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GTM245Complex Analysis
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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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