Complex Variables (Dover Books on Mathematics) New edition

This book has just arrived last week at my home and I'm almost finishing the last chapter. I couldn't stop reading it. It's well written, easy to read, but, at the same time, quite rigorous and complete. I never saw a book about complex variables like this. It's true--as some other reviews have said--that some theorems are just stated and not entirely proved (e.g. the Riemann mapping theorem). But, by the other side, there is a great discussion about harmonic functions, the Cauchy integral theorem, the argument principle, conformal mappings, and many other topics.It's important to notice that the approach to complex variables adopted by Flanigan is different from the standard textbooks. The main difference is that he starts discussing calculus on the real plane and only later he develops the complex calculus. His intention is to present first real harmonic functions, which he uses later to define analytic complex functions. Harmonic functions on the real plane become analytic functions on the complex plane, the Green theorem becomes the Cauchy integral theorem, analytic functions are seen as conformal maps, and so on. If you already know real calculus on the plane, this is probably the best way to approach complex calculus. Flanigan is quite convincing in his defense of this approach.It's also important to notice that this is an introductory book designed to beginner students (like a second year undergraduate student in sciences or math). But the book is not interesting only to beginners, since the excellent explanations provided by Flanigan not only clarify many usually obscure points in complex analysis, but also furnish the reader with intuition about how things work in the complex plane. This kind of intuition is useful to any kind of student, at any level.(Comment added in 2013: A few years after I wrote this review, I took a course in complex analysis at the graduate level and this "elementary book" was an absolutely great companion to Ahlfors's "Complex Analysis"! Now, having finished my PhD, I still have the same opinion about Flanigan that I had many years ago. If I had to chose a textbook to teach introductory complex analysis to undergraduate students in math or physics, I would definitely chose Flanigan's. At the undergraduate level, this book is second to none.)To sum up, this is an extraordinary book, extremely well written, which has an interesting (and quite unusual) approach to the complex variables.T. Hartz* * * * *Since there is no "search inside" for this book (actually, there wasn't when I wrote the review), these are the chapters:1. Calculus in the plane (in the real plane, i.e. R^2)2. Harmonic Functions in the Plane (once again, real plane)3. Complex Numbers and Complex Functions4. Integrals and Analytic Functions5. Analytic Functions and Power Series6. Singular Points and Laurent Series7. The Residue Theorem and the Argument Principle8. Analytic Functions as Conformal Mappings

If you're like me, you're looking to supplement whatever textbook was chosen by the professor. As I'm now studying for my final in Complex Analysis, I can say confidently that this book was perfect for that. Not only is it very affordable, but Dr. Flanigan writes in such a way that makes the most difficult concepts seem easy to understand. He doesn't mind putting in those extra few words you need to really turn up that lightbulb where other authors would just assume your "mathematical maturity".The textbook we're using is Complex Variables and Applications, by Brown & Churchill, and is actually a pretty good book on its own. I have also purchased Shilov's Elementary Real and Complex Analysis, which was not much help at all for this course, as well as Palka's An Introduction to Complex Function Theory, which was much thicker and yet still not nearly as easy to follow as Flanigan. I also bought the amazing pair of books Theory of Functions of a Complex Variable, by A. I. Markushevich, and yeah, those really are wonderful books for anyone studying complex analysis at any level, but they are also pretty expensive (although totally worth it if you like variety). Flanigan's book is actually more readable than even Markushevich in some places, and yet is more precisely restricted to undergraduate topics and written in the most relaxed style a math book can be written in. It is actually reminiscent of Sylvanus P. Thompson's wonderful Calculus Made Easy.

Not only is this book is remarkably clear, but it also makes important connections between complex analysis and geometry, harmonic functions, and other branches of mathematics. There are problems at the end of each section that have a broad range in difficulty so that the reader many challenge themselves as much or as little as they wish. Solutions are provided for many of the problems so that the reader may check themselves for correctness and/or work backwards when they are stuck on such a problem. This book has everything that I look for in an introductory text on a subject in math or science.

I have couple of other books on Complex analysis or Complex variables.I like this book best.

It was everything that I had.hoped and more!.

The other reviewers have already done a terrific job describing the content. I'll just add that while a profesional/accademic mathematician might find this book a bit informal, as an amateur I really appreciate its appeal to intuition and the author's tendency to review the meaning of terminology for a few of it's subsequent uses immediately following it's introduction. Not a difficult read, but a working knowledge of calculus for one variable is prerequisite. Seems to be out of print as of this date -- I've had good luck with used book purchases through Amazon marketplace from dealers with positive ratings better than 95%.

Sure, if you get Complex Analysis this just might be a good book. It's certainly not if it's the first time you're taking the course. Find yourself another complex analysis book, because this one just isn't all that great.