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 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