This book is less about complex integration per se than it is about areas of mathematics where complex integration is a useful technique. It is a posthumous work of the author, who was at University College London and died in 2011.

The first chapter develops the basics of complex integrals and the Cauchy theorem, and shows how to evaluate several types of real integrals (but not infinite series) using them. It also uses contour integration to develop the theory of Fourier transforms and inverse transforms. The treatment is similar to what is found in introductory texts on complex variables, and is representative rather than comprehensive.

Chapters 2 and 3 deal with hypergeometric functions and with integral equations, respectively. These develop the basics of these subjects and show some applications. The amount of complex integration is relatively small. The most common technique is to use Laplace or Fourier transforms to convert the problem from an integral form to an algebraic form, and then use the Mellin transform or ad hoc complex integration to recover the solution to the original problems. Often there is some ingenuity applied to express the given problem as an integral first.

Chapter 4 deals with orthogonal polynomials. This is done in some generality, with specializations to the functions of mathematical physics. Complex integration makes only a brief appearance here, in the use of the Schläfli integral to represent the polynomial.

The book is aimed at physics undergraduates, but has a good level of rigor and would also be useful for math majors interested in these subjects. There is a set of representative exercises and the end of each chapter, with complete solutions in the back of the book.

Bottom line: well-written and logical, but with a small percentage of complex integration.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.