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Visual Complex Analysis: 25th Anniversary Edition Anniversary Edition

by Tristan Needham (Author), Roger Penrose (Foreword)

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FROM THE AUTHOR:

The new Foreword by Sir Roger Penrose is only available through the "Read Sample" attached to the Kindle edition; for some reason, it is not included in the free sample for the paperback.

All legitimate copies of VCA produced by Oxford University Press are crisply printed on high-quality paper. If you obtain a shoddily printed copy, it's a fake: please return it and purchase a genuine OUP copy.

****************************************************************************************************
The fundamental advance in the new 25th Anniversary Edition is that the original 501 diagrams now include brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way―as a highbrow comic book!

The 25th Anniversary Edition also features a new Foreword by Sir Roger Penrose, as well as a new Preface by the author.

Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years.

This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality.

REVIEWS FROM THE BACK COVER:

"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, former president of Pixar and Disney Animation Studios

"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

"If your budget limits you to only buying one mathematics book in a year then make sure that this is the one that you buy."
Mathematical Gazette

". . . it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer."
Newsletter of the European Mathematical Society

". . . an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas 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

"Newton would have approved . . . a fascinating and refreshing look at a familiar subject . . . essential reading for anybody with any interest at all in this absorbing area of mathematics. "
Times Higher Education Supplement

"It is . . . a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach."
Ian Stewart, New Scientist

"The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory . . . this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs . . . Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes."
Frank Farris, American Mathematical Monthly

"The arguments constructed are highly innovative; even veterans of the field will find new ideas here. . . .This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics but has presented it in a clear and compelling way. Visual Complex Analysis is worthy of the accolades it has received."
MAA Reviews

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ISBN-10

0192868926
ISBN-13

978-0192868923
Edition

Anniversary
Publisher

Oxford University Press
Publication date

February 28, 2023
Language

English
Dimensions

9.99 x 1.3 x 7 inches
Print length

720 pages

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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 "..it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer." -- Newsletter of the European Mathematical Society "Newton would have approved... a fascinating and refreshing look at a familiar subject... 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. 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 tradition 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 "an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas 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 "If your budget limits you to only buying 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, former president of Pixar and Disney Animation Studios "The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory... this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs: Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes." -- Frank Farris, American Mathematical Monthly "This informal style is excellently judged and works extremely well. Many of the arguments presented will be new even to experts, and the book will be of great interest to professionals working in either complex analysis or in any field where complex analysis is used." -- David Armitage, Mathematical Reviews "The arguments constructed are highly innovative; even veterans of the field will find new ideas here. This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics, but has presented it in a clear and compelling way. Visual Complex Analysis is worthy of the accolades it has received" -- MAA Reviews "This new edition of Visual Complex Analysis applies Newton's geometrical methods from the Principia and his concept of ultimate equality to Complex Analysis." -- MathSciNet

From the Author

VCA@25 was officially published on Feb 28, 2023

Publisher ‏ : ‎ Oxford University Press
Publication date ‏ : ‎ February 28, 2023
Edition ‏ : ‎ Anniversary
Language ‏ : ‎ English
Print length ‏ : ‎ 720 pages
ISBN-10 ‏ : ‎ 0192868926
ISBN-13 ‏ : ‎ 978-0192868923
Item Weight ‏ : ‎ 3.7 pounds
Dimensions ‏ : ‎ 9.99 x 1.3 x 7 inches
Best Sellers Rank: #116,529 in Books (See Top 100 in Books)
- #6 in Topology (Books)
- #15 in Mathematical Analysis (Books)
- #42 in Calculus (Books)
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(114)

