Why this book?

Since 1985, I have been fortunate to have taught the theory of complex variables for several courses in both the USA and the UK. In the USA I lectured a course on advanced calculus for engineers and scientists at MIT, and in the UK I have given tutorials on the subject to undergraduate students in mathematics at both Cambridge and Oxford. Indeed, draft versions of this text have been inflicted on my students at Balliol and, more recently, at St. Catherine's over the last fourteen years. Few topics have given me such pleasure to teach, given the rich yet highly accessible structure of the subject, and it has at times formed the subject of my research, notably in its development into twistor theory, and latterly in its applications to financial mathematics. A parallel thread of my work has been in the applications of computer algebra and calculus systems, and in particular Mathematica®, to diverse topics in applied mathematics. This book is in part an attempt to use Mathematica to illuminate the topic of complex analysis, and draws on both these threads of my experience.

The book attempts also to inject some new mathematical themes into the topic and the teaching of it. These themes I feel are, if not actually missing, under-emphasized in most traditional treatments. It is perfectly possible for students to have had a formal training in mathematics that leaves them unaware of many key and/or beautiful topics.