[教材书籍] [下载]原版普林斯顿复分析：Complex Analysis [推广有奖]

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Princeton Lectures in Analysis系列教材的第二册复分析Complex Analysis,原版PDF格式，很清晰Complex Analysis Elias M. Stein &amp; Rami Shakarchi 2003 | 392 pp. TABLE OF CONTENTS:Foreword vii Introduction xv Chapter 1. Preliminaries to Complex Analysis 1 &nbsp;&nbsp;&nbsp; &nbsp;1 Complex numbers and the complex plane 1 &nbsp;&nbsp;&nbsp;&nbsp; 2 Functions on the complex plane 8 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3 Integration along curves 18 &nbsp;&nbsp;&nbsp; &nbsp;4 Exercises 24 Chapter 2. Cauchy's Theorem and Its Applications 32 &nbsp;&nbsp;&nbsp;&nbsp; 1 Goursat's theorem 34 &nbsp;&nbsp;&nbsp; &nbsp;2 Local existence of primitives and Cauchy's theorem in a disc 37 &nbsp;&nbsp;&nbsp;&nbsp; 3 Evaluation of some integrals 41 &nbsp;&nbsp;&nbsp;&nbsp; 4 Cauchy's integral formulas 45 &nbsp;&nbsp;&nbsp;&nbsp; 5 Further applications 53 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6 Exercises 64 &nbsp;&nbsp;&nbsp; &nbsp;7 Problems 67 Chapter 3. Meromorphic Functions and the Logarithm 71 &nbsp;&nbsp;&nbsp; &nbsp;1 Zeros and poles 72 &nbsp;&nbsp;&nbsp;&nbsp; 2 The residue formula 76 &nbsp;&nbsp;&nbsp;&nbsp; 3 Singularities and meromorphic functions 83 &nbsp;&nbsp;&nbsp;&nbsp; 4 The argument principle and applications 89 &nbsp;&nbsp;&nbsp;&nbsp; 5 Homotopies and simply connected domains 93 &nbsp;&nbsp;&nbsp; &nbsp;6 The complex logarithm 97 &nbsp;&nbsp;&nbsp;&nbsp; 7 Fourier series and harmonic functions 101 &nbsp;&nbsp;&nbsp;&nbsp; 8 Exercises 103 &nbsp;&nbsp;&nbsp; &nbsp;9 Problems 108 Chapter 4. The Fourier Transform 111 &nbsp;&nbsp;&nbsp; &nbsp;1 The class F 113 &nbsp;&nbsp; &nbsp; 2 Action of the Fourier transform on F 114 &nbsp;&nbsp;&nbsp;&nbsp; 3 Paley-Wiener theorem 121 &nbsp;&nbsp;&nbsp; &nbsp;4 Exercises 127 &nbsp;&nbsp;&nbsp; &nbsp;5 Problems 131 Chapter 5. Entire Functions 134 &nbsp;&nbsp;&nbsp; &nbsp;1 Jensen's formula 135 &nbsp;&nbsp;&nbsp; &nbsp;2 Functions of finite order 138 &nbsp;&nbsp;&nbsp;&nbsp; 3 Infinite products 140 &nbsp;&nbsp;&nbsp;&nbsp; 4 Weierstrass infinite products 145 &nbsp;&nbsp;&nbsp; &nbsp;5 Hadamard's factorization theorem 147 &nbsp;&nbsp;&nbsp;&nbsp; 6 Exercises 153 &nbsp;&nbsp;&nbsp; &nbsp;7 Problems 156 Chapter 6. The Gamma and Zeta Functions 159 &nbsp;&nbsp;&nbsp;&nbsp; &nbsp;1 The gamma function 160 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 The zeta function 168 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Exercises 174 &nbsp;&nbsp;&nbsp; &nbsp; 4 Problems 179 Chapter 7. The Zeta Function and Prime Number Theorem 181 &nbsp;&nbsp;&nbsp; &nbsp;1 Zeros of the zeta function 182 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2 Reduction to the functions v and v1 188 &nbsp;&nbsp;&nbsp;&nbsp; 3 Exercises 199 &nbsp;&nbsp;&nbsp;&nbsp; 4 Problems 203 Chapter 8. Conformal Mappings 205 &nbsp;&nbsp;&nbsp;&nbsp; 1 Conformal equivalence and examples 206 &nbsp;&nbsp;&nbsp;&nbsp; 2 The Schwarz lemma; automorphisms of the disc and upper half-plane 218 &nbsp;&nbsp;&nbsp;&nbsp; 3 The Riemann mapping theorem 224 &nbsp;&nbsp;&nbsp;&nbsp; 4 Conformal mappings onto polygons 231 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;5 Exercises 248 &nbsp;&nbsp;&nbsp; &nbsp;6 Problems 254 Chapter 9. An Introduction to Elliptic Functions 261 &nbsp;&nbsp;&nbsp; &nbsp;1 Elliptic functions 262 &nbsp;&nbsp;&nbsp;&nbsp; 2 The modular character of elliptic functions and Eisenstein series 273 &nbsp;&nbsp;&nbsp;&nbsp; 3 Exercises 278 &nbsp;&nbsp;&nbsp; &nbsp;4 Problems 281 Chapter 10. Applications of Theta Functions 283 &nbsp;&nbsp;&nbsp; &nbsp;1 Product formula for the Jacobi theta function 284 &nbsp;&nbsp;&nbsp;&nbsp; 2 Generating functions 293 &nbsp;&nbsp;&nbsp;&nbsp; 3 The theorems about sums of squares 296 &nbsp;&nbsp;&nbsp;&nbsp; 4 Exercises 309 &nbsp;&nbsp;&nbsp; &nbsp;5 Problems 314 Appendix A: Asymptotics 318 &nbsp;&nbsp; &nbsp; 1 Bessel functions 319 &nbsp;&nbsp; &nbsp; 2 Laplace's method; Stirling's formula 323 &nbsp;&nbsp;&nbsp;&nbsp; 3 The Airy function 328 &nbsp;&nbsp; &nbsp; 4 The partition function 334 &nbsp;&nbsp;&nbsp;&nbsp; 5 Problems 341 Appendix B: Simple Connectivity and Jordan Curve Theorem 344 &nbsp;&nbsp;&nbsp;&nbsp; 1 Equivalent formulations of simple connectivity 345 &nbsp;&nbsp;&nbsp; &nbsp;2 The Jordan curve theorem 351 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Notes and References 365 Bibliography 369 Symbol Glossary 373 Index 375 &nbsp;