Complex Analysis

by Stein, Elias M.; Shakarchi, Rami

• ISBN13: 9781400831159
• ISBN10: 1400831156
• Copyright: 2010-04-15
• Publisher: Princeton Univ Pr

Summary

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis,Complex Analysiswill be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of whichComplex Analysisis the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

Table of Contents

Foreword	p. vii
Introduction	p. xv
Preliminaries to Complex Analysis	p. 1
Complex numbers and the complex plane	p. 1
Basic properties	p. 1
Convergence	p. 5
Sets in the complex plane	p. 5
Functions on the complex plane	p. 8
Continuous functions	p. 8
Holomorphic functions	p. 8
Power series	p. 14
Integration along curves	p. 18
Exercises	p. 24
Cauchy's Theorem and Its Applications	p. 32
Goursat's theorem	p. 34
Local existence of primitives and Cauchy's theorem in a disc	p. 37
Evaluation of some integrals	p. 41
Cauchy's integral formulas	p. 45
Further applications	p. 53
Morera's theorem	p. 53
Sequences of holomorphic functions	p. 53
Holomorphic functions defined in terms of integrals	p. 55
Schwarz reflection principle	p. 57
Runge's approximation theorem	p. 60
Exercises	p. 64
Problems	p. 67
Meromorphic Functions and the Logarithm	p. 71
Zeros and poles	p. 72
The residue formula	p. 76
Examples	p. 77
Singularities and meromorphic functions	p. 83
The argument principle and applications	p. 89
Homotopies and simply connected domains	p. 93
The complex logarithm	p. 97
Fourier series and harmonic functions	p. 101
Exercises	p. 103
Problems	p. 108
The Fourier Transform	p. 111
The class F	p. 113
Action of the Fourier transform on F	p. 114
Paley-Wiener theorem	p. 121
Exercises	p. 127
Problems	p. 131
Entire Functions	p. 134
Jensen's formula	p. 135
Functions of finite order	p. 138
Infinite products	p. 140
Generalities	p. 140
Example: the product formula for the sine function	p. 142
Weierstrass infinite products	p. 145
Hadamard's factorization theorem	p. 147
Exercises	p. 153
Problems	p. 156
The Gamma and Zeta Functions	p. 159
The gamma function	p. 160
Analytic continuation	p. 161
Further properties of T	p. 163
The zeta function	p. 168
Functional equation and analytic continuation	p. 168
Exercises	p. 174
Problems	p. 179
The Zeta Function and Prime Number Theorem	p. 181
Zeros of the zeta function	p. 182
Estimates for 1/s(s)	p. 187
Reduction to the functions v and v1	p. 188
Proof of the asymptotics for v1	p. 194
Note on interchanging double sums	p. 197
Exercises	p. 199
Problems	p. 203
Conformal Mappings	p. 205
Conformal equivalence and examples	p. 206
The disc and upper half-plane	p. 208
Further examples	p. 209
The Dirichlet problem in a strip	p. 212
The Schwarz lemma; automorphisms of the disc and upper half-plane	p. 218
Automorphisms of the disc	p. 219
Automorphisms of the upper half-plane	p. 221
The Riemann mapping theorem	p. 224
Necessary conditions and statement of the theorem	p. 224
Montel's theorem	p. 225
Proof of the Riemann mapping theorem	p. 228
Conformal mappings onto polygons	p. 231
Some examples	p. 231
The Schwarz-Christoffel integral	p. 235
Boundary behavior	p. 238
The mapping formula	p. 241
Return to elliptic integrals	p. 245
Exercises	p. 248
Problems	p. 254
An Introduction to Elliptic Functions	p. 261
Elliptic functions	p. 262
Liouville's theorems	p. 264
The Weierstrass p function	p. 266
The modular character of elliptic functions and Eisenstein series	p. 273
Eisenstein series	p. 273
Eisenstein series and divisor functions	p. 276
Exercises	p. 278
Problems	p. 281
Applications of Theta Functions	p. 283
Product formula for the Jacobi theta function	p. 284
Further transformation laws	p. 289
Generating functions	p. 293
The theorems about sums of squares	p. 296
The two-squares theorem	p. 297
The four-squares theorem	p. 304
Exercises	p. 309
Problems	p. 314
Asymptotics	p. 318
Bessel functions	p. 319
Laplace's method; Stirling's formula	p. 323
The Airy function	p. 328
The partition function	p. 334
Problems	p. 341
Simple Connectivity and Jordan Curve Theorem	p. 344
Equivalent formulations of simple connectivity	p. 345
The Jordan curve theorem	p. 351
Proof of a general form of Cauchy's theorem	p. 361
Notes and References	p. 365
Bibliography	p. 369
Symbol Glossary	p. 373
Index	p. 375
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