Tristan Needham’s *Visual Complex Analysis* has been around for a while now and has earned a significant following. It is meant as an introduction to complex analysis, but the approach is genuinely unique. The subject is developed wholly geometrically — there are no epsilons to be found. The arguments constructed are highly innovative; even veterans of the field will find new ideas here. Many arguments boil down to navigating through a series of pictures that are thoughtfully constructed and explained.

A nice example of the approach comes from one of the many fantastic exercises (Ch. 2, #22). It explores the function *P*_{n}(*z*) =(1 + *z*/*n*)^{n} geometrically. The reader is asked to break down this function into a translation, a contraction, and a power mapping, and then explore the effect of these transformations on circular arcs and rays. For large *n*, of course, we see behavior consistent with what we should expect from the exponential map. This limit is discussed in all analysis texts, but is rarely considered as a geometric statement. The book is filled with uncommonly insightful geometric interpretations like this.

Much attention is devoted to inversive geometry and Möbius transformations before advancing to the featured topic, the geometry of differentiation. Indeed, much of the point of the book is to show just how much theory one can extract from simple preservation-of-angle arguments. The effort pays off when the author does define the derivative (although he calls it “amplitwist”), as many traditional analytic results become highly sensible when presented on a solid geometric foundation. For most results involving analytic functions, the goal is to understand deeply why they follow from the fact that having a derivative means that the function behaves locally as a composition of a rotation and a dilation.

Historical and physical context play important roles in the book and are integrated into the narrative in very natural way. The author notes that many ideas in complex analysis developed from physical intuition and works to impart that intuition on the reader. This is well exemplified in his development of vector fields and harmonic flows in the context of complex integration. There are also plenty of interesting topics included as optional sections throughout the text, including an extended discussion of Möbius transformations, non-euclidean geometry, curvature, analytic continuation, and celestial mechanics.

When teaching from this book, one must be prepared for some deviation from traditional texts. The unique approach helps make the topic more accessible and intuitive, but at the cost of some rigor. I have had great success using it as a text, supplementing with some more formal analysis and traditional exercises where appropriate. Alternatively, this book could inform a traditional course as a teacher’s or student’s supplement.

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.

Bill Wood is a Visiting Assistant Professor of Mathematics at Hendrix College in Conway, Arkansas.

## Comments

## Allen Stenger

This book develops a theory of complex analysis based on geometric ideas instead of on analytic concepts such as power series or the Cauchy-Riemann equations. Compared to many current math textbooks, this one is very concrete, and spends most of its time working with examples rather than developing and applying general theorems.

The author summarizes his intentions by saying (p. 222), "The basic philosophy of this book is that while it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by bringing you nearer to the Truth." The book's key concept is the "amplitwist" (p. 194), a portmanteau of amplify and twist. The amplitwist is a geometric view of the complex derivative: an analytic function will map an infinitesimal vector at a point to another infinitesimal vector that is grown or shrunk and rotated compared to the original.

The book is very leisurely and discursive. We don't even get to the definition of an analytic function until p. 197. The approach is conceptual rather than rigorous, and the book tends to argue by example and picture rather than giving formal proofs. The lack of rigor is not a problem in itself, but it does raise the question of audience: for a first course, math majors would want more rigor, and physics and engineering majors would want more mechanics. The book has many charms and fresh outlook, but I think it may be more successful as a monograph, and I'm skeptical that it can be the sole textbook for an introductory course.

I also felt that this is more of a geometry book than a complex analysis book. The geometry is relevant and interesting, but it's not what I call complex analysis. As (perhaps) a confession of bias, I'll state that my all-time favorite complex analysis book is E. C. Titchmarsh's Theory of Functions, followed not too far behind by the similar but more modern

Complex Analysisof Bak and Newman (Springer, 1999).