Visual Complex Analysis (Paperback)

~ (Author) "Four and a half centuries have elapsed since complex numbers were first discovered..." (more)
Key Phrases: power series centred, dilative rotations, infinitesimal arrow, Cauchy's Theorem, Fundamental Theorem, Symmetry Principle (more...)

Editorial Reviews

Review

"Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis." -- Sir Roger Penrose

"For the more mathematically inclined who also have an interest in keeping one foot in physical reality, I highly recommend Tristan Needham, Visual Complex Analysis." -- Paul Nahin, in An Imaginary Tale, Princeton Univ. Press, 1998

"I can only describe this book as amazing... it is not an exaggeration to say that there are gems in every section ... even familiar facts are frequently explained in refreshingly new ways in this wonderful book. ... If your budget limits you to buying only one mathematics book in a year then make sure that this is the one that you buy." -- MATHEMATICAL GAZETTE

"I was delighted when I came across Visual Complex Analysis. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago." -- Ed Catmull, Founder and Chief Technology Officer of Pixar Animation Studios [makers of Toy Story and A Bug's Life]

"Newton would have approved. ...a fascinating and refreshing look at a familiar subject. ... It is essential reading for anybody with any interest at all in this absorbing area of mathematics." -- TIMES HIGHER EDUCATION SUPPLEMENT

"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. ... to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices-but his are interesting." -- Ian Stewart, NEW SCIENTIST

"The pictures in this note were produced directly in PostScript. ... There are other possibilities-one very fine recent example of mathematical graphics at a high level is Needham's Visual Complex Analysis, which used quite different tools." -- Bill Casselman, NOTICES OF THE AMS

"Too many textbooks neglect the historical foundations; this book is rich with history. The author has taken full advantage of modern computer graphics to provide a variety of lovely and helpful pictures... There is a rich bibliography at the end of the book spanning the gamut of both historical and modern references." -- ZENTRALBLATT FR MATHEMATIK

"an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas (non-Euclidean geometry, harmonic functions, etc.) from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g., the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject." -- Paul Zorn, AMERICAN MATHEMATICAL MONTHLY

"it is comparable with Feynman's lectures in Physics. At every point it asks 'why' and finds a beautiful visual answer. ...I believe that this book can make every student understand and enjoy complex analysis. If its methods could be applied in teaching more generally, mathematics would become a flourishing subject." -- NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY

Review

"Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis." --Roger Penrose
"Tristan Needham's Visual Complex Analysis will show you the field of complex analysis in a way you almost certainly have not seen before. Drawing on historical sources and adding his own insights, Needham develops the subject from the ground up, drawing us attractive pictures at every step of the way. If you have time for a year course, full of fascinating detours, this is the perfect text; by picking and choosing, you could use it for a variety of shorter courses. I am tempted to hide the book from my own students, in order to appear more clever for popping up with crisp historical anecdotes, great exercises, and pictures that explain things like that mysterious 2pi that crops up in integrals. Whether you use Visual Complex Analysis as a text, a resource, or entertaining summer reading, I highly recommend it for your bookshelf."--American Mathematical Monthly
"Delivers what its title promises, and more: an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas. . .A truly unusual and notably creative look at a classical subject." --American Mathematical Monthly
"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's 'Visual Complex Analysis' with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices--but his are interesting." --New Scientist
"Committed to the exclusive use of geometrical arguments and content to pay the price of 'an initial lack of rigour', he has produced a radically new text. The author writes "as though [he] were explaining the ideas directly to a friend". This informal style is excellently judged and works extremely well."--Mathematical Review
"This is a book in which the author has been willing to make himself available as our teacher. His own voice enters in a rather charming way....I recommend Visual Complex Analysis, as something to read and enjoy, to share with students, and perhaps to inspire other books in which the voice of the author is vividly present to teach and explain."--American Mathematical Monthly

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Product Details

• Paperback: 616 pages
• Publisher: Oxford University Press, USA (February 18, 1999)
• Language: English
• ISBN-10: 0198534469
• ISBN-13: 978-0198534464
• Product Dimensions: 9.1 x 6.2 x 1.4 inches
• Shipping Weight: 1.9 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (40 customer reviews)
• Amazon.com Sales Rank: #91,092 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

101 of 103 people found the following review helpful:

5.0 out of 5 stars A fresh and insightful perspective on a beautiful subject, November 13, 2001

By	mzb "mzb" (Winchester, MA United States) - See all my reviews

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 "ampli-twist" 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 ampli-twist 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.

58 of 60 people found the following review helpful:

5.0 out of 5 stars A marvel, eye-popping, fun. More than five stars!, January 9, 2004

By	Paul J. Papanek "latoxdoc" (Los Angeles, CA United States) - See all my reviews (REAL NAME)

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 reader--a 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 ex-mathematician 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 eye-popping 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.

26 of 26 people found the following review helpful:

5.0 out of 5 stars A Tremendously Insightful Presentation of Complex Analysis, June 14, 2000

By	Eze (Valencia, Spain) - See all my reviews

Although mathematical visualization has not been as implicitly forbidden in modern mathematics as claimed by Needham, his work is nonetheless highly innovative even besides his wonderful graphs. The reason is that his prose accompanies very well his extraordinary insight and intuition for the subject. It is purposely not extremely rigorous in order to make the presentation smoother. (This is not so bad as many think. Complex analysis is the target of many excellent books which, fortunately, do not all take the same approach. For more rigor see Ahlfors' "Complex Analysis.")

This book can therefore be an ideal way to get started with complex analysis or even to further one's understanding in the subject. If you are looking for a very affordable predecessor with a similar intuitive style, check Flanigan's "Complex Variables."