Concise Complex Analysis (Revised Edition)

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. Cover......Page 1 Title page......Page 3 Date-line......Page 4 Dedication......Page 5 Preface to the Revised Edition......Page 7 Preface to the First Edition......Page 9 Foreword......Page 11 Contents......Page 17 1.1 A Brief Review of Calculus......Page 21 1.2 The Field of Complex Numbers, The Extended Complex Plane and Its Spherical Representation......Page 28 1.3 Derivatives of Complex Functions......Page 31 1.4 Complex Integration......Page 37 1.5 Elementary Functions......Page 39 1.6 Complex Series......Page 46 Exercise I......Page 49 2.1 Cauchy-Green Formula (Poinpeiu Formula)......Page 59 2.2 Cauchy-Goursat Theorem......Page 64 2.3 Taylor Series and Liouville Theorem......Page 72 2.4 Some Results about the Zeros of Holomorphic Functions......Page 79 2.5 Maximum Modulus Principle, Schwarz Lemma and Group of Holomorphic Automorphisms......Page 84 2.6 Integral Representation of Holomorphic Functions......Page 89 Exercise II......Page 95 Appendix I Partition of Unity......Page 102 3.1 Laurent Series......Page 105 3.2 Isolated Singularity......Page 110 3.3 Entire Functions and Meromorphic Functions......Page 113 3.4 Weierstrass Factorization Theorem, Mittag-Lefller Theorem and Interpolation Theorem......Page 117 3.5 Residue Theorem......Page 126 3.6 Analytic Continuation......Page 133 Exercise III......Page 137 4.1 Conformal Mapping......Page 143 4.2 Normal Family......Page 148 4.3 Riemann Mapping Theorem......Page 151 4.4 Symmetry Principle......Page 154 4.5 Some Examples of Riemann Surface......Page 156 4.6 Schwarz-Christoffel Formula......Page 158 Exercise IV......Page 161 Appendix II Riemann Surface......Page 163 5.1 Metric and Curvature......Page 165 5.2 Ahlfors-Schwarz Lemma......Page 171 5.3 The Generalization of Liouville Theorem and Value Distribution......Page 173 5.4 The Little Picard Theorem......Page 174 5.5 The Generalization of Normal Family......Page 176 5.6 The Great Picard Theorem......Page 179 Exercise V......Page 182 Appendix III Curvature......Page 183 6.1 Introduction......Page 189 6.2 Cartan Theorem......Page 192 6.3 Groups of Holomorphic Automorphisms of The Unit Ball and The Bidisc......Page 194 6.4 Poincare Theorem......Page 199 6.5 Hartogs Theorem......Page 201 7.1 The Concept of Elliptic Functions......Page 205 7.2 The Weierstrass Theory......Page 211 7.3 The Jacobi Elliptic Functions......Page 217 7.4 The Modular Function......Page 220 8.1 The Gamma Function......Page 227 8.2 The Rieniann $\\zeta$-function......Page 231 8.3 The Prime Number Theorem......Page 238 8.4 The Proof of The Prime Number Theorem......Page 242 Bibliography......Page 251 Index......Page 255 "This is a concise textbook of complex analysis for undergraduate and graduate students. Written from the viewpoint of modern mathematics - the d-equation, differential geometry, Lie group, etc. it contains all the traditional material on complex analysis. However, many statement and proofs of classical theorems in complex analysis have been made simpler, shorter and more elegant due to modern mathematical ideas and methods. For example, the Mittag-Leffer theorem is proved by the d-equation, the Picard theorem is proved using the methods of differential geometry, and so on."--BOOK JACKET