Functional Equations And How to Solve Them

Over the years, a number of books have been written on the theory of functional equations. However, very little has been published which helps readers to solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap. The student who encounters a functional equation on a mathematics contest will need to investigate solutions to the equation by finding all solutions (if any) or by showing that all solutions have a particular property. Our emphasis will be on the development of those tools which are most useful in giving a family of solutions to each functional equation in explicit form.

At the end of each chapter, readers will find a list of problems associated with the material in that chapter. The problems vary greatly diffculty, with the easiest problems being accessible to any high school student who has read the chapter carefully. The most diffcult problems will be a reasonable challenge to advanced students studying for the International Mathematical Olympiad at the high school level or the William Lowell Putnam Competition for university undergraduates.

The modern theory of functional equations can occur in a very abstract setting that is quite inappropriate for the most high school students. However, the abstraction of some parts of the modern theory reflects the fact that functional equations can occur in diverse settings: functions on the natural numbers, the integers, the reals, or the complex numbers can all be studied within the subject area of functional equations. Most of the time, the functions in this book are real-valued functions of a single real variable. However, readers will also find functions with complex arguments and functions defined on natural numbers in these pages. In some cases, equations for functions between circles will also crop up. The book ends with an appendix containing topics that provide a springboard for further investigation of the concepts of limits, infinite series and continuity.