Visual 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.


A marvel, eye-popping, fun. More than five stars!   January 10, 2004
Paul J. Papanek (Los Angeles, CA United States)
49 out of 51 found this review helpful

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.


Gee, a math book that really teaches, how unusual   April 23, 2003
Carl F. Mclaren Jr. (Haines City, Florida USA)
25 out of 26 found this review helpful

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.


Best math book I've read in years!   September 29, 2002
Gerald Kaiser (Austin, TX USA)
21 out of 21 found this review helpful

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.


Destined to be a Classic   June 19, 2000
James M. Cargal (Montgomery, AL USA)
14 out of 14 found this review helpful

This is a large informal work devoted to making complex analysis intuitive. There is no other book on complex analysis roughly like it. It is roughly at the junior-senior level and of course requires a background in calculus and preferably real analysis. I suspect many students who have already had the C.A. course might enjoy reading this to find out what they should have learned. It is a must-have for anyone interested in complex analysis.