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Summary: A Tremendously Insightful Presentation of Complex Analysis
Review: Although mathematical visualization has not been as implicitly forbidden in modern mathematics as claimed by Needham, his work is nonetheless highly innovative even besides his wonderful graphs. The reason is that his prose accompanies very well his extraordinary insight and intuition for the subject. It is purposely not extremely rigorous in order to make the presentation smoother. (This is not so bad as many think. Complex analysis is the target of many excellent books which, fortunately, do not all take the same approach. For more rigor see Ahlfors' "Complex Analysis.") This book can therefore be an ideal way to get started with complex analysis or even to further one's understanding in the subject. If you are looking for a very affordable predecessor with a similar intuitive style, check Flanigan's "Complex Variables."
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Summary: excellent book on this subject
Review: i am not a philisopher like some reviewers here...this book is really wonderful and even better.
the subject is very clearly explained in a very educational manner.
the illustrations are phenomenal and better than some books in color.
the printing is superb,what anybody need more in a book?
in short this book is a marvel and insightful,thanx alot prof. Needham.
so sad he only has wrote one book,
i recommend it highly.
Rating: 
Summary: Overly geometric
Review: I have enjoyed working through this book. As others have pointed out, it's a nice geometric introduction to complex analysis. I do, however, have a couple gripes. One major and one very minor. First, Needham seems to strive for a kind of geometric "purity". He tries to give the impression that geometric arguments are more valid than standard logic or algebra. While some might feel this is a needed correction to alleged anti-geometric trends in math, Needham's correction can, at times, be an overreaction. The result is that the book is excellent in those areas that are well-suited to a geometric approach (e.g. Mobius transformations and hyperbolic geometry), but fails in areas for which algebraic approaches are simpler and easier to understand. Parts of the chapters on differentiation are unnecessarily cumbersome. (While the "amplitwist" thing --- a geometric version of complex-differentiation-as-local-multiplication --- is neat, it's a little overdone, and ends up making differentiation sound more mysterious than it is.) Insisting on "pure" geometric arguments is a nice exercise. But when it obscures the subject and makes it more difficult to follow, one begins to see why math has moved away from that kind of reasoning over the past several hundred years. By the time I finished Needham's book, my appreciation for non-geometric mathematics had increased quite a bit. In any case, I think students could learn well from certain chapters of this book (the more geometric ones), but should definitely be steered away from others (differentiation and integration). This would make a great supplementary text but not a good main text. If used as the main text it should certainly be supplemented with less roundabout approaches to differentiation and integration. My second, very tiny gripe: Needham seems to be obsessed with Roger Penrose. Nothing against Penrose, who I'm sure does great things in physics, but the constant references get a little tedious after a while. Having said all of that, I should repeat that I still enjoyed the book very much. It's definitely worth the money, espcially if you've already had some complex analysis and want to see this geometric way of doing things, or if you're currently studying complex analysis and want to develop some intuition. If you're studying complex analysis for the first time, Churchill and Brown would be a better book at a similar level. The book by Ahlfors is the standard more advanced introduction.
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Summary: Lots of material, great pictures but too chatty
Review: I purchased this book as a reference and because of it's coverage on Mobius Transformations, which is great! My qualms are with the other parts of the book, however. I'll reach for this book or Churchill and Brown when I'm dealing with complex numbers. Browns is much more direct and to the point. There are times that I'll have to flip through several pages jsut to get to the point. Needham often includes a history of the topic and several applications before getting to the mathematics of it. I like reading about applications at the end of the chapters and histories as footnotes (or both in a completely seperate part of the book, i.e. the appendix). If you buy this book, you'll get a lot of great mathematics and wonderful visualizations, but expect a lot of reading that may not be immidiately necessary to your studies.
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Summary: Hm.......
Review: I tried to learn complex analysis from Ahlfors, I wouldn't recommend you try it although it is a good book. The problem is there are certain subtleties in complex variables that are NOT obvious. There are few authors of math books that remember that we do not know these subtleties. I could go on a tirade about the general state of math literature for hours, but my only remark here is that in my view most authors seem to be trying to impress someone other than the students, maybe other professors ? Anyhow, this book is a definite departure from this nonsense. There are 12 chapters each with many exercises. The first couple of chapters have over forty and since I try to do them all, well ... If you read this book carefully and do the execises you WILL know this subject. You could teach it. You don't see Thm 1.2.3.5.8 followed by Proof. What you do see is a clear presentation of the ideas with PICTURES and EXPLANATIONS that you can understand, of course you really find out about that "understand" part when you get to the exercises. The biggest problem I had was getting out of the old way of thinking and into a more geometric way of thinking. Couldn't recommend it more highly. Another author who writes to teach is Victor Bryant. His book Yet Another Introduction to Analysis is great for a highschool senior or 1st year college. (He is with me on the state of math literature.) Also, Hans Schwerdtfeger's book Geometry of Complex Numbers goes well with Needham and is very cheap ! I'm surprised Needham didn't include it in the bibliography. It's a little gem and covers some of the same material.
