Synopses & Reviews
This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.
Review
"The author employs a comfortable writing style throughout. Every chapter and many sections start with a chatty discussion of what lies ahead. We are frequently cautioned against jumping to conclusions, and alerted to what are the points-at-issue. Sources of approaches to topics are acknowledged, and references cited where the researcher will find them convenient. Topics studied include Green functions, the Dirichlet Principle (via Sobolev spaces), harmonic measure, logarithmic capacity, and the finite topology. Students who make this trip through Conway II should find themselves ready for the research literature of complex analytic functions of a single variable. Zentralblatt Math"
Synopsis
This is the sequel to my book Functions of One Complex Variable I, and probably a good opportunity to express my appreciation to the mathemat ical community for its reception of that work. In retrospect, writing that book was a crazy venture. As a graduate student I had had one of the worst learning experiences of my career when I took complex analysis; a truly bad teacher. As a non-tenured assistant professor, the department allowed me to teach the graduate course in complex analysis. They thought I knew the material; I wanted to learn it. I adopted a standard text and shortly after beginning to prepare my lectures I became dissatisfied. All the books in print had virtues; but I was educated as a modern analyst, not a classical one, and they failed to satisfy me. This set a pattern for me in learning new mathematics after I had become a mathematician. Some topics I found satisfactorily treated in some sources; some I read in many books and then recast in my own style. There is also the matter of philosophy and point of view. Going from a certain mathematical vantage point to another is thought by many as being independent of the path; certainly true if your only objective is getting there. But getting there is often half the fun and often there is twice the value in the journey if the path is properly chosen."
Synopsis
This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.
Description
Includes bibliographical references (p. [384]-388) and index.
Table of Contents
x Return to Basics
x Conformal Equivalence for Simply Connected Regions
x Conformal Equivalence for Finitely Connected Regions
x Analytic Covering Maps
x De Branges' Proof of the Bieberbach Conjecture
x Some Fundamental Concepts from Analysis
x Harmonic Functions Redux
x Hardy Spaces on the Disk
x Potential Theory in the Plane