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4 of 4 people found the following review helpful
5.0 out of 5 stars
Gee, a math book that really teaches, how unusual,
By Carl Mclaren (Haines City, Florida USA)  See all my reviews
This review is from: Visual Complex Analysis (Paperback)
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.
3 of 3 people found the following review helpful
5.0 out of 5 stars
A marvel, eyepopping, fun. More than five stars!,
By
This review is from: Visual Complex Analysis (Paperback)
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 readera 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 exmathematician 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 eyepopping 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.
2 of 2 people found the following review helpful
5.0 out of 5 stars
Best math book I've read in years!,
By
This review is from: Visual Complex Analysis (Paperback)
I have recently finished reading this book covertocover 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 postdoc, claiming it was a "friendly" introduction to suchandsuch. Without even glancing at the evidence, the hotshot 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.
2 of 2 people found the following review helpful
3.0 out of 5 stars
Decent Book for Graduate work but not good main text,
By Christopher G. Moore (Worcester, MA United States)  See all my reviews
This review is from: Visual Complex Analysis (Paperback)
I used this book in an introductory Complex Variables course in a top 20 ranked US college. I enjoyed the authors clear explaination of material and clearly british sense of humor. Unfortunately I felt it lacked a great deal of rigor. Proofs were often either just sketched or pictorally shown. I understand that it was the author's objective to give a purely goemetric approach, but I felt that more detail was needed. When I needed to use ideas such as residue classes and other important complex variable conecepts in later math courses, my background was weak.
I agree that the book does have merits. It takes the field of complex variables and looks at it in another way. I do feel though, that a more traditional book would be better to first secure undersatanding of the material. If a student were to continue to graduate work or want to learn more about complex variables, this would be a good supplement. I do not feel though, that this book is a good main, first text.
1 of 1 people found the following review helpful
3.0 out of 5 stars
Overly geometric,
By A Customer
This review is from: Visual Complex Analysis (Paperback)
I have enjoyed working through this book. As others have pointed out, it's a nice geometric introduction to complex analysis. I do, however, have a couple gripes. One major and one very minor.
First, Needham seems to strive for a kind of geometric "purity". He tries to give the impression that geometric arguments are more valid than standard logic or algebra. While some might feel this is a needed correction to alleged antigeometric trends in math, Needham's correction can, at times, be an overreaction. The result is that the book is excellent in those areas that are wellsuited to a geometric approach (e.g. Mobius transformations and hyperbolic geometry), but fails in areas for which algebraic approaches are simpler and easier to understand. Parts of the chapters on differentiation are unnecessarily cumbersome. (While the "amplitwist" thing  a geometric version of complexdifferentiationaslocalmultiplication  is neat, it's a little overdone, and ends up making differentiation sound more mysterious than it is.) Insisting on "pure" geometric arguments is a nice exercise. But when it obscures the subject and makes it more difficult to follow, one begins to see why math has moved away from that kind of reasoning over the past several hundred years. By the time I finished Needham's book, my appreciation for nongeometric mathematics had increased quite a bit. In any case, I think students could learn well from certain chapters of this book (the more geometric ones), but should definitely be steered away from others (differentiation and integration). This would make a great supplementary text but not a good main text. If used as the main text it should certainly be supplemented with less roundabout approaches to differentiation and integration. My second, very tiny gripe: Needham seems to be obsessed with Roger Penrose. Nothing against Penrose, who I'm sure does great things in physics, but the constant references get a little tedious after a while. Having said all of that, I should repeat that I still enjoyed the book very much. It's definitely worth the money, espcially if you've already had some complex analysis and want to see this geometric way of doing things, or if you're currently studying complex analysis and want to develop some intuition. If you're studying complex analysis for the first time, Churchill and Brown would be a better book at a similar level. The book by Ahlfors is the standard more advanced introduction.
1 of 1 people found the following review helpful
4.0 out of 5 stars
Insightful book about Complex Analysis via Geometry!,
By
This review is from: Visual Complex Analysis (Paperback)
This book attracted my interest mainly because of its geometry content, and sustained it with its informal approach . More than the Complex Analysis that I learnt, and which is peripheral to my main interest, I learnt a lot about Geometric approach to solving many mathematics problems.
