Complex Analysis with Mathematica

News: "Complex Analysis with Mathematica" honoured by American Library Association

Every year The American Library Association gives a number of books the award of "Outstanding Academic Title" status in their review publication Choice. "Complex Analysis with Mathematica" has been given this accolade in the January 2008 edition of Choice. The book was also reviewed in their September 2007 edition, where it was described as a "truly next-generation book" and "Essential" (see below for full review).


September 2007: Review in American Library Association “Choice” magazine

Reprinted with permission from CHOICE, © American Library Association. Contact for permission to reproduce or redistribute. CHOICE reviews are available on-line at

Traditionally, teaching complex analysis requires overcoming steep pedagogical hurdles. Complex analytic functions, the objects of study, have graphs that form two-dimensional subsets of four-dimensional space, so their visualization (if it happens at all) requires indirect approaches--the more the better. Also, many important results in complex analysis (conformal mapping) express themselves in only very limited examples such as one may work out by hand. So computers should transform the teaching of the subject, as Tristan Needham pointed out in his breakthrough 1997 Visual Complex Analysis (CH, Jan'98, 35-2767). But Needham avoided integrating computer technology into his exposition, presumably because of limited distribution of powerful hardware and awkwardness of then-available software. Shaw, taking advantage of considerable technological progress, offers a truly next-generation book: Mathematica's graphics make old ideas leap off the page, and Mathematica's computational powers permit students to interact effectively with complicated and typical examples. Technology aside, Shaw expands typical undergraduate complex analysis itinerary with excursions through dynamics and fractals, solution of cubic and quartic equations, hyperbolic tessellations, and even an introduction to spinors and twistors, important tools in contemporary approaches to relativity. A vital library resource. Summing Up: Essential. General readers; lower- and upper-division undergraduates; professionals; two-year technical program students. -- D. V. Feldman, University of New Hampshire

July 2007: New notebooks with enhanced Version 6 information

Having played with Mathematica 6.0.1 for a few days, I have now produced updates to the material on the CD with improved features. These can now be downloaded as rather large files with all the output, or smaller files with no output included (recommended)

Smaller files are here; larger files are here. Coming soon… Demonstrations…

May 2007 : Mathematica 6 released!

Tips and hints for making the most of version 6 will be posted here during the course of summer 2007. To get you going here are the passwords for accessing the files in the “Goodies” directory. Note that these were produced with an early test version of Mathematica 6 so there will probably be a few issues. Check back at the end of May for an update and any revised notebooks. The book is essentially Mathematica 6 compatible and only a few changes are needed in a very small number of places, and most of this is covered in the Goodies zone.

The first password is:


The second password is:


I recommend you copy and paste where possible, otherwise type carefully and note there are 1s and Is.

E-mail me with any issues!

June 2006

The ISBN is 0521836263

This was published by Cambridge University Press in the UK in mid-April of 2006 - in the US it was published at the end of May. You can see the CUP catalogue information and buy direct from the publisher - click on the right geographical area in the CUP global site to get pricing and ordering information.

Tips on using the CD – if you have problems.

Errata – printed book

The table of contents is available.

Here are the editorial reviews:

"William Shaw's Complex Analysis with Mathematica is a remarkable achievement. It masterfully combines excellent expositions of the beauties and subtlety of complex analysis, and several of its applications to physical theory, with clear explanations of the flexibility and the power of Mathematica for computing and for generating marvellous graphical displays." 
Roger Penrose, University of Oxford

"This is an innovative text in which the basic ideas of complex analysis are skillfully interwoven with geometry, chaos and physics through the learning and repeated application of Mathematica. This text moves from complex numbers, quadratic and cubic equations through to the Schwarz-Christoffel transformation and four-dimensional physics, and at each stage promotes understanding through geometric intuition and reader participation. It should appeal to anyone with an interest in the geometric side of complex analysis." 
Alan F. Beardon, University of Cambridge

"This book provides an inspiring way to learn complex analysis thanks to the inclusion of many topics of current interest in the field as well as the integration of highly visual Mathematica routines throughout. The book is sure to excite students about the field." 
Bob Devaney, Boston University

The preface contains a summary of the book and some suggestions for its use in courses, and is available here. The index is available here. A picture of the front and back of the book is available here.  You can also see some short excerpts from sample chapters as follows – these illustrate some of the ways the book can enhance a traditional complex variable course (note that these are PDFs produced by the author to a downloadable size and do not have as good quality graphics as the published work):

The first few pages of Chapter 4 are available here. They show how the classical theory of polynomial equation-solving can be extended via Newton-Raphson iteration, and give a brief explanation of Cayley’s solution for a quadratic and the reason for his difficulty with the cubic (fractals!).

In Chapter 21 I look at using Mathematica to illuminate the theory of Schwarz-Christoffel transformations.

Chapter 22 contains some hyperbolic tilings – this excerpt shows the “Ghosts and Birdies” tiling in one representation of the hyperbolic plane and the full version of the cover image in another (see below as well).

You can also order the book on-line from,, ,,, Be warned that there are some fake listings of “used” books from booksellers selling new books in the Amazon marketplace! Some booksellers are even trying to charge above list price for brand new books passed off as used (do not ask me why someone would do it this way round). The UK list price is £40.00 and the US list is $75.00. Do NOT pay more than this, shipping aside.

Check back for information on:

Solutions manual for  “Complex Analysis with Mathematica”;

 “Complex Analysis with Mathematica” and Mathematica technologies beyond version 5.2.

Another view of a hyperbolic tiling, from "Complex Analysis with Mathematica: