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.

Table of Contents

1 Geometry and Complex Arithmetic

2 Complex Functions as Transformations

3 Mobius Transformations and Inversion

4 Differentiation: The Amplitwist Concept

5 Further Geometry of Differentiation

6 Non-Euclidean Geometry*

7 Winding Numbers and Topology

8 Complex Integration: Cauchy's Theorem

9 Cauchy's Formula and Its Applications

10 Vector Fields: Physics and Topology

11 Vector Fields and Complex Integration

12 Flows and Harmonic Functions

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

About the Author

Tristan Needham, Emeritus Professor of Mathematics, University of San Francisco

Tristan Needham (son of the distinguished social anthropologist Rodney Needham) grew up in Oxford, England. 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 continuing mission is to seek out new intuitive forms of understanding, and new visualizations.

He is also the author of Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts.