This book describes a classical introductory part of complex analysis for university students in the sciences and engineering and could serve as a text or reference book. It places emphasis on rigorous proofs, presenting the subject as a fundamental mathematical theory. The volume begins with a problem dealing with curves related to Cauchy's integral theorem. To deal with it rigorously, the author gives detailed descriptions of the homotopy of plane curves. Since the residue theorem is important in both pure and applied mathematics, the author gives a fairly detailed explanation of how to apply it to numerical calculations; this should be sufficient for those who are studying complex analysis as a tool.
Author(s): Junjiro Noguchi
Series: TMM168
Publisher: AMS
Year: 1998
Language: English
Pages: 266
Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Dedication......Page 8
Contents......Page 10
Preface......Page 12
1.1. Complex Numbers......Page 14
1.2. Plane Topology......Page 16
1.3. Sequences and Limits......Page 22
Problems......Page 28
2.1. Complex Functions......Page 30
2.2. Sequences of Complex Functions......Page 32
2.3. Series of Functions......Page 36
2.4. Power Series......Page 38
2.5. Exponential Functions and Trigonometric Functions......Page 43
2.6. Infinite Products......Page 47
2.7. Riemann Sphere......Page 50
2.8. Linear Transformations......Page 53
Problems......Page 59
3.1. Complex Derivatives......Page 62
3.2. Curvilinear Integrals......Page 67
3.3. Homotopy of Curves......Page 75
3.4. Cauchy's Integral Theorem......Page 80
3.5. Cauchy's Integral Formula......Page 85
3.6. Mean Value Theorem and Harmonic Functions......Page 94
3.7. Holomorphic Functions on the Riemann Sphere......Page 100
Problems......Page 102
4.1. Laurent Series......Page 104
4.2. Meromorphic Functions and Residue Theorem......Page 107
4.3. Argument Principle......Page 112
4.4. Residue Calculus......Page 119
Problems......Page 127
5.1. Analytic Continuation......Page 130
5.2. Monodromy Theorem......Page 137
5.3. Universal Covering and Riemann Surface......Page 143
Problems......Page 151
6.1. Linear Transformations......Page 154
6.2. Poincare Metric......Page 157
6.3. Contraction Principle......Page 162
6.4. The Riemann Mapping Theorem......Page 166
6.5. Boundary Correspondence......Page 170
6.6. Universal Covering of C {0, 1}......Page 176
6.7. The Little Picard Theorem......Page 180
6.8. The Big Picard Theorem......Page 183
Problems......Page 187
7.1. Approximation Theorem......Page 192
7.2. Existence Theorems......Page 198
7.3. Riemann-Stieltjes' Integral......Page 205
7.4. Meromorphic Functions on C......Page 207
7.5. Weierstrass' Product......Page 215
7.6. Elliptic Functions......Page 223
Problems......Page 239
Hints and Answers......Page 242
References......Page 254
Index......Page 258
Symbols......Page 262
Correction List......Page 264
Back Cover......Page 266
