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complex analysis, elias stein, rami shakarchi [PDF] .

### Complex Analysis by Elias M. Stein, Rami Shakarchi

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Complex Analysis written by Elias M. Stein, Rami Shakarchi
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)

Book Detail :-
Title: Complex Analysis
Edition:
Author(s): Elias M. Stein, Rami Shakarchi
Publisher: Princeton University Press
Series: Princeton Lectures in Analysis, No. 2
Year: 2003
Pages: 398
Type: PDF
Language: English
ISBN: 0691113858,9780691113852
Country: US
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Author Elias Menachem Stein was an American mathematician who was famous because of his work in the field of harmonic analysis. He was professor of Mathematics at Princeton University from 1963 until his death in 2018.
Author Menachem Stein was born in Antwerp Belgium, to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium. In 1940, the Stein family move to the United States. He graduated from Stuyvesant High School in 1949, where he was classmates with future Fields Medalist Paul Cohen, before moving on to the University of Chicago for college. In 1955, Stein earned a Ph.D. from the University of Chicago under the direction of Antoni Zygmund. He began teaching in MIT in 1955, moved to the University of Chicago in 1958 as an assistant professor, and in 1963 became a full professor at Princeton.
Stein worked primarily in the field of harmonic analysis, and made contributions in both extending and clarifying Calderón–Zygmund theory. These include Stein interpolation, the Stein maximal principle, Stein complementary series representations, Nikishin–Pisier–Stein factorization in operator theory, the Tomas–Stein restriction theorem in Fourier analysis, the Kunze–Stein phenomenon in convolution on semisimple groups, the Cotlar–Stein lemma concerning the sum of almost orthogonal operators, and the Fefferman–Stein theory of the Hardy space and the space of functions of bounded mean oscillation.

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Book Contents :-
Complex Analysis written by Elias M. Stein, Rami Shakarchi cover the following topics.
Foreword
Introduction
1. Preliminaries to Complex Analysis
2. Cauchy’s Theorem and Its Applications
3. Meromorphic Functions and the Logarithm
4. The Fourier Transform
5. Entire Functions
6. The Gamma and Zeta Functions
7. The Zeta Function and Prime Number Theorem
8. Conformal Mappings
9. An Introduction to Elliptic Functions
10. Applications of Theta Functions
Appendix A: Asymptotics
Appendix B: Simple Connectivity and Jordan Curve Theorem
Notes and References
Bibliography
Symbol Glossary
Index

Note:-

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