Good, insightful expositions of relationship between Geometry and Complex Arithmetic, Mobius Transformations, and Vector Fields. The mathematics content is at about the level of freshman undergraduate and the book is fairly easy "read". In fact, you don't "read" this book; you work throgh it by drawing pictures after pictures to understand the logic. This is a good preparation for physics graduate students before first courses in Electrodynamics, Mechanics, and Relativity. In addition, those interested in Geometry, Graphics, Visualization will also appreciate the book.
1 of 1 people found the following review helpful
4.0 out of 5 stars
Lots of material, great pictures but too chatty,
By A Customer
This review is from: Visual Complex Analysis (Hardcover)
I purchased this book as a reference and because of it's coverage on Mobius Transformations, which is great! My qualms are with the other parts of the book, however. I'll reach for this book or Churchill and Brown when I'm dealing with complex numbers. Browns is much more direct and to the point. There are times that I'll have to flip through several pages jsut to get to the point. Needham often includes a history of the topic and several applications before getting to the mathematics of it. I like reading about applications at the end of the chapters and histories as footnotes (or both in a completely seperate part of the book, i.e. the appendix). If you buy this book, you'll get a lot of great mathematics and wonderful visualizations, but expect a lot of reading that may not be immidiately necessary to your studies.
1 of 1 people found the following review helpful
4.0 out of 5 stars
amazing visual representations,
By UNPINGCO (Los Angeles, CA)  See all my reviews
This review is from: Visual Complex Analysis (Paperback)
This text provides sometimes amazing visual representations of concepts in complex analysis that I have never seen anywhere else. For example, I have never seen a complex contour integral interpreted geometrically. Also, the text presents many very important conformal maps. Having said that, however, I would not recommend this as a first book in complex analysis. A better first book along these same visual lines, but with more rigor is Flanigan's "Complex Analysis". On the other hand, if you already have a more traditional grounding in complex analysis and want to motivate many of the results geometrically, then this book is uniquely suited for that. This book provides useful "explanations" and not rigorous proofs.
5.0 out of 5 stars
A fresh and insightful perspective on a beautiful subject,
By mzb "mzb" (Winchester, MA United States)  See all my reviews
This review is from: Visual Complex Analysis (Paperback)
Needham's book is a masterpiece which will be appreciated by anyone who already has gained (or is simultaneously gaining) a firm knowledge of the traditional, i.e. more algebraic, approach to complex analysis. In addition to reading it for pleasure, I have used the book extensively in teaching 18.04 Complex Variables with Applications at MIT, not as a required textbook, but rather as inspiration for lectures and homework problems. The book helps me give the students (mostly undergraduates in applied mathematics, science, and engineering) the geometrical insights needed for a deeper understanding of the subject, beyond what is found in various standard texts, such as Churchill and Brown or Saff and Snider (the required textbook for 18.04). As a prelude or companion to Needham's book, however, I would recommend reading one of these other books and working through more straightforward examples of algebra and calculus with complex functions. With that said, Needham's book is a perfect supplement to a first course in complex analysis.
Needham's book is unique in its clear explanation of how the rich properties of analytic functions all follow from the "amplitwist" 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 amplitwist 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.
3.0 out of 5 stars
Hm.......,
By A Customer
This review is from: Visual Complex Analysis (Paperback)
Th merits of this book has been well talked by others, I have few to add to. My question is, in order to distinguish the text, does the author distort the history of the subject? In the preface, after discussed a geometric resoning of Newton, the author claimed that such line of reasoning was rediscovered only 300 years later (by himself?). And after many standard geometric (and great!) arguments, the author voices excitement like it's discovered yesterday. Truth is, geometric ideas is never forgotten in complex analysis, just think of the greatest figure in the subject, Riemann, and Rieman surface. It's doubtless that author emphasizes geometry in a rightly way, and he does make some novel expositins. Besides, the text flows smoothly and consists delightful reading. But, under any circumstances, you cannot give the beginners a vastly wrong impression of the historial settings of the subject in order to achieve an effect!

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Visual Complex Analysis by Tristan Needham (Paperback  Jan. 1 1999)
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