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Complex Analysis : Princeton Lectures in Analysis - Elias M. Stein

Complex Analysis

Princeton Lectures in Analysis

Hardcover

Published: 7th April 2003
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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 Analysis" will 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 which "Complex Analysis" is 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.

Elias M. Stein, Winner of the 2005 Stefan Bergman Prize, American Mathematical Society

Forewordp. vii
Introductionp. xv
Preliminaries to Complex Analysisp. 1
Complex numbers and the complex planep. 1
Basic propertiesp. 1
Convergencep. 5
Sets in the complex planep. 5
Functions on the complex planep. 8
Continuous functionsp. 8
Holomorphic functionsp. 8
Power seriesp. 14
Integration along curvesp. 18
Exercisesp. 24
Cauchy's Theorem and Its Applicationsp. 32
Goursat's theoremp. 34
Local existence of primitives and Cauchy's theorem in a discp. 37
Evaluation of some integralsp. 41
Cauchy's integral formulasp. 45
Further applicationsp. 53
Morera's theoremp. 53
Sequences of holomorphic functionsp. 53
Holomorphic functions defined in terms of integralsp. 55
Schwarz reflection principlep. 57
Runge's approximation theoremp. 60
Exercisesp. 64
Problemsp. 67
Meromorphic Functions and the Logarithmp. 71
Zeros and polesp. 72
The residue formulap. 76
Examplesp. 77
Singularities and meromorphic functionsp. 83
The argument principle and applicationsp. 89
Homotopies and simply connected domainsp. 93
The complex logarithmp. 97
Fourier series and harmonic functionsp. 101
Exercisesp. 103
Problemsp. 108
The Fourier Transformp. 111
The class Fp. 113
Action of the Fourier transform on Fp. 114
Paley-Wiener theoremp. 121
Exercisesp. 127
Problemsp. 131
Entire Functionsp. 134
Jensen's formulap. 135
Functions of finite orderp. 138
Infinite productsp. 140
Generalitiesp. 140
Example: the product formula for the sine functionp. 142
Weierstrass infinite productsp. 145
Hadamard's factorization theoremp. 147
Exercisesp. 153
Problemsp. 156
The Gamma and Zeta Functionsp. 159
The gamma functionp. 160
Analytic continuationp. 161
Further properties of Tp. 163
The zeta functionp. 168
Functional equation and analytic continuationp. 168
Exercisesp. 174
Problemsp. 179
The Zeta Function and Prime Number Theoremp. 181
Zeros of the zeta functionp. 182
Estimates for 1/s(s)p. 187
Reduction to the functions v and v1p. 188
Proof of the asymptotics for v1p. 194
Note on interchanging double sumsp. 197
Exercisesp. 199
Problemsp. 203
Conformal Mappingsp. 205
Conformal equivalence and examplesp. 206
The disc and upper half-planep. 208
Further examplesp. 209
The Dirichlet problem in a stripp. 212
The Schwarz lemma; automorphisms of the disc and upper half-planep. 218
Automorphisms of the discp. 219
Automorphisms of the upper half-planep. 221
The Riemann mapping theoremp. 224
Necessary conditions and statement of the theoremp. 224
Montel's theoremp. 225
Proof of the Riemann mapping theoremp. 228
Conformal mappings onto polygonsp. 231
Some examplesp. 231
The Schwarz-Christoffel integralp. 235
Boundary behaviorp. 238
The mapping formulap. 241
Return to elliptic integralsp. 245
Exercisesp. 248
Problemsp. 254
An Introduction to Elliptic Functionsp. 261
Elliptic functionsp. 262
Liouville's theoremsp. 264
The Weierstrass p functionp. 266
The modular character of elliptic functions and Eisenstein seriesp. 273
Eisenstein seriesp. 273
Eisenstein series and divisor functionsp. 276
Exercisesp. 278
Problemsp. 281
Applications of Theta Functionsp. 283
Product formula for the Jacobi theta functionp. 284
Further transformation lawsp. 289
Generating functionsp. 293
The theorems about sums of squaresp. 296
The two-squares theoremp. 297
The four-squares theoremp. 304
Exercisesp. 309
Problemsp. 314
Asymptoticsp. 318
Bessel functionsp. 319
Laplace's method; Stirling's formulap. 323
The Airy functionp. 328
The partition functionp. 334
Problemsp. 341
Simple Connectivity and Jord
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691113852
ISBN-10: 0691113858
Series: Princeton Lectures in Analysis
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 400
Published: 7th April 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 24.3 x 16.4  x 2.9
Weight (kg): 0.72