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31 of 32 people found the following review helpful
5.0 out of 5 stars Some books are worth their weight in gold.
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...
Published on 9 Sept. 1998

versus
3.0 out of 5 stars Valuable
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...
Published 2 months ago by André Gargoura


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31 of 32 people found the following review helpful
5.0 out of 5 stars Some books are worth their weight in gold., 9 Sept. 1998
By A Customer
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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6 of 6 people found the following review helpful
5.0 out of 5 stars Part of the books that help crack complex analysis, 15 Aug. 2007
By 
ab..c (england) - See all my reviews
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This review is from: Visual Complex Analysis (Paperback)
Hi,

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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20 of 21 people found the following review helpful
5.0 out of 5 stars One of the best maths textbooks ever, 30 April 2001
This review is from: Visual Complex Analysis (Paperback)
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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12 of 13 people found the following review helpful
5.0 out of 5 stars Absorbing , reflective and highly interesting, 2 Jan. 2002
This review is from: Visual Complex Analysis (Paperback)
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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2 of 2 people found the following review helpful
5.0 out of 5 stars How it should be done!, 1 Jun. 2011
This review is from: Visual Complex Analysis (Paperback)
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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9 of 10 people found the following review helpful
5.0 out of 5 stars Complex analysis as you never studied before, 10 Aug. 2006
By 
This review is from: Visual Complex Analysis (Paperback)
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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3.0 out of 5 stars Valuable, 13 Mar. 2015
By 
André Gargoura "André Gargoura" (Paris, France) - See all my reviews
(REAL NAME)   
This review is from: Visual Complex Analysis (Paperback)
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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3 of 4 people found the following review helpful
5.0 out of 5 stars What a Beauty!, 21 July 2011
By 
Jay Jina "jjina" (UK) - See all my reviews
(REAL NAME)   
This review is from: Visual Complex Analysis (Paperback)
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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2 of 3 people found the following review helpful
5.0 out of 5 stars je parle des mathématiques, 21 Mar. 2010
This review is from: Visual Complex Analysis (Paperback)
I remember French class at school mainly for the endless, pointless recitation of verbs and nouns. Like many, two things stuck with me from those classes- how to order a crepe and how to ask the way to the station. Its almost a tautology that the best way to learn French is to go to France- but what is the best way to learn mathematics? Judging by most of the textbooks I've attempted to decipher, the answer is the mathematical equivalent to reciting verbs- dense notation and dry examples leaving an impression that there is no intuitive explanation of the material. This book is different, it is the mathematical-textbook-equivalent of a trip to Paris. It really opened my eyes to the beauty of the subject. Take for example the treatment of the Dirichlet problem- something that I've had to look at recently for some University coursework. Needham explains this, with plenty of clear and concise diagrams, in terms of figuring out the temperature somewhere on the surface of a disc of metal given the temperature on the disc's circumference- easy! Try comprehending that from the reams of integrals in my set textbook!
Don't mistake this for a popular science book (you hardly could from the title), it covers most of the material for my Masters course in Complex Variables and I'd say you need a solid mathematical background to 'get it'. But that said, it's just about possible to read this book page by page, especially the early chapters. For me, as an Engineer, doing this really helped me form a wider appreciation for some of the stuff necessarily taken for granted in undergraduate courses (what exactly _is_ a complex derivative?!?).
This book is how maths should be presented, with its help I can almost say 'je parle des mathématiques'.
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15 of 34 people found the following review helpful
1.0 out of 5 stars What a disappointment, 18 Dec. 2007
By 
L. Hurman - See all my reviews
(REAL NAME)   
This review is from: Visual Complex Analysis (Paperback)
I stood in the bookshop and read the reviews printed on the back cover and with mounting excitement the introduction. I bought the book and took it home. Imagine my disappointment when I actually started reading it. I got to page 18 before the combination of unexplained terms and unexplained steps brought me to a halt.

In my experience books like this which are not written as text books fall into two groups. There are those in which the author has truly translated the mathematics into English so as to make the topic accessible to those not already familiar with it (try Eli Maor), and there are those that require the reader to be familiar with the topic in order to be able to follow the explanations and are presumably written for the sake of the ego of the author. This book definitely falls into the latter category. Notice how all of the other reviewers talk of their previous knowledge of the subject.

Of course you may think me stupid but I have an engineering degree and an Msc in dynamics and I do not know any mathematics which I think of now as being conceptually difficult. And yet think how hard we had to work to learn some of it when we were at school and college. Things that we think of as simple and ordinary now were such a struggle to learn, and why? Because we didn't have the language. Well in order to gain any insight from this book you are already going to have to have the language because this author does not think of it as his job to give it to you. What a pity. What a missed opportunity. What a disappointment.
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Visual Complex Analysis
Visual Complex Analysis by Tristan Needham (Paperback - 26 Nov. 1998)
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