Concise Complex Analysis

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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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Product Details

  • ISBN-13: 9789810243791
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 3/9/2001
  • Pages: 196
  • Product dimensions: 6.30 (w) x 8.52 (h) x 0.45 (d)

Table of Contents

Ch. I Calculus 1
1.1 A glimpse of calculus 1
1.2 Complex number field, extended complex plane and spherical representation 8
1.3 Complex differentiation 11
1.4 Complex integration 16
1.5 Elementary functions 19
1.6 Complex series 26
Ch. II Cauchy integral theorem and Cauchy integral formula 37
2.1 Cauchy-Green formula (Pompeiu formula) 37
2.2 Cauchy-Goursat theorem 41
2.3 Taylor series and Liouville theorem 48
2.4 Some results on zero points 55
2.5 Maximum modulus principle, Schwarz lemma, group of holomorphic automorphism 60
2.6 Integral representation of holomorphic function 65
App Partition of unity 77
Ch. III Theory of series of Weierstrass 81
3.1 Laurent series 81
3.2 Isolate singularity 86
3.3 Entire functions and meromorphic functions 89
3.4 Weierstrass factorization theorem, Mittag-Leffler theorem and interpolation theorem 93
3.5 Residue theorem 102
3.6 Analytic continuation 107
Ch. IV Riemann mapping theorem 115
4.1 Conformal mapping 115
4.2 Normal family 120
4.3 Riemann mapping theorem 123
4.4 Symmetric principle 126
4.5 Examples of Riemann surface 128
4.6 Schwarz-Christoffel formula 130
App Riemann surface 135
Ch. V Differential geometry and Picard theorem 137
5.1 Metric and curvature 137
5.2 Ahlfors-Schwar lemma 142
5.3 Extension of Liouville theorem and value distribution 144
5.4 Picard little theorem 146
5.5 Extension of normal family 147
5.6 Picard great theorem 151
App Curvature 154
Ch. VI Elementary facts on several complex variables 159
6.1 Introduction 159
6.2 Cartan theorem 162
6.3 Groups of holomorphic automorphisms of unit ball and unit bidisk 164
6.4 Poincare theorem 168
6.5 Hartogs theorem 170
References 175
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