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Summary: Decent Book for Graduate work but not good main text
Review: I used this book in an introductory Complex Variables course in a top 20 ranked US college. I enjoyed the authors clear explaination of material and clearly british sense of humor. Unfortunately I felt it lacked a great deal of rigor. Proofs were often either just sketched or pictorally shown. I understand that it was the author's objective to give a purely goemetric approach, but I felt that more detail was needed. When I needed to use ideas such as residue classes and other important complex variable conecepts in later math courses, my background was weak. I agree that the book does have merits. It takes the field of complex variables and looks at it in another way. I do feel though, that a more traditional book would be better to first secure undersatanding of the material. If a student were to continue to graduate work or want to learn more about complex variables, this would be a good supplement. I do not feel though, that this book is a good main, first text.
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Summary: essential
Review: It is possible to memorize definitions, master proofs and work endless exercises and still feel that you don't understand what's going on. This is especially true in complex analysis. This book emphasizes the visual/geometric aspect of analytic functions at the expense of some loss of rigor. These insights are priceless.
Needham employs a wider range of mathematical tools than other books aimed at the upper level undergraduate, e.g., Palka and Brown/Churchill. This would include simple group theory, linear algebra, vector calculus and obviously geometry. Often this works very well at the expense of some digression. At other times, the more traditional algebraic approach is better.
This book is unique and fills an important need.
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Summary: A fresh and insightful perspective on a beautiful subject
Review: Needham's book is a masterpiece which will be appreciated by anyone who already has gained (or is simultaneously gaining) a firm knowledge of the traditional, i.e. more algebraic, approach to complex analysis. In addition to reading it for pleasure, I have used the book extensively in teaching 18.04 Complex Variables with Applications at MIT, not as a required textbook, but rather as inspiration for lectures and homework problems. The book helps me give the students (mostly undergraduates in applied mathematics, science, and engineering) the geometrical insights needed for a deeper understanding of the subject, beyond what is found in various standard texts, such as Churchill and Brown or Saff and Snider (the required textbook for 18.04). As a prelude or companion to Needham's book, however, I would recommend reading one of these other books and working through more straightforward examples of algebra and calculus with complex functions. With that said, Needham's book is a perfect supplement to a first course in complex analysis. Needham's book is unique in its clear explanation of how the rich properties of analytic functions all follow from the "ampli-twist" concept of complex differentiation. In my class, I use this crucial, geometrical idea from the first mention of the derivative, where it goes hand in hand with the concept of conformal mapping (which is often at the back of introductory texts, but which I think should appear near the beginning). Perhaps the most delighful section of Needham's book is the one where he uses the same ampli-twist concept to give a very intuitive, unified proof of Cauchy's theorem, Morera's theorem, and the fact that a loop integral of the conjugate gives 2i times the area enclosed. The book also contains many clever and challenging problems, which are appropriate to give students to help them "think outside the box", as it were. The most amazing thing about Needham's book is that it is sure to delight and edify both beginners and experts alike with its simple, geometrical explanations. This is all the more impressive because geometry in mathematics education is more traditionally a vehicle to teach rigorous proofs rather than intuitive understanding.
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Summary: Where do I send the 50-cent soda for the book?!
Review: This book does an exceptional job of giving a hands-on person an opportunity to "see" what is happening in complex analysis. Its a shame that one person did not enjoy the book. Unfortunately, I loaned my copy out to someone and it got ruined. Please find out how I can get the unhappy person's copy and I'll gladly send them the 50-cent soda.
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Summary: A barely tolerable door stop
Review: This is a marvelous presentation of the subject. After looking at many analysis books which require a great deal of coffee to accompany them, I was surprised to find a book that would keep me awake till 3 a.m. without a caffeine supplement. The concepts are visually presented and clearly explained. Needham even shows how Mobius transformation is connected to Einstein's great idea. This is definitely a great book!
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