Conformal Mapping on Riemann Surfaces

The subject matter loosely called ""Riemann surface theory"" has been the starting point for the development of topology, functional analysis, modern algebra, and any one of a dozen recent branches of mathematics; it is one of the most valuable bodies of knowledge within mathematics for a student to learn.Professor Cohn's lucid and insightful book presents an ideal coverage of the subject in five parts. Part I is a review of complex analysis analytic behavior, the Riemann sphere, geometric constructions, and presents (as a review) a microcosm of the course. The Riemann manifold is introduced in Part II and is examined in terms of intuitive physical and topological technique in Part III. In Part IV the author shows how to define real functions on manifolds analogously with the algebraic and analytic points of view outlined here. The exposition returns in Part V to the use of a single complex variable z. As the text is richly endowed with problem material - 344 exercises - the book is perfect for self-study as well as classroom use.Harvey Cohn is well-known in the mathematics profession for his pedagogically superior texts, and the present book will be of great interest not only to pure and applied mathematicians, but also engineers and physicists. Dr. Cohn is currently Distinguished Professor of Mathematics at the City University of New York Graduate Center.

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. Mobius 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' A 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-Poincare 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

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