Concise Complex Analysis

Concise Complex Analysis

by Sheng Gong
     
 

ISBN-10: 9810243782

ISBN-13: 9789810243784

Pub. Date: 03/28/2001

Publisher: World Scientific Publishing Company, Incorporated

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…  See more details below

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:
9789810243784
Publisher:
World Scientific Publishing Company, Incorporated
Publication date:
03/28/2001
Pages:
196

Table of Contents

Preface
Foreword
Ch. ICalculus1
1.1A glimpse of calculus1
1.2Complex number field, extended complex plane and spherical representation8
1.3Complex differentiation11
1.4Complex integration16
1.5Elementary functions19
1.6Complex series26
Ch. IICauchy integral theorem and Cauchy integral formula37
2.1Cauchy-Green formula (Pompeiu formula)37
2.2Cauchy-Goursat theorem41
2.3Taylor series and Liouville theorem48
2.4Some results on zero points55
2.5Maximum modulus principle, Schwarz lemma, group of holomorphic automorphism60
2.6Integral representation of holomorphic function65
AppPartition of unity77
Ch. IIITheory of series of Weierstrass81
3.1Laurent series81
3.2Isolate singularity86
3.3Entire functions and meromorphic functions89
3.4Weierstrass factorization theorem, Mittag-Leffler theorem and interpolation theorem93
3.5Residue theorem102
3.6Analytic continuation107
Ch. IVRiemann mapping theorem115
4.1Conformal mapping115
4.2Normal family120
4.3Riemann mapping theorem123
4.4Symmetric principle126
4.5Examples of Riemann surface128
4.6Schwarz-Christoffel formula130
AppRiemann surface135
Ch. VDifferential geometry and Picard theorem137
5.1Metric and curvature137
5.2Ahlfors-Schwar lemma142
5.3Extension of Liouville theorem and value distribution144
5.4Picard little theorem146
5.5Extension of normal family147
5.6Picard great theorem151
AppCurvature154
Ch. VIElementary facts on several complex variables159
6.1Introduction159
6.2Cartan theorem162
6.3Groups of holomorphic automorphisms of unit ball and unit bidisk164
6.4Poincare theorem168
6.5Hartogs theorem170
References175

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