复分析中的若干论题 内容简介

This book provides a concise treatment of topics in complex analysis, suitable for a one-semester course. It is an outgrowth of lectures given by the author over the last ten years at the University of Göteborg and Chalmers University of Technology. While treating classical complex function theory, the author emphasizes connections to real and harmonic analysis, and presents general tools that basic ideas in beginning complex analysis. The book introduces all of the basic ideas in beginning complex analysis and still has time to reach many topics near the frontier of the subject. It covers classical highlights in the field such as Fatou theorems and some Nevanlinna theory, as well as more recent topics, for example, The corona theorem and the H1-BMO duality. The reader is expected to have an understanding of basic integration theory and functional analysis. Many exercises illustrate and sharpen the theory, and extended exercises give the reader an active part in complementing the material presented in the text.

复分析中的若干论题 目录

Preface
Preliminaries
1. Notation
2. Some Facts
1 Some Basic Properties of Analytic Functions
1. Conformal Mappings
2. Power Series Expansions and Residues
3. Global Cauchy Theorems
2 Properties of Analytic Mappings
1. Conformal Mappings
2. The Riemann Sphere and Projective Space
3. Univalent Functions
4. Picard's Theorems
3 Analytic Approximation and Continuation
1. Approximation with Rationals
2. Mittag-Leffler's Theorem and the Inhomogeneous Cauchy-Riemann Equation
3. Analytic Continuation
4. Simply Connected Domains
5. Analytic Functionals and the Fourier-Laplace
6. Mergelyan's Theorem
4 Harmonic and Subharmonic Functions
1. Harmonic Functions
2. Subharmonic Functions
5 Zeros, Growth, and Value Distribution
1. Weierstrass' Theorem
2. Zeros and Growth
3. Value Distribution of Entire Functions
6 Harmonic Functions and Fourier Series
1. Boundary Values of Harmonic Functions
2. Fourier Series
7 Hp Spaces
1. Factorization in Hp Spaces
2. Invariant Subspaces of H2
3. Interpolation of H8
4. Carleson Measures
8 Ideals and the Corona Theorem
1. Ideals in A
2. The Corona Theorem
9 H1 and BMO
1. Bounded Mean Oscillation
2. The Duality of H1 and BMO
Bibliography
List of Symbols
Index