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Integration with complex numbers: a primer on complex analysis
McMaster B., McCluskey A., Oxford University Press, Oxford, UK, 2022. 288 pp. Type: Book (978-0-192846-43-3)
Date Reviewed: Feb 9 2023

Complex analysis is a rich subject that can bewilder the beginning student with the nonintuitive behavior of its objects and the wide range of applications to which it has been applied. This highly readable introduction engages only a severely restricted part of the subject, but does so in a way that will equip the student to expand into other areas.

Students of physics and engineering often study complex analysis to learn how conformal mapping or the Schwartz-Christoffel transformation can help solve physical problems, but the reader will search this volume in vain for any reference to them. Riemann surfaces are central to some areas of quantum physics, but this volume says nothing about them. It is built around one simple but intriguing feature of complex variables: the integration of a function of real variables, particularly one involving transcendental components, can often become much simpler if the function is first transformed into a function of complex variables. In complex space, Cauchy’s theorem allows arbitrary contours to be replaced with simple circles, and the residue theorem then leads immediately to the required integral simply by identifying the singularities enclosed in the curve.

This result, while intrinsically interesting, may have a limited audience. Many engineering students learn to integrate in their college calculus courses, but then work productively for years without ever having to evaluate an integral. But the authors’ purpose is not primarily to make integration easier. This application serves as a concrete example to introduce the student to complex analysis. Even this limited objective requires the student to understand the structure and algebra of complex numbers, the complex analogs of familiar structures such as power series and functions, and the definition of paths in the complex plane, and with this background, the full scope of the subject becomes much more accessible.

The first 40 percent of the book consists of a review in three parts of mathematical fundamentals, each of which is then expanded to include complex variables. Chapter 1, a refresher on sets, sequences, and series, leads to chapter 2, defining complex numbers. Chapter 3, on real functions and their derivatives, similarly leads to chapter 4, complex functions. The message throughout is on emphasizing the commonalities between the two domains.

Chapter 5, discussing real integration, finds its complex counterpart in the rest of the book. Chapter 6 introduces the notion of paths in the complex plane, allowing exposition of Cauchy’s theorem in chapter 7, Taylor series of complex variables in chapter 8, and the residue theorem in chapter 9. Then chapter 10 pulls these pieces together by exhibiting a real function whose integration is extremely subtle, and showing how transforming it to a function of complex variables leads directly to a solution.

Throughout the book, the authors defer details of some proofs, and these details are provided in chapter 11.

This volume, aimed at the undergraduate student, is a pedagogical masterpiece. The explanations are lucid and well motivated, and the text naturally leads the student from familiar concepts to new ones. Instructors who do not feel compelled to offer a breadth-first discussion of the field can expect excellent results in classes built around this volume, and it is also a great tool for self-study.

Reviewer:  H. Van Dyke Parunak Review #: CR147548
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