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复分析与Riemann曲面教程（影印版）

样章
作者：
Wilhelm Schlag
定价：
169.00元
ISBN：
978-7-04-056979-7
版面字数：
620.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
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
  • 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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