Concise complex analysis

"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.

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Gong (Univ. of Science and Technology of China) offers a concise, well-written introduction for readers studying classical complex analysis using modern mathematical notation and techniques. The use of modern techniques allow for shorter, more elegant statements and proofs of many of the more important theorems of the subject. Gong, in chapter 1, explains the basics of the calculus of complex numbers, then treats Cauchy's theorem and integral formula along with Liouville's theorem, the Schwarz lemma, and integral representations in chapter 2. Chapter 3 discusses Laurent series, the Mittag-Leffler theorem, residue calculus, and analytic continuation. The main topic of chapter 4 is the Riemann mapping theorem, while chapter 5 explores differential geometry, concluding with Picard's big theorem. Gong concludes with a brief introduction to several complex variables in chapter 6. The book contains more than 130 exercises and appendixes on partitions of unity, Riemann surfaces, and curvature. Unfortunately, there are only a very few illustrations. Upper-division undergraduates through professionals. D. P. Turner Faulkner University

Preface to the Revised Edition	p. vii
Preface to the First Edition	p. ix
Foreword	p. xi
1 Calculus	p. 1
1.1 A Brief Review of Calculus	p. 1
1.2 The Field of Complex Numbers, The Extended Complex Plane and Its Spherical Representation	p. 8
1.3 Derivatives of Complex Functions	p. 11
1.4 Complex Integration	p. 17
1.5 Elementary Functions	p. 19
1.6 Complex Series	p. 26
Exercise I	p. 29
2 Cauchy Integral Theorem and Cauchy Integral Formula	p. 39
2.1 Cauchy-Green Formula (Pompeiu Formula)	p. 39
2.2 Cauchy-Goursat Theorem	p. 44
2.3 Taylor Series and Liouville Theorem	p. 52
2.4 Some Results about the Zeros of Holomorphic Functions	p. 59
2.5 Maximum Modulus Principle, Schwarz Lemma and Group of Holomorphic Automorphisms	p. 64
2.6 Integral Representation of Holomorphic Functions	p. 69
Exercise II	p. 75
Appendix I Partition of Unity	p. 82
3 Theory of Series of Weierstrass	p. 85
3.1 Laurent Series	p. 85
3.2 Isolated Singularity	p. 90
3.3 Entire Functions and Meromorphic Functions	p. 93
3.4 Weierstrass Factorization Theorem, Mittag-Leffler Theorem and Interpolation Theorem	p. 97
3.5 Residue Theorem	p. 106
3.6 Analytic Continuation	p. 113
Exercise III	p. 117
4 Riemann Mapping Theorem	p. 123
4.1 Conformal Mapping	p. 123
4.2 Normal Family	p. 128
4.3 Riemann Mapping Theorem	p. 131
4.4 Symmetry Principle	p. 134
4.5 Some Examples of Riemann Surface	p. 136
4.6 Schwarz-Christoffel Formula	p. 138
Exercise IV	p. 141
Appendix II Riemann Surface	p. 143
5 Differential Geometry and Picard Theorem	p. 145
5.1 Metric and Curvature	p. 145
5.2 Ahlfors-Schwarz Lemma	p. 151
5.3 The Generalization of Liouville Theorem and Value Distribution	p. 153
5.4 The Little Picard Theorem	p. 154
5.5 The Generalization of Normal Family	p. 156
5.6 The Great Picard Theorem	p. 159
Exercise V	p. 162
Appendix III Curvature	p. 163
6 A First Taste of Function Theory of Several Complex Variables	p. 169
6.1 Introduction	p. 169
6.2 Cartan Theorem	p. 172
6.3 Groups of Holomorphic Automorphisms of The Unit Ball and The Bidisc	p. 174
6.4 Poincare Theorem	p. 179
6.5 Hartogs Theorem	p. 181
7 Elliptic Functions	p. 185
7.1 The Concept of Elliptic Functions	p. 185
7.2 The Weierstrass Theory	p. 191
7.3 The Jacobi Elliptic Functions	p. 197
7.4 The Modular Function	p. 200
8 The Riemann [zeta]-Function and The Prime Number Theorem	p. 207
8.1 The Gamma Function	p. 207
8.2 The Riemann [zeta]-function	p. 211
8.3 The Prime Number Theorem	p. 218
8.4 The Proof of The Prime Number Theorem	p. 222
Bibliography	p. 231
Index	p. 235

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