The author of this slender but useful volume sets the stage by writing, “Functional equations are often the most difficult problems to be found on mathematics competitions because they require a minimal amount of background theory and a maximal amount of ingenuity.” His aim is to provide participants and coaches with enough theory to serve as a framework for successful problem solving in contests such as the International Mathematical Olympiad and the Putnam. As the author points out, books on problem solving usually include some discussion of functional equations, but these analyses are often buried within a chapter on algebra problems and fail to develop enough methodology to be more than a bag of tricks. One of the better treatments can be found in Problem-Solving Strategies by Arthur Engel (Springer, 1998).
The book under review deals mostly with real-valued functions of a single real variable, but also looks at functions with complex arguments and functions defined on the natural numbers. The author makes the case for setting the chapter on functional equations in two or more variables before a discussion of functional equations in a single variable, pointing out that an equation in two or more variables is formally equivalent to a family of simultaneous equations in one variable. In the context of high school competitions, a knowledge of limits and continuity cannot be assumed, so the author discusses other regularity conditions that can be substituted (monotonicity, boundedness, and so forth).
Small opens with an interesting chapter on the history of functional equations, beginning with the work of Nicole Oresme in the fourteenth century and then moving on to contributions by Cauchy, d’Alembert, Babbage, and Ramanujan. The last official chapter (“Some closing heuristics”) is a list of nine rules of thumb for dealing with functional equations. The material in between consists of good exposition, illuminating examples, and excellent problems, many from math competitions. The influence of Janos Aczél, an emeritus colleague of the author at the University of Waterloo, is obvious; and this book serves as an excellent introduction to more comprehensive works such as Aczél’s Lectures on Functional Equations and Their Applications. In summary, Functional Equations and How to Solve Them fills a need and is a valuable contribution to the literature of problem solving.
Henry Ricardo (firstname.lastname@example.org) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.