Visual Complex Analysis (Paperback)

This book is easy to read and full of many carefully-drawn pictures. Beautiful pictures! I went through almost a hundred pages at each sitting, doing all the in-line exercises, and a few of those at the chapter ends. Hardly any other math book has ever been such a piece of cake or so much fun. I do remember having read Grossman and Magnus' wonderful little "Groups and their Graphs" all at one go, one night, long ago --- but then its subject is quite elementary.

The exercises in VCA are very well-designed. The in-line exercises stretch the mind very slightly, never breaking the flow of thought. Never asking more than a minute or two of the reader. The exercises at chapter ends arenot the sort that even the author's butler could solve. Nor are they the sort that would frustrate you for hours and days and lead to fits of weeping and withdrawal.

I should perhaps mention that I did not come to VCA cold. As a signal processing person that works for a telephone company I use complex analysis every day --- at least in a manner of speaking. Usually with as much thought and imagination as a cobbler his awl. I have suffered stoically through the venerable "Complex Variables and Applications" by Churchill and Brown, and also Flanigan's "Complex Variables: Harmonic and Analytic Functions". That was many years ago.

Like all electricalengineers I am familiar with the usual brutal treatment meted out to complex analysis in the leading American signal processing text-books used in India and the US, whose authors betray little taste and less feeling for the subject. Why won't engineers write decently? I have read exactly one good book in engineering. That was "Structures: Or Why Things Don't Fall Down". There is so much extraordinarily-good writing in math --- even I can name at least ten golden books right off the top of my head, though I am no mathematician. Even physics is not entirely devoid of beauty in exposition. Is it just us engineers that won't write anything but horse-gobur?

The wonderful thing about Professor Needham is that he approaches even things I thought I knew well from so many fresh andunexpected directions that they become new and sweet all over again. For example, if you read about the Riemann sphere in Churchill and Brown, you'd say: so what's the big bloody deal? But Needham's treatment of Riemann spheres in the context of isogonal mappings and inversions in the sphere gives a rich idea of their power and their beauty. To give another example, at the very close of Chapter Four he suddenly springs the Cauchy-Riemann equations on the reader, pulling them out of a Jacobian of transformation rather suddenly, like a magician a rabbit. That was delicious! There are a whole bunch of things like that that will make you fall off your chair.

Likewise, despite a certain uneasy acquaintance with it, I had never appreciatedthe wonders of the Mobius transform, till I read Needham's account of it, and saw it come in to bat in the context of inversions in circles and in non-Euclidean geometries. As a onetime student of Roger Penrose, Needham brings with him the fresh breeze of physics in to the musty hallways of mathematics. As an engineer, and one not as imaginative as he would like to be, I much appreciate the application perspective. I am still saving the last three entirely physics-oriented chapters for a nice rainy day. They are like the candy my daughter hides away behind her books.

The Cauchy integral theorem is one result of immediate use to the electrical engineer. For many electrical engineers all they need the fearful djinn of complexanalysis for is to invert their Laplace and zee transforms. And then they can get going with their life. Needham gets to Cauchy's theorem in a rather leisurely way --- following discussions on the Mobius group, celestial mechanics, the Gaussian measure of curvature, the automorphisms of a disk, and everything else besides. The scenery along the way couldn't possibly be more seductive. But for a person in a big hurry this may not be the fastest route to work. That is about the only gripe I have.

There are a few typos. An errata is available at the author's web-page.

The bottom-line: Buy today, read tomorrow.

Now who is going to do a job like this for real analysis? And functional analysis?

The scope of the book is very broad. It covers 2D and 3D geometry, Mobius transforms, non-Euclidean geometry, analytic functions, complex differentiation and integration, winding numbers, vector fields and harmonic functions. But it is the approach that makes this text so unusual and so accessible.

Needham believes that geometric arguments reveal underlying connections which algebraic proofs diguise. In his own words: "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". Needham gloriously justifies his assertion in this text. Geometric proofs are used wherever possible, with the final conclusions translated back into algebraic terms. A variety of effective techniques are introduced for visualising the effect of Mobius transforms, analytic functions, complex differentiation etc.

One small word of warning - as Needham says himself in the Introduction, the arguments in this book are not formally rigorous. He bypasses the usual scaffolding of convergence and limits, and treats continuity as an intuitive concept. He uses phrases such as "the effect on an infinitesimal vector" which would cause a sharp intake of breath from a purist. This is not a problem, as long as you are happy to take it on trust that a formal framework can be provided if required. However, if you are studying for a conventional complex analysis exam, then you will need to fill in the formal structure from a more "standard" text once you know the landscape.

Definitely one of the best maths textbooks that I have ever read.

Absorbing, explanatory and fun to read the reader takes an active part.
There are 12 main chapters and each has exercises at the end. There are no solutions however, this book takes a visual insight into the world of complex numbers so the more you reflect the more your understanding grows.

There are plenty of well-illustrated and annotated diagrams. This book also has a few topics linked with Physics such as Riemann Mapping theorem, and Mobus transformation with Einstein's theory of relativity.

If you are serious about Mathematics and love logical and abstract thinking as well as visualising then this book is definitely worth a thorough look.