Suitable for a one year course in complex analysis, at the advanced undergraduate or graduate level, this is a pretty good introduction to the subject, with well-written, detailed proofs and lots of exercises. If you take the time to work the exercises, you will learn the subject, and you will learn it well, and if you have trouble with some of them, you can look in the back for answers, though if you have this as a textbook, your professor will probably assign problems whose solutions are not in the book.
The chapter on Cauchy's integral theorem is particularly good, describing the material in detail and providing a nice illustration of homotopy of curves. Studied dutifully, this can help a student through one of the thornier areas of the subject, often an area that trips those new to complex analysis. That alone could make the book worth the price.
This is, though, only an introduction, and a student hoping to cover more advanced material is urged to consider William Veech's "A Second Course in Complex Analysis" or some other books suitable for that purpose.
Author(s): Junjiro Noguchi
Series: Translations of mathematical monographs 168
Publisher: American Mathematical Society
Year: 2008
Language: English
Pages: 266
City: Providence, R.I
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
