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Title: Visual Complex Analysis by Tristan Needham ISBN: 0-19-853446-9 Publisher: Clarendon Pr Pub. Date: June, 2000 Format: Paperback Volumes: 1 List Price(USD): $65.00 |
Average Customer Rating: 4.23 (22 reviews)
Rating: 5
Summary: A fresh and insightful perspective on a beautiful subject
Comment: 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.
Rating: 5
Summary: A Tremendously Insightful Presentation of Complex Analysis
Comment: 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."
Rating: 4
Summary: Lots of material, great pictures but too chatty
Comment: 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.
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Title: Geometry of Complex Numbers by Hans Schwerdtfeger ISBN: 0486638308 Publisher: Dover Pubns Pub. Date: 01 February, 1980 List Price(USD): $12.95 |
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Title: Counterexamples in Analysis by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Pubns Pub. Date: 04 June, 2003 List Price(USD): $14.95 |
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Title: Counterexamples in Topology by Lynn Arthur Steen, J. Arthur Seebach ISBN: 048668735X Publisher: Dover Pubns Pub. Date: 22 September, 1995 List Price(USD): $11.95 |
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Title: Riemann's Zeta Function by Harold M. Edwards ISBN: 0486417409 Publisher: Dover Pubns Pub. Date: 13 June, 2001 List Price(USD): $14.95 |
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Title: Problems and Solutions for Complex Analysis by Rami Shakarchi, Serge Lang ISBN: 0387988319 Publisher: Springer Verlag Pub. Date: December, 1999 List Price(USD): $44.95 |
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