Complex Variables (Dover Books on Mathematics) by Francis J. Flanigan (PDF)

Description

A caution to mathematics professors: Complex Variables does not follow conventional outlines of course material. One reviewer noting its originality wrote: “A standard text is often preferred [to a superior text like this] because the professor knows the order of topics and the problems, and doesn’t really have to pay attention to the text. He can go to class without preparation.” Not so here — Dr. Flanigan treats this most important field of contemporary mathematics in a most unusual way. While all the material for an advanced undergraduate or first-year graduate course is covered, discussion of complex algebra is delayed for 100 pages, until harmonic functions have been analyzed from a real variable viewpoint. Students who have forgotten or never dealt with this material will find it useful for the subsequent functions. In addition, analytic functions are defined in a way which simplifies the subsequent theory. Contents include: Calculus in the Plane, Harmonic Functions in the Plane, Complex Numbers and Complex Functions, Integrals of Analytic Functions, Analytic Functions and Power Series, Singular Points and Laurent Series, The Residue Theorem and the Argument Principle, and Analytic Functions as Conformal Mappings. Those familiar with mathematics texts will note the fine illustrations throughout and large number of problems offered at the chapter ends. An answer section is provided. Students weary of plodding mathematical prose will find Professor Flanigan’s style as refreshing and stimulating as his approach.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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

⭐, by Brown & Churchill, and is actually a pretty good book on its own. I have also purchased Shilov’s

⭐, which was not much help at all for this course, as well as Palka’s

⭐, which was much thicker and yet still not nearly as easy to follow as Flanigan. I also bought the amazing pair of books

⭐, 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

⭐.

⭐Good book on complex subject, even though old, covers the topic well.

⭐Complex analysis can seem like witchcraft with a language of its own. Students can have a hard time resolving theory with physical application, and the way this text meets that objective makes it one of a kind as far as I know. Any geek who views complex analysis with less than full confidence ought to give this book a shot.The author designed his own class on the subject and this book reads like a well-organized collection of lecture notes. Even if you’ve had a course already, his treatment is comfortably painless and, on the connection between theorems of Green and Cauchy, even enlightening. One of Dover’s finest reprints.

⭐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.

⭐Complex analysis is the most beautiful subject in math, and also easy to forget. Students often find that he/she forgot “everything”(no kidding!) about complex analysis even after one or two semesters in school, including me. After reading this book I found out why: I didn’t have a good understanding of plane calculus. The first 1/3 of this book is all about plane calculus, and it is the right way to do.You don’t even have to know so called “calculus on manifold”, the only manifold you need to know is a curve. Every theorems of complex variables in this book follows naturally from the contents of plane calculus, that’ why you won’t forget.

⭐Lo mejor del libro es que está bien escrito y es muy didáctico. No sigue el orden tradicional en los libros de variable compleja, pero está bien, porque comienza con un recordatorio de Cálculo Diferencial e Integral de funciones de 2 variables, y luego pasa a funciones de variable compleja. La edición es aceptable, aunque solamente tiene una pequeña pega, y es que algunas imágenes están algo borrosas. De todas formas se pueden distinguir suficientemente bien.I had to return a new copy of this excellent text because of unacceptable typographical quality. New re-issues of old Dover titles sold by Amazon as of 2020 are “printed on demand” and manufactured very shoddily: the print is uneven and fuzzy (like a b/w photocopy of a photocopy of a photocopy), figures are smudgy, and the paper is low-grade – on each page you can see what is printed on reverse. This is a far cry from Dover’s usually crisp editions of the previous years. Any piece of junk mail has a higher typographical quality than “digitally printed” science books sold today by publishers’ like Dover and Springer.If you want to read Flanigan’s “Complex Variables” – look for a used copy or an older printing.

⭐The books looks good but the quality of the figures is the worst I have ever seen in my entire life!

⭐gift

⭐Very Informative.

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