Functional Equations and How to Solve Them
This book is devoted to functional equations of a special type, namely to those appearing in competitions like the International Mathematical Olympiad for high school students or in the William Lowell Putnam Competition for undergraduates. Its aim is to present methods of solving functional equations and related problems and to provide basic information on its history. The intention limits the generality of the treatment; more complicated cases are omitted to keep the exposition understandable for secondary school students. To give a feeling of how it is done I will describe the content of the first two chapters. In the introductory part, contributions by Nicole Oresme, Gregory of Saint Vincent, Cauchy, d’Alembert, Babbage and Ramanujan are presented together with some facts from their lives. Also, a simultaneous solution of two equations is found and then the terminology is fixed. The chapter ends with some simple problems testing understanding of the subject.
The second chapter starts with the Cauchy equation for additive functions and with Jensen’s equation. Some generalizations like Pexider’s and Vincze’s equations are treated. Cauchy’s inequality for subadditive functions is studied as well as the Euler and d’Alembert equations. The author is able in such a way to show important types of equations and ways/tricks necessary for dealing with problems containing them. The book contains many solved examples and problems at the end of each chapter. More than 25 pages at the end of the book offer hints and briefly formulated solutions of those problems. Also a short appendix on Hammel basis is included. The book has 130 pages, 5 chapters and an appendix, a Hints/Solutions section, a short bibliography and an index. It has a nice and clear exposition and is therefore a very readable book, accessible without special or highly advanced mathematics. The book will be valuable for instructors working with young gifted students in problem solving seminars.