Arun Kumar (kumar@tri.sbc.com) from Austin, Texas , September 9, 1998. Some books are worth their weight in gold. This is one of those. I bet it will make a mathematician of a young person that happens to pick it up at the golden moment of his or her life. 

    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 are not 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 electrical engineers 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 and unexpected 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 appreciated the 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 complex analysis 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?