Complex Analysis
By addebook • Oct 3rd, 2008 • Category: Mathematics
Complex Analysis (Princeton Lectures in Analysis, Volume 2)
Complex Analysis (Princeton Lectures in Analysis)
By Elias M. Stein, Rami Shakarchi
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Publisher: Princeton University Press
Number Of Pages: 392
Publication Date: 2003-04-07
ISBN-10 / ASIN: 0691113858
ISBN-13 / EAN: 9780691113852
Binding: Hardcover
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.
Summary: it is just good
Rating: 4
I got a copy of this book. It is a text for undergraduate students in pure mathematics. It is a good reference for elementary proofs of most common theorems in complex variables. However, some important theorems (ej: Three lines lemma and Picard theorem) are placed as exercises and problems. It is not a book for applications in engineering, its applications are taken from number theory. At some places it refers to sections or chapters in other books in the Princeton lectures in analysis. I think this is a four starts book.
Summary: The exercises are not very good
Rating: 3
I used this book in a first year graduate course. I found the exposition not very clear, and the exercises particularly uninteresting. If you have the choice, I definitely recommend Gamelin’s Complex Analysis instead.
Summary: bad book
Rating: 1
This book is not helpful. There are no answers to problems. Symbols used in the problems are not explained. It is difficult to learn unless you have someone explaining the concepts for you. I would not buy this book now.
Summary: Beautifully written !
Rating: 5
This is a very beautifully written book on complex analysis. It is not very easy to read though, especially if you’ve never been exposed to the subject before. Most proofs are clearly presented, and can be easily understood by the mature reader. Other proofs require filling in the gaps to get the whole picture. As far as problems go, there’s a list of relatively easy exercises at the end of each chapter. Following the exercises is a list of problems which require some head scratching. Overall, I had a fun time reading and learning from this book.
Summary: A Gem
Rating: 5
In reviewing a textbook, one should consider the background of the book’s audience. I believe that this text by Stein and Shakarchi on complex analysis is outstanding, and is appropriate for a student who has the background of a course in real analysis at the level of Rudin’s “Principles of Mathematical Analysis”.
The text has a number of strengths. Some of these are the following:
1. The choice of material and the order of presentation are superb. Just to give you a sample, within the first 100 pages, the authors cover Runge’s Theorem, the Schwarz Reflection Principle, Riemann’s Theorem on Removable Singularities, the Casorati-Weierstrass Theorem, Rouche’s Theorem, and the homotopy version of Cauchy’s Integral Theorem. The novice is thus treated to some beautiful mathematics very quickly.
2. The statements of theorems and definitions are simple and clear. The authors carefully avoid unnecessary technicalities that would only tend to confuse the beginner and obfuscate the essential concepts.
3. The proofs are very clear and elegant. The main ideas are emphasized, and just enough details are given so that a diligent student with the background stated above will be able to grasp the arguments.
4. The examples are nontrivial, and worked out in detail. Some may prefer a greater number and variety of examples, but I found that there were enough to illustrate the theory.
5. The authors pay considerable attention to motivating the development of ideas. It seems to me that the authors were keen to enhance the reader’s intuition for the subject and to impart an appreciation for the inherent beauty of complex function theory.
6. The book is very well edited. There are very few typos, none of which should cause difficulty for a beginner.
For these reasons and others, I highly recommend this book to anyone who desires to learn complex analysis, or who simply desires to learn some beautiful mathematics, and who has the suggested background. Stein and Shakarchi have written a book which is a joy to read!
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