This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
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Editorial Reviews
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"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 2*pi 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
From the Author
The book recently won First Prize in the National Jesuit Book Award Contest for the best mathematics or computer science book published in 1994, 1995, or 1996.
--This text refers to an out of print or unavailable edition of this title.
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Most Helpful Customer Reviews
128 of 130 people found the following review helpful
Format: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 "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.
72 of 74 people found the following review helpful
Format: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 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.
31 of 31 people found the following review helpful
Format:Paperback
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."
Most Recent Customer Reviews
Indispensable
Complex analysis can challenge the intuition of the new student. This text is unique, among high quality textbooks, in giving a careful and thorough exploration of the geometric... Read more
Published 5 months ago by Jim Curry
Beautiful
It's not just beautiful because the author included heaps of beautiful pictures (which he did, by the way - but not just to make the book look pretty), it is just a very beautiful... Read more
Published 16 months ago by mgill
A fun math book!
This book is a real treat. It gives a beautiful visual description of complex differentiation. I was happy to learn how to visualize this due to my "lack of four-dimensional... Read more
Published 23 months ago by Kenneth C. Crandall
The best mathematics book I've ever read
This book lives true to its name. Most other complex analysis books put so much focus on the algebra of complex analysis that the geometry of complex numbers are almost forgotten. Read more
Published on April 28, 2010 by Jonathan Nacionales
Visual Complex Analysis
This text ended up being the only required reading for the undergraduate complex analysis course which I am taking this spring. Read more
Published on January 7, 2010 by R. Smith
Illuminating
It is one of the most illuminating text on Complex Analysis. Visualization is missing in almost all theoretical texts on this subject. Compliment to the author. Read more
Published on October 7, 2009 by Materassi Aldo
Recomendado
A typical judgement about this book would be as follows; "works good on the concepts and intuition, specially the visual/geometrical, but lacks mathematical rigor". Read more
Published on September 3, 2009 by Paco Zuniga
Good book for the intermediate learner
This is a great book that has earned my respect, but it is not for a beginner. It is intended for the crowd who already has some good mathematics background (at least the basic... Read more
Published on August 31, 2009 by D. Becker
Excellent
It's tragic that so many other books are so pedantic and lacking in insight. This book teaches genuine understanding, so that the symbols that you manipulate never become detached... Read more
Published on June 29, 2009 by Muppet Master
Excellent Presentation
Having struggled when first taking complex analysis in college, I wanted to find a book that presented the topic in a much more clear and concise manner. Read more
Published on June 1, 2009 by Diamond20
Inside This Book
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First Sentence:
Four and a half centuries have elapsed since complex numbers were first discovered. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
power series centred, dilative rotations, infinitesimal arrow, infinitesimal disc, topological multiplicity, semicircle orthogonal, harmonic duals, infinitesimal squares, new temperature distribution, deformation theorem, use the previous part, central dilation, modular surface, given harmonic function, second intersection point, angular excess, small circle centred, infinitesimal neighbourhood, geometric equality, equidistant curves, complex inversion, disc centred, hyperbolic plane, infinitesimal circles, hemisphere model Key Phrases - Capitalized Phrases (CAPs): (learn more)
Cauchy's Theorem, Fundamental Theorem, Symmetry Principle, Binomial Theorem, Mean Value Theorem, Residue Theorem, General Cauchy Theorem, Schwarz's Lemma, Taylor's Theorem, Einstein's Theory of Relativity, Hopf's Degree Theorem, Liouville's Theorem, Riemann's Mapping Theorem, Further Reading, Index Theorem, Sir Roger Penrose, Annular Laurent Series, Brouwer's Fixed Point Theorem, Local Description of Mappings, The Logarithm Function, The Real Integral New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Four and a half centuries have elapsed since complex numbers were first discovered. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
power series centred, dilative rotations, infinitesimal arrow, infinitesimal disc, topological multiplicity, semicircle orthogonal, harmonic duals, infinitesimal squares, new temperature distribution, deformation theorem, use the previous part, central dilation, modular surface, given harmonic function, second intersection point, angular excess, small circle centred, infinitesimal neighbourhood, geometric equality, equidistant curves, complex inversion, disc centred, hyperbolic plane, infinitesimal circles, hemisphere model Key Phrases - Capitalized Phrases (CAPs): (learn more)
Cauchy's Theorem, Fundamental Theorem, Symmetry Principle, Binomial Theorem, Mean Value Theorem, Residue Theorem, General Cauchy Theorem, Schwarz's Lemma, Taylor's Theorem, Einstein's Theory of Relativity, Hopf's Degree Theorem, Liouville's Theorem, Riemann's Mapping Theorem, Further Reading, Index Theorem, Sir Roger Penrose, Annular Laurent Series, Brouwer's Fixed Point Theorem, Local Description of Mappings, The Logarithm Function, The Real Integral New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 47
books:
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35
books
cite this book:
See all 35 books citing this book
- The Theory of Relativity: & Other Essays by Albert Einstein on page 122, page 123, and page 158
- Mathematics and its History by John Stillwell in Front Matter, and Back Matter
- The Schwarz Function and Its Generalization to Higher Dimensions by Harold S. Shapiro in Back Matter
- Fuchsian Groups (Chicago Lectures in Mathematics) by Svetlana Katok in Back Matter
- Algebraic Topology: A First Course (Mathematics Lecture Note Series) by Marvin J. Greenberg in Back Matter
See all 47 books this book cites
- Visualization, Explanation and Reasoning Styles in Mathematics (Synthese Library) by P. Mancosu on page 29, and page 87
- The Art and Craft of Problem Solving by Paul Zeitz on page 131, and Back Matter
- Mathematical Adventures for Students and Amateurs (Spectrum) by David F. Hayes on page 250, and page 257
- Computer Algebra and Geometric Algebra with Applications: 6th International Workshop, IWMM 2004, Shanghai, China, May 19-21, 2004 and International ... Computer Science and General Issues) by Hongbo Li on page 277, and page 296
- Complex Analysis with MATHEMATICA® by William T. Shaw in Front Matter, and Back Matter
See all 35 books citing this book
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