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Roger Penrose
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Tristan Needham
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VISUAL COMPLEX ANALYSIS: 25th ANNIVERSARY EDITION (with a new Foreword by Roger Penrose) was published on February 28th, 2023.
MARCH 2024 UPDATE TO VDGF "DYNAMIC ERRATA": VDGF"dot"space, my website for "Visual Differential Geometry and Forms", contains both a STATIC INITIAL ERRATA (corrected in the current printing) and a DYNAMIC ERRATA, listing errors that were reported *after* PUP froze the current version. The latest update to the DYNAMIC ERRATA was posted in March, 2024.
******************************************************************************************************************************************
Tristan Needham (son of the distinguished social anthropologist Rodney Needham) grew up in Oxford, England, where he attended the Dragon School (with Stephen Wolfram and Hugh Laurie).
He studied physics at Merton College, Oxford, before moving to the Mathematical Institute, where he enjoyed the great privilege of studying black holes under the supervision of Sir Roger Penrose.
Tristan received his DPhil in 1987, and joined the faculty of the University of San Francisco in 1989. His current focus is Differential Geometry, but Complex Analysis, General Relativity, and the history of science are abiding loves. His continuing mission is to seek out new intuitive forms of understanding, and new visualizations.
His book "Visual Complex Analysis" (Oxford University Press) won first prize in the National Jesuit Book Award Competition. An earlier paper received the Mathematical Association of America's Carl B. Allendoerfer Award.
His new book, "Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts", was published by Princeton University Press on July 13, 2021.
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Frederick Lopez
EXCELLENT QUALITY, PROMPT DELIVERY.
Reviewed in the United States on March 17, 2026
Format: HardcoverVerified Purchase

EXCELLENT QUALITY, PROMPT DELIVERY.
THANK YOU SO MUCH.

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DaveM
Great book!!
Reviewed in the United States on January 5, 2024
Format: PaperbackVerified Purchase
The graphics are phenomenal and insightful.

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2 people found this helpful

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Jean M. Gawron
For those who already love the complex plane and want a more intuive understanding
Reviewed in the United States on May 25, 2023
Format: Paperback
Quite simply, I love this book. I came to it by way of Roger Penrose's Road to Reality, which says some very flattering things about Needham, and being a fan of Penrose's book, I tried out Needham's. If you loved Road to Reality, and learned something from it, there is a good chance you will also love Visual Complex Analysis. Having said that, they are not the same kind of book. Penrose has an end in mind, and the book arcs toward it, supplying you at each step with what you need to take the next. Needham's book is a leisurely tour through a particularly beautiful part of the landscape of math. You are there for the views, to stand lingering at the precipice watching the sun set. If you an engineering or physics student interested in complex analysis because you need to get on with your physics and engineering work, this book is not for you. Complex Differentiation and the Cauchy-Riemann equations are not introduced until Chapter 4. There is way more time spent on topics like the Riemann sphere and Mobius transformations than any reasonable textbook could afford. There are extended discussions of individual functions on the complex plane, like the exponential function, the log function, the trigonometric functions. Why? Because they are beautiful. Because these are all places where there is an exploision of hidden structure in the complex plane. Needham has a gift for revisiting elementary topics and bringing out their beauty. The discussion (and proof?) of Euler's Equation is inspiring. All this is very untextbooklike. Also untextbooklike: there are few exercises, and they are not really designed to work the reader through a minimal skill set. Like the rest of the book, they focus on the areas where the views are good. Readers turning to complex analysis for practical reasons are better off with Gamelin. But if you want insight, and a fresh look at old ideas, and the occasional flash of purely geometric intuition, then this book is for you. Or if you too believe that the complex plane is the doorway to some beautiful mathematics, then this book is especially for you.

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mitchus k.
Beautiful!
Reviewed in the United States on April 22, 2023
Format: Kindle
This is a great book for getting visual intuitions for the universe of complex numbers. The 25th anniversary edition has been refreshed with new content and improvements all around. Can heartily recommend

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5 people found this helpful

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Leeber Cohen
I wish the solved answered to the excercies were available.
Reviewed in the United States on February 10, 2025
Format: PaperbackVerified Purchase
This is a classic publication in complex analysis. Unfortunately the worked out solutions to the excercises are not available. It is clearly worth reading the text and having it in your library. It gave me a little better understanding of differentia forms. It would be nice to have a version written for lifetime learners with fully solved problems. I am not sure if there is enough of an audience to make this possible.

