复分析

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

2 action of the fourier transform on

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 γ

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 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

i 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

i 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

《复分析》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。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。《复分析》已被哈佛大学和加利福尼亚理工学院选为教材。目次：复分析预备知识；柯西定理及其应用；亚纯函数和对数；傅立叶转换；Gamma和Zeta函数；Zeta函数和素数定理；共形映射；椭圆函数；Theta函数。