A Friendly Approach to Complex Analysis

This book has emerged from a course in an undergraduate programme for mathematics with economics at the London School of Economics. Perhaps for this reason it omits coverage of many familiar topics such as conformal mappings, analytic continuation, Schwarz-Chrisoffel transformations, Green’s theorem and Gamma functions. And, although clues are provided regarding the practical importance of complex analysis, I could find no examples of its use in economics (does it have any?).

In essence, therefore, the book is restricted to chapters on complex numbers and elementary functions, differentiability, complex integration, Taylor-Laurent series and harmonic functions. Indeed, the authors summarize the core contents by proclaiming the equivalence of various central ideas. Specifically, if \(D\) is an open path-connected set and \(f:D\subset\mathbb{C}\longrightarrow\mathbb{C}\), then the following statements are equivalent:

1. \(f'(z)\) exists on \(D\).
2. \(f^{(n)}(z)\) exists on \(D\) for all \(n \geq 0\).
3. The Cauchy-Riemann equations are satisfied on \(D\).
4. For each simply connected subdomain \(S\) of \(D\), there exists a holomorphic function \(F:S\longrightarrow\mathbb{C}\), such that \(F'(z)=f(z)\) for all \(z\) in \(S\).
5. \(f\) satisfies the first Cauchy integral theorem
6. \(f\) has a Taylor series expansion in \(D\).

To my mind, this book is ‘friendly’ because the treatment is rigorous and makes no concessions to lazy-mindedness. Another reason is that the narrative always conveys a sense of direction, and it makes many valuable comparisons with real and complex analysis. Finally, for those committed to self-study, more than one third of the book consists of complete solutions to all the exercises (more than 150 of them). Some of these are routine, but many are very challenging.

Overall, this is a very nice addition to the existing literature on complex analysis. It is rich in ideas such as the Dirichlet problem, Riemann hypothesis, winding numbers, and Picard’s theorem. At times, the style is almost conversational and there is a very effective use of diagrams at key points in the text (e.g. in the proof of the Cauchy integral formula).

Peter Ruane has taught mathematics across the age-range (5 year olds to 55 year olds). That is, from school arithmetic to transfinite arithmetic etc.