Conformal Mapping on Riemann Surfaces

by Cohn, Harvey

• ISBN13: 9780486640259
• ISBN10: 0486640256
• Format: Paperback
• Copyright: 2010-10-18
• Publisher: Dover Publications

Summary

This lucid and insightful exploration reviews complex analysis and introduces the Riemann manifold. It also shows how to define real functions on manifolds analogously with algebraic and analytic points of view. Richly endowed with more than 340 exercises, this book is perfect for classroom use or independent study. 1967 edition.

Author Biography

A Distinguished Professor of Mathematics at the City University of New York Graduate Center, Harvey Cohn is well known for his pedagogically superior texts.

Table of Contents

Preface
PART ONE Review of Complex Analysis
  Introductory Survey
    Chapter 1. Analytic Behavior
      Differentiation and Integration
      1-1. Analyticity
      1-2. Integration on curves and chains
      1-3. Cauchy integral theorem
      Topological Considerations
      1-4. Jordan curve theorem
      1-5. Other manifolds
      1-6. Homologous chains
    Chapter 2. Riemann Sphere
      Treatment of Infinity
      2-1. Ideal point
      2-2. Stereographic projection
      2-3. Rational functions
      2-4. Unique specification theorems
      Transformation of the Sphere
      2-5. Invariant properties
      2-6. Möbius geometry
      2-7. Fixed-point classification
    Chapter 3. Geometric Constructions
      Analytic Continuation
      3-1. Multivalued functions
      3-2. Implicit functions
      3-3. Cyclic neighborhoods
      Conformal Mapping
      3-4. Local and global results
      3-5. Special elementary mappings
PART TWO Riemann Manifolds
  Definition of Riemann Manifold through Generalization
    Chapter 4. Elliptic Functions
      Abel's Double-period Structure
      4-1. Trigonometric uniformization
      4-2. Periods of elliptic integrals
      4-3. Physical and topological models
      Weierstrass' Direct Construction
      4-4. Elliptic functions
      4-5. Weierstrass' Ã function
      4-6. The elliptic modular function
      Euler's Addition Theorem
      4-7. Evolution of addition process
      4-8. Representation theorems
    Chapter 5. Manifolds over the z Sphere
      Formal Definitions
      5-1. Neighborhood Structure
      5-2. Functions and differentials
      Triangulated Manifolds
      5-3. Triangulation structure
      5-4. Algebraic Riemann manifolds
    Chapter 6. Abstract Manifolds
      6-1. Punction field on M
      6-2. Compact manifolds are algebraic
      6-3. Modular functions
PART THREE Derivation of Existence Theorems
  Return to Real Variables
    Chapter 7. Topological Considerations
      The Two Canonical Models
      7-1. Orientability
      7-2. Canonical subdivisions
      7-3. The Euler-Poincaré theorem
      7-4. Proof of models
      Homology and Abelian Differentials
      7-5. Boundaries and cycles
      7-6. Complex existence theorem
    Chapter 8. Harmonic Differentials
      Real Differentials
      8-1. Cohomology
      8-2. Stokes' theorem
      8-3. Conjugate forms
      Dirichlet Problems
      8-4. The two existence theorems
      8-5. The two uniqueness proofs
    Chapter 9. Physical Intuition
      9-1. Electrostatics and hydrodynamics
      9-2. Special solutions
      9-3. Canonical mappings
PART FOUR Real Existence Proofs
  Evolution of Some Intuitive Theorems
    Chapter 10. Conformal Mapping
    10-1. Poisson's integral
    10-2. Riemann' s theorem for the disk
    Chapter 11. Boundary Behavior
    11-1. Continuity
    11-2. Analyticity
    11-3. Schottky double
    Chapter 12. Alternating Procedures
    12-1. Ordinary Dirichlet problem
    12-2. Nonsingular noncompact problem
    12-3. Planting of singularities
PART FIVE Algebraic Applications
  Resurgence of Finite Structures
    Chapter 13. Riemann's Existence Theorem
    13-1. Normal integrals
    13-2. Construction of the function field
    Chapter 14. Advanced Results
    14-1. Riemann-Roch theorem
    14-2. Abel's theorem
    Appendix A. Minimal Principles
    Appendix B. Infinite Manifolds
    Table 1: Summary of Existence and Uniqueness Proofs
    Bibliography and Special Source Material
    Index