Customer Reviews

Complex Analysis (Princeton Lectures in Analysis)

5 star:		(6)
4 star:		(3)
3 star:		(1)
2 star:		(0)
1 star:		(1)

 

 

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!

The two authors are indeed very good writers. This book presents the elements of complex analysis at the graduate level (so the assumption is that the reader has gone through undergraduate real and complex analysis). All the topics covered are covered well (I especially like their treatment of the Prime Number Theorem and Elliptic Functions). Note: theorems of Picard and Mittag-Leffler are not proved in the textbook - they are actually assigned as exercises for the reader to prove). If you need the proofs of these theorems, look them up elsewhere. Overall, a very solid book.

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.

If possible, I would give this book 4.5 starts, just because I don't think it is quite as good as the great classic by Conway, "Functions of One Complex Variable" (which treats all the standard topics). On the other hand, Stein and Shakarchi's book is beautiful, lucid, and obviously written by one of the grandmaster analysts of our time. Also, I give it five stars for including a beautiful treatment of the Paley-Wiener theorem, a topic that doesn't usually make its way into elementary complex analysis texts. Finally, this book has the best treatment I've seen of the Hadamard factorization theorem.

Very good first book for complex analysis.

Pros:
- Get to the point real quick. All the basic big theorems (Cauchy integral formula, open mapping theorem etc) are in the first three chapters. Even talks about Runge's theorem early on, which is nice.
- Is also a good introduction to analytic number theory. Built up basics of Gamma function and zeta function which cumulates to the proof of prime number theorem.

Cons:
- If you only want to learn about computations, this is not the book for you.

I strongly prefer this to Ahlfors. (I never understand why people like it)

This book is an excellent treatment of complex analysis, and ranks with the top math books on any subject. As a self-studier and a very isolated, I am particularly dependent on the books I use. This book far surpassed any that I have encountered for c.a. (although I have not looked at the pricey Ahlfors).

It's developments are elegant, self-contained, and accessible. The problems are relevant and leading. The presentation of material (as in the usual curriculum) of applications following Cauchy's integral formulas is outstanding. I really came away with a feeling of confidence in my understanding.

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.

I used this book as a first year graduate student in pure mathematics. The book covers a big part of what we can call classical material, but in a modern treatment. The treatment is beautiful comparing to many other books. The authors do not assume more than Rudin's book, and maybe some elementary course in complex analysis. I also liked this book because , at the end of each chapter, it contains a lot of exercises which are divided into Exercises and Problems. However, tThere are no answers included, and some important topics are not included in this book: Mobius transformation, analytic continuation. I believe buying this book with Lang's book as well will allow the student to have a comprehensive study of complex analysis. If you have a limited budget, I would recommend Lang's book instead.

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.

Exactly same item as I expected.
This book is a little hard for undergrad student.
There is no solution for exercise question.