复分析

     Elias M.Stein、RamiShakarchi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。

ForewordIntroductionChapter 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 ExercisesChapter 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 ProblemsChapter 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 ProblemsChapter 4. The Fourier Traform 1 The class ξ 2 Action of the Fourier traform on ξ 3 Paley-Wiener theorem 4 Exercises 5 ProblemsChapter 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 ProblemsChapter 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 ProblemsChapter 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 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 upperhalf-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 ProblemsAppendix A: Asymptotics 1 Bessel functio 2 Laplace's method; Stirling's formula 3 The Airy function 4 The partition function 5 ProblemsAppendix 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 theoremNotes and ReferencesBibliographySymbol GlossaryIndex