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基础复分析：分析综合教程（第2A部分）（影印版）

样章
作者：
Barry Simon
定价：
269.00元
ISBN：
978-7-04-059300-6
版面字数：
1060.000千字
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：
2023-03-08
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. Preliminaries
    • §1.1. Notation and Terminology
    • §1.2. Complex Numbers
    • §1.3. Some Algebra, Mainly Linear
    • §1.4. Calculus on R and Rn
    • §1.5. Differentiable Manifolds
    • §1.6. Riemann Metrics
    • §1.7. Homotopy and Covering Spaces
    • §1.8. Homology
    • §1.9. Some Results from Real Analysis
  • Chapter 2. The Cauchy Integral Theorem: Basics
    • §2.1. Holomorphic Functions
    • §2.2. Contour Integrals
    • §2.3. Analytic Functions
    • §2.4. The Goursat Argument
    • §2.5. The CIT for Star-Shaped Regions
    • §2.6. Holomorphically Simply Connected Regions, Logs, and Fractional Powers
    • §2.7. The Cauchy Integral Formula for Disks and Annuli
  • Chapter 3. Consequences of the Cauchy Integral Formula
    • §3.1. Analyticity and Cauchy Estimates
    • §3.2. An Improved Cauchy Estimate
    • §3.3. The Argument Principle and Winding Numbers
    • §3.4. Local Behavior at Noncritical Points
    • §3.5. Local Behavior at Critical Points
    • §3.6. The Open Mapping and Maximum Principle
    • §3.7. Laurent Series
    • §3.8. The Classification of Isolated Singularities
    • Casorati–Weierstrass Theorem
    • §3.9. Meromorphic Functions
    • §3.10. Periodic Analytic Functions
  • Chapter 4. Chains and the Ultimate Cauchy Integral Theorem
    • §4.1. Homologous Chains
    • §4.2. Dixon’s Proof of the Ultimate CIT
    • §4.3. The Ultimate Argument Principle
    • §4.4. Mesh-Defined Chains
    • §4.5. Simply Connected and Multiply Connected Regions
    • §4.6. The Ultra Cauchy Integral Theorem and Formula
    • §4.7. Runge’s Theorems
    • §4.8. The Jordan Curve Theorem for Smooth Jordan Curves
  • Chapter 5. More Consequences of the CIT
    • §5.1. The Phragm´en–Lindel¨of Method
    • §5.2. The Three-Line Theorem and the Riesz–Thorin Theorem
    • §5.3. Poisson Representations
    • §5.4. Harmonic Functions
    • §5.5. The Reflection Principle
    • §5.6. Reflection in Analytic Arcs
    • §5.7. Calculation of Definite Integrals
  • Chapter 6. Spaces of Analytic Functions
    • §6.1. Analytic Functions as a Fr´echet Space
    • §6.2. Montel’s and Vitali’s Theorems
    • §6.3. Restatement of Runge’s Theorems
    • §6.4. Hurwitz’s Theorem
    • §6.5. Bonus Section: Normal Convergence of Meromorphic Functions and Marty’s Theorem
  • Chapter 7. Fractional Linear Transformations
    • §7.1. The Concept of a Riemann Surface
    • §7.2. The Riemann Sphere as a Complex Projective Space
    • §7.3. PSL(2,C)
    • §7.4. Self-Maps of the Disk
    • §7.5. Bonus Section: Introduction to Continued Fractions and the Schur Algorithm
  • Chapter 8. Conformal Maps
    • §8.1. The Riemann Mapping Theorem
    • §8.2. Boundary Behavior of Riemann Maps
    • §8.3. First Construction of the Elliptic Modular Function
    • §8.4. Some Explicit Conformal Maps
    • §8.5. Bonus Section: Covering Map for General Regions
    • §8.6. Doubly Connected Regions
    • §8.7. Bonus Section: The Uniformization Theorem
    • §8.8. Ahlfors’ Function, Analytic Capacity and the Painlev´e Problem
  • Chapter 9. Zeros of Analytic Functions and Product Formulae
    • §9.1. Infinite Products
    • §9.2. A Warmup: The Euler Product Formula
    • §9.3. The Mittag-Leffler Theorem
    • §9.4. The Weierstrass Product Theorem
    • §9.5. General Regions
    • §9.6. The Gamma Function: Basics
    • §9.7. The Euler–Maclaurin Series and Stirling’s Approximation
    • §9.8. Jensen’s Formula
    • §9.9. Blaschke Products
    • §9.10. Entire Functions of Finite Order and the Hadamard Product Formula
  • Chapter 10. Elliptic Functions
    • §10.1. A Warmup: Meromorphic Functions on C
    • §10.2. Lattices and SL(2, Z)
    • §10.3. Liouville’s Theorems, Abel’s Theorem, and Jacobi’s Construction
    • §10.4. Weierstrass Elliptic Functions
    • §10.5. Bonus Section: Jacobi Elliptic Functions
    • §10.6. The Elliptic Modular Function
    • §10.7. The Equivalence Problem for Complex Tori
  • Chapter 11. Selected Additional Topics
    • §11.1. The Paley–Wiener Strategy
    • §11.2. Global Analytic Functions
    • §11.3. Picard’s Theorem via the Elliptic Modular Function
    • §11.4. Bonus Section: Zalcman’s Lemma and Picard’s Theorem
    • §11.5. Two Results in Several Complex Variables: Hartogs’ Theorem and a Theorem of Poincar´e
    • §11.6. Bonus Section: A First Glance at Compact Riemann Surfaces
  • Bibliography
  • Symbol Index
  • Subject Index
  • Author Index
  • Index of Capsule Biographies

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