Concise Complex Analysis (Revised Edition) - Sheng Gong

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Concise Complex Analysis (Revised Edition)

By: Sheng Gong, Youhong Gong

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Hardcover | 27 January 2007

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Hardcover
260 Pages

Edition Type
Revised

Dimensions(cm)
22.76 x 16.79 x 1.88

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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.

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

Concise Complex Analysis (Revised Edition)

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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