Complex Variables (Dover Books on Mathematics)

Rating:
Summary: Excellent excercises, poor explainations
Review: I used this book for a complex variable course as part of my engineering study. I found this book very insufficient to explain things well as a beginner in the subject. There were often problems I could not understand because the book only offered a couple line explaination and no example. In fact almost all the class thought the book was terrible and didn't even read it. Now my school Rochester Institute of Technology has returned to its previous book. If you are well versed in math and want to explore another realm, go ahead with this book, but if you are no mathematician try something else.

Rating:
Summary: Great text
Review: It is wonderful to see this great book on undergraduate complex analysis back in print at even a more affordable price. I've used it in in one of my junior level courses and been totally satisfied with it. I will use a part of it again in a continuation course.

What is nice about this book is that it is a textbook, and not a cookbook nor a book that tries to include everything and fails at all of them. This book never lists too many results; instead it aims at the understanding of the subject matter. Its treatment of Cauchy's theorem clearly exposes the fact that different points of view (derivatives, series, integrals) in the complex plane lead to the same object, analytic functions. The sections on geometric and applied topics, such as linear fractional transformations and fluid mechanics, are a delight to read.

The book assumes nothing other than calculus (Green's theorem) as background. Topological concepts are kept at a reasonable level and some are introduced later when necessary so as not to hinder the development of its main topic. Some short side issues are discussed in tiny sections within the exercises. There are also plenty of regular exercises ranging from elementary calculations to rigorous proofs. This book also contains an appendix that I love on the zeros of polynomials, including the cubic and the quartic.

What attracted me most in this book is that one can read it straight through. There are no secondary undeveloped paths, sections to omit, unnecessary details, or long list of formulas. I recomend it for any course or self-study at the introductory level complex analysis.

Rating:
Summary: mediocre
Review: This book contains a wealth of information, and at this price, one really shouldn't complain, but if you're looking to really understand complex analysis as a mathematical subject, keep looking.
My main problem with the book is that while it states plenty of facts on the subject, the explanations for them (i.e. proofs, examples, counterexamples) are unclear, incomplete, or absent. There's enough "theorem"... "proof" talk to scare off those unaccustomed to it, but the information contained therein is often of little use to those who ARE. In trying to cover both bases, I think the author ends up failing both.
If you're looking to learn complex variables for applications, find a book that covers just that. If you're looking for analysis, do likewise.
I was impressed by Stein and Shakarchi's "Complex Analysis", which gives an introductory, but thorough and sufficiently rigorous, treatment of the subject.