Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable

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 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 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.

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.