Complex Analysis

Princeton Lectures in Analysis

Elias M. Stein, Rami Shakarchi

380 pages, parution le 02/06/2003

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Résumé

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.

Introduction
Preliminaries to Complex Analysis
Cauchy's theorem and it's applications
Meromorphic functions and the Logarithm
The Fourier Transform
Entire functions
The gamma and Zeta functions
The Zeta function and prime number theorem
Conformal mappings
An introduction to elliptic functions
Applications of theta functions

A:Asymptotics
B:Simple connectivity and Jordan curve theorem
Notes and reference

L'auteur - Elias M. Stein

Elias M. Stein is Professor of Mathematics at Princeton University.

L'auteur - Rami Shakarchi

Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Elias M. Stein, Rami Shakarchi
Parution	02/06/2003
Nb. de pages	380
Format	16,5 x 24
Couverture	Relié
Poids	715g
Intérieur	Noir et Blanc
EAN13	9780691113852
ISBN13	978-0-691-11385-2