Customer Reviews

Complex Analysis

5 star:		(10)
4 star:		(4)
3 star:		(3)
2 star:		(2)
1 star:		(6)

 

 

This is a classic complex analysis text, a pleasure to read and covering all the usual topics. The prerequisites are modest; ideally, one will be familiar with the material in Rudin's "Princples of Mathematical Analysis," but a good, mathematically oriented calculus course (Spivak's "Calculus" is beautiful) is quite sufficient.

That said, the price tag is ridiculous. It was bad enough at $90 (judging by previous reviewers, that was back in the ancient days of 2001). The last edition of this book is dated 1979. It's used in graduate courses all around the world. That means that used copies are not hard to come by.

For $143, one can buy a used copy of Ahlfors, and *new* copies of Conway's and Needham's complex analysis books, and still have pocket change left. That's the course I would recommend.

This book has for decades been THE classic graduate level text in Complex Analysis. It is important to point out that it is not for beginners. To learn complex analysis from the ground up, my own recommendations are the book by Saff & Snider or the somewhat dated, but delightfully conversational book by Stewart and Tall.

Not only does this book require some previous understanding of Complex Analysis, but it also requires that mysterious ability called "mathematical maturity" - the ability to fill in omitted steps and details when following an argument. But, for a person posessing the prerequisites, this is a fine book.

However, any review of this book would be incomplete if it didn't address the issue of price. Advanced math books are all expensive, it is true. But this book is a particularly egregious case of price-gouging. For one thing, the book was written many years ago, so the publisher is not trying to recover any recent high cost of paying the author for his work. Secondly, the book is only something like 336 pages long (much shorter, for example, than a mystery novel by Elizabeth George). It comes out to about 40 cents per page!

Math students, as a rule, are not wealthy people. The price of this book is simply offensive. You can save more than 25% off the price of this book and get BOTH volumes of the Conway book, "Functions of One Complex Variable". I'm not thoroughly familiar with that Conway book, but I've browsed it online. It seems to be well written and has more material (in the two volumes together) than this (Ahlfors) book has. Furthermore, just in principle, I don't think a publisher should be rewarded for this kind of unwarranted greed and price-gouging. Refuse to buy this until the price becomes more reasonable.

I was a (French)graduate student in France some 25 years ago and I would have been delighted to use this book if translated in French; I had to rely on Cartan's book which is a very good book too but which takes for granted that one already knows quite a lot on complex numbers, series, convergence and topology...As a substitute to Cartan, there was a translation of Rudin's real and complex analysis which begins with measure theory...Anyway, it is very difficult to learn this subject in any book without advice from instructors and attending lectures.

There could be more worked examples in this book but it is not a self teaching book (neither is Cartan's...which is very similar in essence to Ahlfors but more narrow minded). For a more "basic" book in the subject, see Marsden's Basic complex analysis but proofs are often mixed up with exercises...which does not suit everybody. My final point is the following: this book contains much more stuff to work at or to think about than its French counterpart; moreover,in this book, efforts are made to avoid formalism (Bourbaki?). US maths students are very lucky indeed. But the book is certainly too expensive.

This book has been, since its first edition in 1953, the standard textbook for rigorously learning complex analysis, and not without a reason. The wonderful theory of this branch of mathematics is appropriately emphasized and thoroughly constructed, leading to more general and precise results than most textbooks. While the constant appearance of new texts on the field can only help appreciate the subject from a different perspective, few give you such a deep and serious treatment like this gem.

Postscript: An earlier reviewer claims that Ahlfors never defines the set of complex numbers, while this is indeed done in the fourth through sixth pages in a much more analytical way than generally found elsewhere. It is quite possible to dislike this author's style or approach (or anybody's for that matter), but it would be difficult to charge Ahlfors with being sloppy with his writing.

This classic is a brilliant exposition of the Riemann (geometrical) method of complex analysis as opposed to the Weierstrassian (power series) method. The latter approach is done well by Whittaker & Watson or Henrici. Ahlfors book is the best I know of for the geometrical approach. It is written for senior undergraduates or graduate students majoring in mathematics.

I picked up this book as a text to my complex functions class. The topics presented in the book is the classic need-to-know materials for undergraduates (complex functions, analytic functions as mappings, complex integration, series and products, etc), plus other topics which undergraduate complex analysis course usually omits: Weirstrass theory, Picard's theorem and zeta function (from complex analysis point of view). The presentation is clear, the mathematic is well presented (but with a few gaps in the proofs), the examples are motivated and useful and the exercises are ok (some of them are pretty challenging!). The book should serve as a text very well.

PS: Lars V. Ahlfors was the first recipient of the Fields Medal (in 1936, along with Jesse Douglas).

How can anyone fail to read this book? The exposition is rigorous, coherent, precise without being either pedantic or overwhelming. A certain level of mathematical maturity is requisite, such as one might acquire in the course of digesting Rudin's "Principles of Mathematical Analysis" or Apostol's book. This is not a compendium of results and exercises for engineers or physicists, it is a concise introductory text in pure mathematics. In that sense it is too abstract and proof oriented for that aforementioned audience which would be better served by a text in mathematical methods. Even pure mathematics students would benefit from supplementing this book with more detailed, computationally oriented books such as Conway or Boas. It's unrealistic to expect to find everything in one text and to further expect it to remain cogent and approachable. Ahlfor's beautiful little book has justifiably remained a classic for four decades.

This is a superb book for anyone who is already familiar with complex analysis but wants to strengthen thier knowledge (although I learnt the subject from scratch thanks to it). In comparison with other books in the field, it is wonderful in that it emphasises important theorems for what they are. The author does not shy away from introducing and using topological methods and choosing the most natural definitions.

Another classic text from graduate school (text for class taught by P.L. Duren) providing a background in introductory complex analysis. This book is nicely written with some elegant exploration of the motivations and backgound for a number of the central concepts. This may be surprising given the physical slimness of the text (I noticed elegance of the exposition and attention to motivation on a recent reread of some of the book after nearly twenty years -- I had not remembered this exposition, perhaps because the reading in graduate school was not quite as "liesurely" (unless "fear driven" and "pressured" are synonyms for "liesurely"). The theory topics are nicely covered -- if, however, you are an engineer looking for methods of calculating complex intgral there are other texts.

The subtitle of this text ought to be "A Rigorous Ode to the Theory of Analytic Functions of One Complex Variable." It enthusiastically emphasizes the most important theoretical aspects which, when understood, render examples presented in more elementary treatises illustrative. Given its occasional terseness and the inexhaustible and good bibliography available on the subject, it probably should not be the first text to buy. My piece of advice: go to a university library (a non-virtual bookstore seldom has more than a few), check the QA 331 section and see for yourself what fits your needs best. If you are genuinely interested in the theory, however, do keep this book in mind. Almost half a century after its first appearance, it is still widely used in introductory graduate courses, and not simply out of habit.