內容大鋼
複分析是數學的基石，是研究生數學研究中的基本元素。本書強調初等複分析的直觀幾何基礎，自然而然地引出Riemann曲面理論。
本書以單復變全純函數的基本理論開篇。前兩章是關於複分析的一個快速但全面的教程。第三章專門研究圓盤和半平面上的調和函數，重點是Dirichlet問題。從第四章起，作者開始較為詳盡和嚴格地介紹Riemann曲面理論：從一開始就強調幾何方面，並以橢圓函數和橢圓積分等經典主題作為抽象理論的例證；解釋了緊Riemann曲面的特殊作用，並建立了它們與代數方程的聯繫。本書的最後三章分別介紹了涉及Riemann曲面理論核心技術內容的三個主要結果：Hodge分解定理、Riemann-Roch定理和單值化定理。
本書旨在提供一個詳細、快速的導引，介紹單復變理論中對數學其他領域最有用的部分，這些領域包括幾何群論、動力學、代數幾何、數論和泛函分析。全書共有70多幅插圖用來闡述相關概念和思想，每章末尾的習題為讀者提供了充分的實踐和獨立學習的機會。
本書適合對於複分析、共形幾何、Riemann曲面、單值化、調和函數、Riemann曲面上的微分形式以及Riemann-Roch定理感興趣的研究生閱讀，也可供相關領域的研究人員參考。
目錄
Preface
Acknowledgments
Chapter 1.From i to z: the basics of complex analysis
§1.1.The field of complex numbers
§1.2.Holomorphic, analytic, and conformal
§1.3.The Riemann sphere
§1.4.M?bius transformations
§1.5.The hyperbolic plane and the Poincar? disk
§1.6.Complex integration, Cauchy theorems
§1.7.Applications of Cauchy's theorems
§1.8.Harmonic functions
§1.9.Problems
Chapter 2.From z to the Riemann mapping theorem: some finer points of basic complex analysis
§2.1.The winding number
§2.2.The global form of Cauchy's theorem
§2.3.Isolated singularities and residues
§2.4.Analytic continuation
§2.5.Convergence and normal families
§2.6.The Mittag-Leffler and Weierstrass theorems
§2.7.The Riemann mapping theorem
§2.8.Runge's theorem and simple connectivity
§2.9.Problems
Chapter 3.Harmonic functions
§3.1.The Poisson kernel
§3.2.The Poisson kernel from the probabilistic point of view
§3.3.Hardy classes of harmonic functions
§3.4.Almost everywhere convergence to the boundary data
§3.5.Hardy spaces of analytic functions
§3.6.Riesz theorems
§3.7.Entire functions of finite order
§3.8.A gallery of conformal plots
§3.9.Problems
Chapter 4.Riemann surfaces: definitions, examples, basic properties
§4.1.The basic definitions
§4.2.Examples and constructions of Riemann surfaces
§4.3.Functions on Riemann surfaces
§4.4.Degree and genus
§4.5.Riemann surfaces as quotients
§4.6.Elliptic functions
§4.7.Covering the plane with two or more points removed
§4.8.Groups of M?bius transforms
§4.9.Problems
Chapter 5.Analytic continuation, covering surfaces, and algebraic functions
§5.1.Analytic continuation
§5.2.The unramified Riemann surface of an analytic germ
§5.3.The ramified Riemann surface of an analytic germ
§5.4.Algebraic germs and functions
§5.5.Algebraic equations generated by compact surfaces
§5.6.Some compact surfaces and their associated polynomials
§5.7.ODEs with meromorphic coefficients
§5.8.Problems
Chapter 6.Differential forms on Riemann surfaces
§6.1.Holomorphic and meromorphic differentials
§6.2.Integrating differentials and residues
§6.3.The Hodge-* operator and harmonic differentials
§6.4.Statement and examples of the Hodge decomposition
§6.5.Weyl's lemma and the Hodge decomposition
§6.6.Existence of nonconstant meromorphic functions
§6.7.Examples of meromorphic functions and differentials
§6.8.Problems
Chapter 7.The Theorems of Riemann-Roch, Abel, and Jacobi
§7.1.Homology bases and holomorphic differentials
§7.2.Periods and bilinear relations
§7.3.Divisors
§7.4.The Riemann-Roch theorem
§7.5.Applications and general divisors
§7.6.Applications to algebraic curves
§7.7.The theorems of Abel and Jacobi
§7.8.Problems
Chapter 8.Uniformization
§8.1.Green functions and Riemann mapping
§8.2.Perron families
§8.3.Solution of Dirichlet's problem
§8.4.Green's functions on Riemann surfaces
§8.5.Uniformization for simply-connected surfaces
§8.6.Uniformization of non-simply-connected surfaces
§8.7.Fuchsian groups
§8.8.Problems
Appendix A.Review of some basic background material
§A.1.Geometry and topology
§A.2.Algebra
§A.3.Analysis
Bibliography
Index
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