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.

72 of 74 found the following review helpful:


A marvel, eye-popping, fun. More than five stars!  Jan 10, 2004 By Paul J. Papanek "latoxdoc"
What a great book this is!

This is a book that any math afficionado must have, and will undoubtedly savor. I frankly don't understand those reviewers who have given this book fewer than five stars. In fact, five stars wouldn't seem to be enough here. This book is among the best math books one will ever find! What else would one want from a such book? It is exciting, friendly, creative, often funny, crystal clear, fresh, deep, and unfailingly courteous to the reader--a quality not always found in math texts.

Additionally, this book succeeds on another level -- it is just plain beautiful. Math, to be great, must be beautiful, while books about great math too often are not. This book is truly beautiful, even artful. The author has taken great care to create beauty here.

I intially bought this book, because as an ex-mathematician whose analysis skills were getting rusty I wanted to revisit complex analysis. This book certainly succeeded in brushing up those old skills, but it also deepened them. The book has marvelous insights and geometric drawings that demonstrate in a clever way the links between complex analysis and other branches of math and physics. How could one not love the lovely and intricate drawings that depict, say, loxodromic transformations on a sphere, or the eye-popping diagrams of rotations in hyperbolic space? They're fabulous! Even the problem sets are delightful.

As a side note, some of the historical glosses about mathematicians are also very lively, and are another source of pleasure here.

On the dust jacket is the blurb--"If you must buy only one math book this year, this is the one to buy." I have to agree. I bought a couple dozen math books last year, and this one outshines the rest. I can't recommend it highly enough, even if you already feel comfortable with complex analysis.

I encourage my fellow readers to pick this up, and see how beautiful a math book can be.

31 of 31 found the following review helpful:


A Tremendously Insightful Presentation of Complex Analysis  Jun 15, 2000 By Eze
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."

24 of 24 found the following review helpful:


Best math book I've read in years!  Sep 29, 2002 By kaiser
I have recently finished reading this book cover-to-cover and, in spite of having worked
in mathematical physics for 40 years, feel compelled to gush like a teenager. It is mighty
therapy for a generation raised on conciseness, abstruseness, abstraction and Bourbaki.
Possibly one cause for this sorry state of affairs (there are others, but I'm in a generous mood!)
is the vast mass of knowledge that has to be mastered by modern devotees. But, like any fashion, this
one has taken on a life of its own. A friend who works at MIT recently showed a book to a young post-doc,
claiming it was a "friendly" introduction to such-and-such. Without even glancing at the evidence, the
hot-shot replied that if it was all that friendly, it couldn't possibly be any good!
Needham takes you back to an earlier sensibility, naive and profound in equal measure,
tackling problems leisurely with nothing but your own intuition and a few simple facts
from geometry. Following his guidance, you understand the solution several times from
different angles and come out with that intoxicating feeling of "owning" the entire
thing, not as a means to an end (publishing, accolades, ...) but as a thing of beauty. It's
hard to believe, but early masters like Newton actually managed to understand vast and
complex fields of science in this very tactile way. That art, largely lost, has been revived
lately by a select few including Needham and Chandrasekhar (Newton's Principia for the
Common Reader, Clarendon Press, 1997). I've made a complete mess of my copy: margin notes,
sketches, ... and probably a few drool marks. Let's hope this starts a movement. If there is a way
to save American math education, this has got to be it! Thanks, Tristan.

69 of 78 found the following review helpful:


Pretty but not a substitute for traditional text.  Oct 09, 2005 By anon2001 "anon2001"
My slightly harsh rating is an antidote to all the gushing about

this book. It is a nice book with lots of pretty pictures and

genuine geometrical insights and is well worth reading as a

supplement to traditional complex analysis texts. The geometrical

topics are actually quite good. If you are a maths major then

this book will be of limited use because its coverage of the

traditional topics is simply too weak. The geometrical approach

quickly runs out of steam, in my opinion, once it gets into

complex integration. Homotopy does not even rate a mention in the

index. My pet dislike was the almost complete omission of the

calculus of residues. The author dimisses that topic as being

old-fashioned. True, the application to computing real integrals

is reduced since the advent of computers. But I think that a

maths major would need to be aware of Jordan's Lemma and other

techniques to estimate the asymptotic behaviour of integrals

along curves. I also found that the treatment of multi-valued

functions and branch cuts quite confusing, which is surprising

in a book which is supposed to have a strong geometrical focus.

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