Visual Complex Analysis

Complex analysis can challenge the intuition of the new student. This text is unique, among high quality textbooks, in giving a careful and thorough exploration of the geometric meaning underlying the usual algebra and calculus of complex numbers. The Cauchy-Riemann equations define what is meant by a holomorphic function. Restricting to these particularly pleasant and useful functions, the formalism of calculus looks very much like ordinary calculus. Too many students muddle through complex analysis with the notion that it looks like calculus, except some extra nice functions also hold. Such students commonly become competent, but they seldom actually get good at complex analysis. In particular, for many engineers, complex calculus remains an unpalatable mystery, even though they know how to do the calculations. That is the difficulty, and Needham corrects the whole of the difficulty perfectly. Ahlfors is a great classical text. Conway (two volumes) is thorough, clear, and modern. Carrier, Krook, and Pearson is especially concise and well oriented to the practical calculations of engineering and applied science. Berenstein/Gay is very modern and oriented to a very high quality undergraduate or beginning graduate who intends to continue in (very) pure mathematics. All these and more (e.g. Saff) are at least very good or perhaps excellent texts. Because there is a body of problems that beginners are expected to be able to work (mathematics is also a culture---there are expectations), it is probably necessary to pick one of these texts and to use Needham's book as one of two texts for an excellent course. I know of no other book that gives the great intuitive and geometric understanding of complex analysis that Needham gives. I would, under no circumstances, teach any beginning course in complex analysis at any school anywhere at any time for any reason without using Needham as one of the texts. If I were feeling particularly self-satisfied, I might possibly use it as the only text. I myself seldom feel so confident. Perhaps you do. This text is used frequently at M.I.T. and at Oxford. That seems to me a great recommendation. The book is very well and clearly written. The prose flows. It is a great joy to read.

The book is well written with excellent illustrations. Reader should have had a course in Complex variables and advanced Calculus before conquering this text. But even if one does not have the math background, the book is worth wading through... it may get one interested in advanced math topics. When in the 8th grade [millions of years ago] I purchased advanced texts in Advanced Engineering, Vector Analysis and Complex Analysis and looked through the books. I was curious about the math symbols and how they worked. Several years later I had taken courses in these subjects. I would encourage others to browse as it may lead you to a career too.

This book is a real treat. It gives a beautiful visual description of complex differentiation. I was happy to learn how to visualize this due to my "lack of four-dimensional imagination". The author elevates the abstract symbolic mathematics into a living picture of vectors transforming through stretches and twists. He takes the reader into the kitchen and uses a knife and rolling pin to really illustrate conformal mapping and its relationship to complex differentiation. Using the author's words to summarize: "The basic philosophy of this book is that while it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by bringing you nearer to the Truth." If that isn't the Richard Feynman philosophy!

Visual Complex Analysis begins with the basics of the complex plane and motivates everything from a visual, geometric viewpoint. The classical concepts of complex variables are developed in an inspiring, geometric, even artistic manner, distinct from the more typical complex variables books I read in grad school. There is a mathematical "poetry" to this book that encourages one to see mathematics as an art. We need more math books like this. Despite the (deliberate) emphasis on mathematical beauty, the book is quite comprehensive, with almost 600 pages and lots of exercises. It would serve as an introductory textbook for a course in complex variables or as a supplement to other courses.

I first read this book in 2001. I have now re-read it and I hope to re-read it many more times. This is an exceptional book and I am dumbfounded by how illuminating it has been with a second read 11 years later. The author makes a distinct effort to provide deep (principally geometric) insights into complex analysis as well as connections between complex analysis and non-Euclidean geometry as well as physics. The power of visualization and the amplitwist concept is clear. I still need to re-read this book to get the most out of it but the explorations of conformal mappings/analyticity/harmonic functions from the multiple viewpoints was a unique and deeply rewarding experience. See the website: [...]

I have held off on writing a review on this book for some time now. After having read it completely, and, more importantly, having worked with complex variables extensively, I am finally ready to deliver a verdict on it. I applaud the author's effort to visually describe the complex plane: in particularly complex multiplication and integration. He also goes into great detail on Mobius transformations and other geometric concepts. However, I think that he missed the opportunity to describe complex differentials completely. While he speaks of analytic functions being "everywhere aplitwist," he doesn't describe the nature of differentials at analytic points: namely, the differential remains the same, regardless of which path we take from the point. This much more clearly explains the rigidity of analytic functions (along with theorems like FTC, maximum modulus, etc. which follow directly from this rigidity). I believe that he forsakes his own thesis in describing the argument principle in generic topological arguments. These arguments are far more involved than they need to be. More than anything, I dislike how he uses results that haven't been proved. It is quite annoying to use Cauchy's Theorem throughout the book, not proving it till very late. All that said, this is an overall great book that will get you thinking about the concepts. His writing style is very skillful, and, obviously, he provides a lot of figures to help get his point across. It is definitely worth adding to your library, but I think that you will need at least one other text to completely grasp the subject. (I personally recommend Gamelin's book.)

This text ended up being the only required reading for the undergraduate complex analysis course which I am taking this spring. I decided to pick it up early to review and its been nothing but a pleasure. I am a mathematics/biology major but definitely not naturally gifted by any means in regards to math, only through diligent work has Needham's "Visual Complex Analysis" made the plunge into this topic pleasant. I feel the author does a good job a using unique examples and geometric illustrations to connect proofs to more visual descriptive explanations. Everyone is bound to find something in this book that they will like. The only thing I noticed is that the author chooses a style that leads to more theory and proofs then others. I do not mind this approach. There tends to be not as many "plug-chug" examples or problems to work through. Background in vector calculus, linear algebra, and differential equations has helped me in reading this book. Two thumbs up!

The product description page is misleading in that this book is described as a "New Ed Edition". It is not the "new" edition which is mentioned if one reads down the page. In fact, the actual new edition refers to the book's "25th Anniversary" in its title.

Excelente. ele sai do padrão dos livros sobre variáveis complexas e trata por exemplo de geometria não euclidiana em um capítulo inteiro.

One of the best books on Complex Analysis. I've searched for a similar book on other topics in mathematics, but didn't find one.

You need this one if you are a maths freak, specially if you are dealing with complex numbers. ENjoy it.

Tolle Darstellung, endlich kommt die Geometrie zu ihrem Recht

It is a great book. Is the best book for complex analysis.