A Course in Complex Analysis: From Basic Results to Advanced Topics

✍ Scribed by Wolfgang Fischer, Ingo Lieb

• Publisher: Vieweg+Teubner
• Year: 2011
• Tongue: English
• Leaves: 281
• Edition: 2012
• Category: Library

✦ Synopsis

This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course. The book presents the fundamental results and methods of complex analysis and applies them to a study of elementary and non-elementary functions elliptic functions, Gamma- and Zeta function including a proof of the prime number theorem ' and ' a new feature in this context! ' to exhibiting basic facts in the theory of several complex variables. Part of the book is a translation of the authors' German text 'Einfuhrung in die komplexe Analysis'; some material was added from the by now almost 'classical' text 'Funktionentheorie' written by the authors, and a few paragraphs were newly written for special use in a master's programme. Content Analysis in the complex plane - The fundamental theorems of complex analysis - Functions on the plane and on the sphere - Integral formulas, residues and applications - Non-elementary functions - Meromorphic functions of several variables - Holomorphic maps: Geometric aspects Readership Advanced undergraduates bachelor students and beginning graduate students master's programme Lecturers in mathematics About the authors Professor Dr. Ingo Lieb, Department of Mathematics, University of Bonn Professor Dr. Wolfgang Fischer, Department of Mathematics, University of Bremen

✦ Table of Contents

Cover......Page 1
A Course in Complex Analysis: From Basic Results to Advanced Topics......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
0. Notations and basic concepts......Page 10
1. Holomorphic functions......Page 12
2. Real and complex differentiability......Page 16
Exercises......Page 21
3. Uniform convergence and power series......Page 22
4. Elementary functions......Page 27
5. Integration......Page 33
Exercises......Page 40
6. Several complex variables......Page 41
Exercises......Page 44
1. Primitive functions......Page 45
2. The Cauchy integral theorem......Page 49
Exercises......Page 52
3. The Cauchy integral formula......Page 53
4. Power series expansions of holomorphic functions......Page 57
Exercises......Page 63
5. Convergence theorems, maximum modulus principle, and open mapping theorem......Page 64
Exercises......Page 69
6. Isolated singularities and meromorphic functions......Page 70
7. Holomorphic functions of several variables......Page 75
Exercises......Page 79
1. The Riemann sphere......Page 80
2. Polynomials and rational functions......Page 84
3. Entire functions......Page 89
4. Möbius transformations......Page 91
5. Logarithms, powers, and roots......Page 96
6. Partial fraction decompositions......Page 104
7. Product Expansions......Page 111
Exercises......Page 116
1. The general Cauchy integral theorem......Page 117
2. The inhomogeneous Cauchy integral formula......Page 126
3. Laurent decomposition and Laurent expansion......Page 128
4. Residues......Page 132
Exercises......Page 135
5. Residue calculus......Page 136
Exercises......Page 145
6. Counting zeros......Page 146
7. The Weierstrass preparation theorem......Page 150
Exercises......Page 156
1. The Γ-function......Page 157
Exercises......Page 164
2. The ζ-function and the Prime Number Theorem......Page 165
Exercises......Page 175
3. The functional equation of the ζ-function......Page 176
4. Elliptic functions......Page 183
Exercises......Page 195
5. Elliptic functions and plane cubics......Page 197
Exercises......Page 201
1. Zero sets of holomorphic functions......Page 202
2. Meromorphic functions......Page 204
3. The inhomogeneous Cauchy-Riemann equation in dimension 1......Page 206
4. The Cauchy-Riemann equations with compact support......Page 208
5. The Cauchy-Riemann equations in a polydisk......Page 210
6. Principal parts: the first Cousin problem......Page 213
Exercises......Page 215
7. Divisors: the second Cousin problem......Page 216
8. Meromorphic functions revisited......Page 220
Exercises......Page 222
1. Holomorphic automorphisms......Page 223
2. The hyperbolic metric......Page 227
3. Hyperbolic geometry......Page 233
Historical remark......Page 243
Exercises......Page 244
4. The Riemann mapping theorem......Page 245
5. Harmonic functions......Page 250
6. Schwarz’s reflection principle......Page 256
Exercises......Page 259
7. The modular map λ......Page 260
Exercises......Page 263
8. Theorems of Picard and Montel......Page 264
Exercises......Page 266
Hints and solutions of selected exercises......Page 267
Bibliography......Page 272
Index of Symbols......Page 274
Index......Page 275