图书信息
图书目录
复分析与Riemann曲面教程（影印版）
暂无简介
作者：
Wilhelm Schlag
定价：
169.00元
出版时间：
2022-02-25
ISBN：
978-7-04-056979-7
物料号：
56979-00
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
重点项目：
暂无
版面字数：
620.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
- 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
