复分析是数学的基石，是研究生数学研究中的基本元素。本书强调初等复分析的直观几何基础，自然而然地引出 Riemann 曲面理论。 本书以单复变全纯函数的基本理论开篇。前两章是关于复分析的一个快速但全面的教程。第三章专门研究圆盘和半平面上的调和函数，重点是 Dirichlet 问题。从第四章起，作者开始较为详尽和严格地介绍 Riemann 曲面理论：从一开始就强调几何方面，并以椭圆函数和椭圆积分等经典主题作为抽象理论的例证；解释了紧 Riemann 曲面的特殊作用，并建立了它们与代数方程的联系。本书的最后三章分别介绍了涉及 Riemann 曲面理论核心技术内容的三个主要结果：Hodge 分解定理、Riemann-Roch 定理和单值化定理。 本书旨在提供一个详细、快速的导引，介绍单复变理论中对数学其他领域最有用的部分，这些领域包括几何群论、动力学、代数几何、数论和泛函分析。全书共有 70 多幅插图用来阐述相关概念和思想，每章末尾的习题为读者提供了充分的实践和独立学习的机会。 本书适合对于复分析、共形几何、Riemann 曲面、单值化、调和函数、Riemann 曲面上的微分形式以及 Riemann-Roch 定理感兴趣的研究生阅读，也可供相关领域的研究人员参考。

前辅文
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¨obius transformations
  §1.5. The hyperbolic plane and the Poincar´e 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¨obius 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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