# Complex Analysis

Princeton University Press, Apr 22, 2010 - Mathematics - 400 pages

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.

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### Contents

 Chapter 1 Preliminaries to Complex Analysis 1 Chapter 2 Cauchys Theorem and Its Applications 32 Chapter 3 Meromorphic Functions and the Logarithm 71 Chapter 4 The Fourier Transform 111 Chapter 5 Entire Functions 134 Chapter 6 The Gamma and Zeta Functions 159 Chapter 7 The Zeta Function and Prime Number Theorem 181 Chapter 8 Conformal Mappings 205
 Chapter 10 Applications of Theta Functions 283 Asymptotics 318 Simple Connectivity and Jordan Curve Theorem 344 Notes and References 365 Bibliography 369 Symbol Glossary 373 Index 375 Copyright

 Chapter 9 An Introduction to Elliptic Functions 261

### About the author (2010)

Elias M. Stein is Professor of Mathematics at Princeton University. Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002.