Function Theory on Planar Domains: A Second Course in Complex Analysis

by Stephen D. Fisher

Overview

A high-level treatment of complex analysis, this text focuses on function theory on a finitely connected planar domain. Clear and complete, it emphasizes domains bounded by a finite number of disjoint analytic simple closed curves.
The first chapter and parts of Chapters 2 and 3 offer background material, all of it classical and important in its own right. The remainder of the text presents results in complex analysis from the far, middle, and recent past, all selected for their interest and merit as substantive mathematics. Suitable for upper-level undergraduates and graduate students, this text is accessible to anyone with a background in complex and functional analysis. Author Stephen D. Fisher, a professor of mathematics at Northwestern University, elaborates upon and extends results with a set of exercises at the end of each chapter.

Product Details

ISBN-13:	9780486151106
Publisher:	Dover Publications
Publication date:	06/10/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	30 MB
Note:	This product may take a few minutes to download.

Table of Contents

Notation and Numbering     xiii
The Dirichlet Problem and Harmonic Measure     1
Introduction     1
The Poisson Formula and Some Preliminaries     1
Subharmonic Functions     6
Solution of the Dirichlet Problem     11
The Green's Function of a Domain     16
Harmonic Measure     19
Logarithmic Capacity     24
Additional Readings and Notes     35
Exercises     35
Uniformization and Conditional Expectation     38
Introduction     38
The Uniformization Theorem     38
Conditional Expectation and the Space N     43
Harmonic Measure and L[superscript 1]/[Characters not reproducible]     48
Additional Readings and Notes     49
Exercises     49
The Hardy Spaces H[superscript p]([Omega])     51
Introduction     51
Basic Properties of H[superscript p]([Omega])     51
H[superscript p] on the Unit Disc     55
H[superscript p] and H[superscript p]([Omega])     62
Null Sets and Essential Boundary Points for H[superscript Infinity]([Omega])     64
H[superscript p]([Omega]) Determines [Omega]     66
Weak Peak Pointsfor H[superscript Infinity]([Omega])     70
Additional Readings and Notes     73
Exercises     74
Domains of Finite Connectivity     77
Introduction     77
The Defect of ReR([Omega]) in C[subscript 1]([Gamma])     78
Measures Orthogonal to R([Omega])     81
H[superscript p]([Omega])     85
N Again     93
Functions with Periods     97
The Factorization of H[superscript p]([Omega]) Functions     103
Additional Readings and Notes     106
Exercises     107
Blaschke Products, Inner Functions, and Extremal Problems     109
The Ahlfors Function     109
Blaschke Products     116
Approximation by Inner Functions     120
Pick-Nevanlinna Interpolation     130
Interpolation Sequences     141
The Maximum Principle for Multiple-Valued Bounded Analytic Functions     151
Additional Readings and Notes     160
Exercises     161
The Maximal Ideal Space of H[superscript Infinity]([Omega])     165
Introduction     165
Peak Points and Parts     165
The Fibers of M([Omega])     171
Distinguished Homomorphisms     177
The Shilov Boundary of H[superscript Infinity]([Omega])     190
The Corona Theorem     195
Additional Readings and Notes     202
Exercises     203
Linear Operators on H[superscript p] Spaces     206
The Isometries of H[superscript p]([Omega])     206
Self-Mappings of a Domain     217
General Properties of Composition Operators     228
Compact Composition Operators on H[superscript Infinity]([Omega])     232
Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 1. The H[superscript Infinity] Case     237
Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 2. The H[superscript 2] Case     251
Additional Readings and Notes     258
Exercises     258
Bibliography     262
Index     267