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VINE VOICEon 15 August 2007

In a way you only see how good this book is when you read a number of other books on this topic? This is a book that works best when other books balance these two approaches, and by doing this it lets you see the whole 'landscape' of complex analysis.

If other books are rich in detailed questions, you slog along and break them down in small steps often without the `big picture' of where it fits in the wider scheme of things. With this book you see a vast sweeping panorama that allows the reader to gain insight with a geometrical approach in conceptualising areas.

The book starts in elemental terms in reflections and translations and complex algebra. Also a common feature is the book has outstanding illustrations and has helpful text to explain in more depth. I found the approach helped my geometrical interpretation of the links between complex numbers projected onto 'Riemann spheres' using 'Möbius transforms' through into 'Hyperbolic geometry' and the Calculus and on further to consider the properties of 3 combinations of two curved mirrors (reflections and translations again) on a Euclidian plane. The book also carries on to cover more general-purpose 'Laurent series' and beyond and how they can be applied in Complex Analysis.

Summary: I.M.H.O. It's a good buy as part of your bookshelf on this gripping topic. A Mathematics professor I knew once (who I will not name) -paraphrased-described the book to me as "the type of book you have at MSc level, without the intensive level of calculation. Its a lovely book to give you a `feel' of the topic".
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on 19 August 2015
If only this book had been available when I did my degree! I got a First, and A grades in all the papers, but two of which I had no proper understanding of: Galois Theory, and Complex Analysis. I am still searching for the book which can explain to me what made Galois set off in the direction he did. As for the latter, this work by Needham more than fulfills the comprehension I have been looking for ever since I finished my degree course, 33 years ago! Admittedly it has been a rather leisurely search. Some reviewers complain that it is not very accessible. I think it is plain right at the outset that it is not an attempt at popularising complex analysis. Nor is it intended to be a text book. The author clearly states on several occasions that sometimes he is only offering a geometric insight rather than a rigorous proof. This book is perfect in complementing a typical text on the subject, which may provide the rigour but possibly totally neglect the geometry. None of this is to suggest that the book is sloppy, or inaccurate in any way. The mathematics is not compromised at all. What was a true revelation for me was the magical world that slowly but surely unfolded with each chapter. When I did my degree, all the results of complex calculus were presented in the usual way, with rigorous proofs, which I could reproduce in exams, and knew that those results which gave complex analysis an advantage over real analysis derived from the definition of an analytic function. But I never understood why until now. People like Cauchy and Riemann clearly understood and saw in a way which is brought to life in CVA. Needham refers to Bach and Wagner in his preface. I think it is no exaggeration to say that exploring the magical world of complex analysis as presented here is just as sublime and beautiful as the music of those two giants.
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on 1 June 2011
All the recommendations are right! So many textbooks on "geometry" - differential and others - in practice are all algebra, and a bit dead. This is pure geometry and pure delight; the elegance and beauty of the methods and proofs reminds me of why I got hooked on maths in the first place. The oft-repeated words conformal etc. sing with meaning in this thorough and illuminating presentation, the antithesis of so many of the texts I've ploughed through recently.

Seeing really is believing here where complex variables are pictured from every conceptual angle so that they become familiar friends, as the complex plane flips into the Riemann sphere and back, Mobius transformation becomes transparent, and derivatives turn into amplitwists and it all ties together and makes sense. The modern view that maths is about transformations is throughout embedded, as complex numbers and their calculus are treated as geometrical transformations.

It's rare to encounter a book so well written that pennies drop almost instantly. The diagrams are carefully drawn and referenced, exercises are plentiful and instructive, and useful cheap software is referenced as an essential teaching aid to building an instinctive feel for this innately subtle and multidimensional subject. As the prototype for deeper geometrical studies it gives a first-class launchpad for the tough stuff, including a lively introduction to non-Euclidean geometries and quaternions, and how the properties of solutions can be inter-related between domains. And physical interpretation and inference is all tied in in the later chapters as vector fields are brought into the fold. This interweaving of so many topics so that they can all be seen as part of a seamless whole is the great delight of the whole book.

