Elias M.Stein、Rami
Shakarchi所著的《复分析》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。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。本书已被哈佛大学和加利福尼亚理工学院选为教材。
目錄：
Foreword
Introduction
Chapter 1. Preliminaries to Complex Analysis
1 Complex numbe and the complex plane
1.1 Basic properties
1.2 Convergence
1.3 Sets in the complex plane
2 Functio on the complex plane
2.1 Continuous functio
2.2 Holomorphic functio
2.3 Power series
3 Integration along curves
4 Exercises
Chapter 2. Cauchy''s Theorem and Its Applicatio
1 Gouat''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 applicatio
5.1 Morera''s theorem
5.2 Sequences of holomorphic functio
5.3 Holomorphic functio defined in terms of integrals
5.4 Schwarz reflection principle
5.5 Runge''s approximation theorem
6 Exercises
7 Problems
Chapter 3. Meromorphic Functio and the Logarithm
1 Zeros and poles
2 The residue formula
2.1 Examples
3 Singularities and meromorphic functio
4 The argument principle and applicatio
5 Homotopies and simply connected domai
6 The complex logarithm
7 Fourier series and harmonic functio
8 Exercises
9 Problems
Chapter 4. The Fourier Traform
1 The class ξ
2 Action of the Fourier traform on ξ
3 Paley-Wiener theorem
4 Exercises
5 Problems
Chapter 5. Entire Functio
1 Jeen''s formula
2 Functio of finite order
3 Infinite products
3.1 Generalities
3.2 Example: the product formula for the sine function
4 Weietrass infinite products
5 Hadamard''s factorization theorem
6 Exercises
7 Problems
Chapter 6. The Gamma and Zeta Functio
1 The gamma function
1.1 Analytic continuation
1.2 Further properties of τ
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 1ζs
2 Reduction to the functio 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 conditio and statement of the theorem
3.2 Montel''s theorem
3.3 Proof of the Riemann mapping theorem
4 Conformal mappings onto polygo
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 Functio
1 Elliptic functio
1.1 Liouville''s theorems
1.2 The Weietrass p function
2 The modular character of elliptic functio and Eisetein series
2.1 Eisetein series
2.2 Eisetein series and divisor functio
3 Exercises
4 Problems
Chapter 10. Applicatio of Theta Functio
1 Product formula for the Jacobi theta function
1.1 Further traformation laws
2 Generating functio
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 functio
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 formulatio 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
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