Conformal Mapping on Riemann Surfaces

Harvey Cohn

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486640259
• eBook ISBN: 9780486153292
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

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.

PART I

REVIEW OF COMPLEX ANALYSIS

INTRODUCTORY SURVEY

We first undertake a critical review of introductory complex function theory for the purpose of examining the “ideal” case of the Riemann sphere (or the ordinary plane with a “point at infinity” adjoined). We make every effort to think of this two-dimensional plane (or sphere) as being represented by a single complex variable, say z.

Here we find that, in a certain sense, those functions of z which “belong in a natural way” to the sphere are rational. This is significant from two seemingly opposite viewpoints:

(a) Algebraic. The rational functions f(z) can first of all be handled formally, through high-school-level identities in z, without any apparent concern for the magnitudes of z or f(z) or for analysis (limiting operations, differentiation, integration, etc.).

(b) Analytic. The rational functions f(z) are now determined by their (limiting) magnitude where they become infinite. This so-called “singular” behavior determines the functions completely, as though “normal” (analytic) behavior were unimportant.

We find that to fully appreciate this condition aesthetically, we must explore the z plane as a two-dimensional (real) xy plane, as is done in Part III. Such considerations were first introduced as “conformal mapping” based on strong ties with applications; but the proper arena for such investigations is the Riemann manifold, which is introduced in Part II and is examined in terms of intuitive physical and topological techniques in Part III. In Part IV we show how to define real functions on manifolds analogously with the algebraic and analytic points of view outlined here. In Part V we return to the use of a single complex variable z.

Thus Part I presents (as a review) the course in microcosm.

CHAPTER 1

ANALYTIC BEHAVIOR

At first we restrict our interest to functions which are well behaved where denned. All well-behaved functions are “equally” well behaved under this restriction, for assumi...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Preface
• Contents
• Part 1. Review of Complex Analysis
• Part 2. Riemann Manifolds
• Part 3. Derivation of Existence Theorems
• Part 4. Real Existence Proofs
• Part 5. Algebraic Applications
• Appendix A. Minimal Principles
• Appendix B. Infinite Manifolds
• Bibliography and Special Source Material
• Index