It should be compulsory reading for every would-be writer of a maths textbook. The main downside of the book is that it doesn't prepare the reader for the notational hell of group classifications, cohomologies, homotopies of the nth kind that is turning modern maths into such a minefield for the learner. If only someone would do a similar service for symplectic geometry!
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on 2 January 2002
This book is a jewel, if only there was a perfect Mathematics lecturer in the world s/he would bother explaining concepts like this fascinating book.
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.
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on 30 April 2001
Tristan Needham has written a wonderful synthesis of geometry, complex analysis and vector fields. Before I read this book I had "studied" complex analysis, but had never truly understood it. Now it all makes sense !
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.
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on 9 September 1998
This is one of those. I bet it will make a mathematician of ayoung 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 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?
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on 13 March 2015
Needham has achieved a unique treatment of complex analysis (CA) : making it visual and thereby attractive.

The challenge was enormous and is definitely met

Yet, there is a cost, to the reader : since it is next to impossible for an author to achieve, simultaneously, a thorough explanation of the figures i.e. the visual treatment of CA and to give a fundamental, classical treatment of the subject, the latter effort is left to the reader who has to fetch for other sources... So, be prepared, with a good load of pencils : my copy is annotated almost at each paragraph.

This obviously cannot be held against Needham who, right from the preface warns us.

Nevertheless, there are a few points that annoyed me :
- bad articulation of chapters/ sections/subsections and no mention of such info at the top of each 2 pages... making searches difficult.
- no table of symbols.
- no synthesis at the end of each chapter e.g. about what we have achieved and where we are heading to.
- poorly contrasted figures, which have to be penciled, if you can guess where the curves and lines are.

I intended to give the book 4 stars, before going through sections IV to VII of chapter 12 which are simply indigestible.

Actually, a good idea would be to follow, simultaneously, the two excellent video series by : Petra Bonfert-Taylor (Coursera) and Herbert Gross (Mit).

Finally, does this book replace a fundamental treatment ? Clearly not and there are many classical treatments around, culminating with Markushevich's "Theory of functions of a complex variable".
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on 10 August 2006
I discovered this book in passing through the bibliography of Penrose "A road to reality" , and suddenly my curiosity brought me to take a look at it (and i thank Sir Penrose for this...).

The subject is treated just as the title says, although not every aspects of complex analysis is covered (for which many standard textbooks do the right and better job).

Of particular interest to me was reading chapter 6 on non euclidean geometry, in which the author gives a concise and insightful description of the main ideas.

I think the book is particularly tailored, other than for mathematicians, for physicists who care of the beautiful links between geometric and algebraic aspects of modern maths.
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on 23 December 2016
I am reading this book from a library copy. I am yet only a few chapters into the book but I can easily say that it is one of the best written books on any topic. It does not follow the dry style of state, define prove cycle that unfortunately many of the mathematics books follows. It motivates everything from at least few different perspectives including the historical perspective of the mathematicians of the time who formulated the result and also a geometric perspective whenever possible. The goal of the book is not seek beauty in abstraction but to seek in simplicity and geometric ideas. Abstraction is used as a necessary tool whenever needed but never becomes the aim. Therefore for instance the existence of such objects as Quaternions are already anticipated very early in the book simply by understanding how complex numbers corresponds to similarity transformations of geometric objects on the plane. It set for me new standards on how one should approach teaching or motivating a topic. Better yet inside the book you will find references to books on other topics who follow the same point of view. It is a real gem.

One warning is in order though, I would not recommend this as a first book on complex analysis and definitely not as a single source for a first lecture. Back of the envelope calculations of Newtonian style in the book can be actually hard to prove formally and the purpose of the book is not to prove theorems or results in a classical text book style to you but rather show you geometric ways in which the proof can be understood. This is why it would be a perfect follow up on a classical, dry complex analysis lecture. Once you have proven theorems previously, informal discussions of those theorems from other points of view will not be so bothersome.
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on 21 July 2011
This is one hell of a book. Needham explains the geometrical concept of complex numbers and functions in a highly visual way.

Classical concepts that are typically described in "dry" terms come to life in this book. I studied Complex Variable several years ago and only picked this book up out of interest. And, what a joy it turned out to be. If only the various theorems attributed to Cauchy, Winding Numbers, the geometrical meaning of Conformal Mappings and their relevance to solving physical problems like fluid flow and electrodynamics had been explained to me in this way when I had to study this subject!

If only.

Students of Mathematics are lucky to have this book at their disposal. It makes a fantastic complement to other excellent books like Complex Variables, Introduction and Applications by Ablowitz and Fokas, ISBN: 0521485231

I have no doubt that this book will be a classic!
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