复分析与Riemann曲面教程（影印版）

样章

作者：

Wilhelm Schlag

定价：

169.00 元

版面字数：

620千字

开本：

特殊

装帧形式：

精装

版次：

最新版次
印刷时间：

2022年

ISBN：

978-7-04-056979-7

物料号：

56979-00

出版时间：

2022-02-25

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

分析

暂无

目录
- 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