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4.5
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