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2 people found this helpful

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Thomas Mitchell
Bad Printing
Reviewed in the United States on August 20, 2023
Format: PaperbackVerified Purchase
The contents are truly unparalleled. Needham gives you eyes to see the beauty of the subject.

The print quality is a crime against books. It is a pain to read this digitally mangled text printed on office copy paper.

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9 people found this helpful

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Amazon Customer
Book has loose pages but no other problems
Reviewed in the United States on November 25, 2023
Format: PaperbackVerified Purchase
See title

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i.i.C.T DR GERHARD KROISS
"OOPS, WE MADE A MISTAKE!"
Reviewed in the United States on April 17, 2025
Format: Paperback
That is what you can sometimes hear from radicals. At the end of the 19th century, there seemed to be a final solution to the program of the ancient Greek EUCLID to transcend human reasoning. He had fixed the starting positions of his mathematical game with axioms and definitions, but he had not yet supplied the rules, the logic, to get to theorems based solely on the starting positions. The German FREGE seemed to have supplied these rules. As a young man, I had been in contact with one his students (CARNAP). I had argued: “Rudolf, if you prove your FIRST logical theorem, on which logic do you base its proof?” Rudolf refused to answer this question in writing, which he had understood very well, but he sent me to his friend STEGMUELLER in Munic, who pretended (?) to not even understand me. But the New Math revolution continued, eradicated or discredited all previous mathematics—unless it was not sanctified in one of their abstract products. One of these guys (WEIL), however, then exclaimed to his colleagues: “OOPS, WE MADE A MISTAKE! Math students could learn much more from Euler’s precalculus books than from any modern one.” COURANT had predicted the downhill slide of mathematics in the 1930s and written his calculus and mathematical physics books to prevent that. However, it is very hard to reach the brains of the brainwashed followers of a revolution. NEEDHAM tries to do that, too, but I think his target audience (people with an interest in mathematics) is too narrow. Mathematics is the unique genetic signature of humans, no other species does it. Needham should better think about reaching humans and handing over thinking skills to them—in order to help them, and, therefore, mathematics, too.

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Pierre Bulens
Remarquable
Reviewed in Belgium on April 11, 2024
Format: PaperbackVerified Purchase

J'avais déjà la première édition de cet excellent ouvrage que je recommande à tous ceux qui connaissent ou veulent apprendre l'analyse complexe. Une façon tout à fait différente d'aborder cette matière. Une autre façon est développée dans un autre ouvrage remarquable : Applied and Computational Complex Analysis de Peter Henrici (3 tomes).
J'ai offert cette édition du 25ème anniversaire à mon fils, et je l'ai achetée pour moi. Un régal de s'y replonger.

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P.J.Walters
Visual Complex Analysis
Reviewed in the United Kingdom on December 29, 2024
Format: PaperbackVerified Purchase
This is not a book you can easily read- you have to work through it and complete the exercises. However, the effort is rewarding because you discover a detailed and visual introduction to complex analysis. Needham draws on Isaac Newton’s development of a geometric calculus in Principia. The book also mentions some of the key figures in the development of the subject. A rewarding visual introduction to a fascinating subject of importance in many other fields of study, such as physics and engineering. Highly recommended for serious study.

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Mario Lemelin
A interresting different approach to complex analysis
Reviewed in Canada on August 26, 2024
Format: PaperbackVerified Purchase
I liked the fact that there are numerous figures, Amplitwist (antiderivative), Mobius transformation and non-euclidean geometry.

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Marc Magrans De Abril
Fantastic
Reviewed in Germany on October 10, 2025
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Fantastic

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Miguel Ángel
Sobre variable compleja
Reviewed in Mexico on July 22, 2025
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Exposición clara y concisa de los temas de variable compleja

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5.0 out of 5 stars Remarquable

5.0 out of 5 stars Visual Complex Analysis

5.0 out of 5 stars A interresting different approach to complex analysis

5.0 out of 5 stars Fantastic

5.0 out of 5 stars Sobre variable compleja

Remarquable

Visual Complex Analysis

A interresting different approach to complex analysis

Fantastic

Sobre variable compleja