Concise Complex Analysis (Revised Edition)

by Sheng Gong, Youhong Gong

Overview

A concise textbook on complex analysis for undergraduate and graduate students, this book is written from the viewpoint of modern mathematics: the Bar {Partial}-equation, differential geometry, Lie groups, all the traditional material on complex analysis is included. Setting it apart from others, the book makes many statements and proofs of classical theorems in complex analysis simpler, shorter and more elegant: for example, the Mittag-Leffer theorem is proved using the Bar {Partial}-equation, and the Picard theorem is proved using the methods of differential geometry.

Product Details

ISBN-13:	9789812706935
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/27/2007
Edition description:	Revised
Pages:	260
Product dimensions:	6.10(w) x 9.00(h) x 0.80(d)

Table of Contents

Preface to the Revised Edition     vii
Preface to the First Edition     ix
Foreword     xi
Calculus     1
A Brief Review of Calculus     1
The Field of Complex Numbers, The Extended Complex Plane and Its Spherical Representation     8
Derivatives of Complex Functions     11
Complex Integration     17
Elementary Functions     19
Complex Series     26
Exercise I     29
Cauchy Integral Theorem and Cauchy Integral Formula     39
Cauchy-Green Formula (Pompeiu Formula)     39
Cauchy-Goursat Theorem     44
Taylor Series and Liouville Theorem     52
Some Results about the Zeros of Holomorphic Functions     59
Maximum Modulus Principle, Schwarz Lemma and Group of Holomorphic Automorphisms     64
Integral Representation of Holomorphic Functions     69
Exercise II     75
Partition of Unity     82
Theory of Series of Weierstrass     85
Laurent Series     85
Isolated Singularity     90
Entire Functions and Meromorphic Functions     93
Weierstrass Factorization Theorem, Mittag-Leffler Theorem and Interpolation Theorem     97
Residue Theorem     106
Analytic Continuation     113
Exercise III     117
Riemann Mapping Theorem     123
Conformal Mapping     123
Normal Family     128
Riemann Mapping Theorem     131
Symmetry Principle     134
Some Examples of Riemann Surface     136
Schwarz-Christoffel Formula     138
Exercise IV     141
Riemann Surface     143
Differential Geometry and Picard Theorem     145
Metric and Curvature     145
Ahlfors-Schwarz Lemma     151
The Generalization of Liouville Theorem and Value Distribution     153
The Little Picard Theorem     154
The Generalization of Normal Family     156
The Great Picard Theorem     159
Exercise V     162
Curvature     163
A First Taste of Function Theory of Several Complex Variables     169
Introduction     169
Cartan Theorem     172
Groups of Holomorphic Automorphisms of The Unit Ball and The Bidisc     174
Poincare Theorem     179
Hartogs Theorem     181
Elliptic Functions     185
The Concept of Elliptic Functions     185
The Weierstrass Theory     191
The Jacobi Elliptic Functions     197
The Modular Function     200
The Riemann [zeta]-Function and The Prime Number Theorem     207
The Gamma Function     207
The Riemann [zeta]-function     211
The Prime Number Theorem     218
The Proof of The Prime Number Theorem     222
Bibliography     231
Index     235