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[资源] GTM245Complex Analysis

Contents
Preface vii
Standard Notation and Commonly Used Symbols ix
Chapter 1. The Fundamental Theorem in Complex Function
Theory 1
1.1. Some motivation 1
1.2. The Fundamental Theorem 3
1.3. The plan for the proof 4
1.4. Outline of text 5
Chapter 2. Foundations 7
2.1. Introduction and preliminaries 7
2.2. Differentiability and holomorphic mappings 14
Exercises 18
Chapter 3. Power Series 23
3.1. Complex power series 24
3.2. More on power series 32
3.3. The exponential function, the logarithm function,
and some complex trigonometric functions 36
3.4. An identity principle 42
3.5. Zeros and poles 47
Exercises 52
Chapter 4. The Cauchy Theory–A Fundamental Theorem 59
4.1. Line integrals and differential forms 60
4.2. The precise difference between closed and exact forms 65
4.3. Integration of closed forms and the winding number 70
4.4. Homotopy and simple connectivity 72
4.5. Winding number 75
4.6. Cauchy Theory: initial version 78
Exercises 80
Chapter 5. The Cauchy Theory–Key Consequences 83
5.1. Consequences of the Cauchy Theory 83
xii CONTENTS
5.2. Cycles and homology 89
5.3. Jordan curves 90
5.4. The Mean Value Property 93
5.5. On elegance and conciseness 96
5.6. Appendix: Cauchy’s integral formula for smooth functions 96
Exercises 97
Chapter 6. Cauchy Theory: Local Behavior and Singularities
of Holomorphic Functions 101
6.1. Functions holomorphic on an annulus 101
6.2. Isolated singularities 103
6.3. Zeros and poles of meromorphic functions 106
6.4. Local properties of holomorphic maps 110
6.5. Evaluation of definite integrals 113
Exercises 117
Chapter 7. Sequences and Series of Holomorphic Functions 123
7.1. Consequences of uniform convergence on compact sets 123
7.2. A metric on C(D) 126
7.3. The cotangent function 130
7.4. Compact sets in H(D) 134
7.5. Approximation theorems and Runge’s theorem 138
Exercises 146
Chapter 8. Conformal Equivalence 147
8.1. Fractional linear (M¨obius) transformations 148
8.2. Aut(D) for D =

C, C, D, and H2 152
8.3. The Riemann Mapping Theorem 154
8.4. Hyperbolic geometry 158
8.5. Finite Blaschke products 167
Exercises 169
Chapter 9. Harmonic Functions 173
9.1. Harmonic functions and the Laplacian 173
9.2. Integral representation of harmonic functions 176
9.3. The Dirichlet problem 179
9.4. The Mean Value Property: a characterization 186
9.5. The reflection principle 186
Exercises 188
Chapter 10. Zeros of Holomorphic Functions
10.4. The field of meromorphic functions 207
10.5. Infinite Blaschke products 209
Exercises 209
BIBLIOGRAPHICAL NOTES 213
Bibliography 215
Index 217

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