ and ψ1
2.1 Proof of the asymptotics for ψ1
Note on interchanging double sums
3 Exercises
4 Problems
Chapter 8.Conformal Mappings
1 Conformal equivalence and examples
1.1 The disc and Upper half—plane
1.2 Further examples
1.3 The Dirichlet problem in a strip
2 The Schwarz lemma; automorphisms of the disc and upper half—plane
2.1 Automorphisms of the disc
2.2 Automorphisms of the upper half—plane
3 The Riemann mapping theorem
3.1 Necessary conditions and statement of the theorem
3.2 Montel's theorem
3.3 Proof of the Riemann mapping theorem
4 Conformal mappings onto polygons
4.1 Some examples
4.2 The Schwarz—Christoffel integral
4.3 Boundary behavior
4.4 The mapping formula
4.5 Return to elliptic integrals
5 Exercises
6 Problems
Chapter 9.An Introduction to Elliptic Functions
1 Elliptic functions
1.1 Liouville's theorems
1.2 The Weierstrass p function
2 The modular character of elliptic functions and Eisenstein series
2.1 Eisenstein series
2.2 Eisenstein series and divisor functions
3 Exercises
4 Problems
Chapter 10.Applications of Theta Functions
1 Product formula for the Jacobi theta function
1.1 Further transformation laws
2 Generating functions
3 The theorems about sums of squares
3.1 The two—squares theorem
3.2 The four—squares theorem
4 Exercises
5 Problems
Appendix A: Asymptotics
1 Bessel functions
2 Laplace's method; Stirling's formula
3 The Airy function
4 The partition function
5 Problems
Appendix B: Simple Connectivity and Jordan Curve Theorem
1 Equivalent formulations of simple connectivity
2 The Jordan curve theorem
2.1 Proof of a general form of Cauchy's theorem
Notes and References
Bibliography
Symbol Glossary
Index

文摘
版权页:

复分析

插图:

复分析

内容简介
《复分析(英文版)》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。《复分析(英文版)》已被哈佛大学和加利福尼亚理工学院选为教材。

购买书籍

当当网购书 京东购书 卓越购书

PDF电子书下载地址

相关书籍

搜索更多