Functional Equations and How to Solve Them
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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, or by showing that all solutions have a particular property. The emphasis here will be on the development of those tools which are most useful in assigning 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, with the easiest problems being accessible to any high school student who has read the chapter carefully. The most difficult 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 book ends with an appendix containing topics that provide a springboard for further investigation of the concepts of limits, infinite series and continuity.
Preface viiAn historical introduction 1Preliminary remarks 1Nicole Oresme 1Gregory of Saint-Vincent 4Augustin-Louis Cauchy 6What about calculus? 8Jean d'Alembert 9Charles Babbage 10Mathematics competitions and recreational mathematics 16A contribution from Ramanujan 21Simultaneous functional equations 24A clarification of terminology 25Existence and uniqueness of solutions 26Problems 26Functional equations with two variables 31Cauchy's equation 31Applications of Cauchy's equation 35Jensen's equation 37Linear functional equation 38Cauchy's exponential equation 38Pexider's equation 39Vincze's equation 40Cauchy's inequality 42Equations involving functions of two variables 43Euler's equation 44D'Alembert's equation 45Problems 49Functional equations with one variable 55Introduction 55Linearization 55Some basic families of equations 57A menagerie of conjugacy equations 62Finding solutions for conjugacy equations 64The Koenigs algorithm for Schroder's equation 64The Levy algorithm for Abel's equation 66An algorithm for Bottcher's equation 66Solving commutativity equations 67Generalizations of Abel's and Schroder's equations 67General properties of iterative roots 69Functional equations and nested radicals 72Problems 75Miscellaneous methods for functional equations 79Polynomial equations 79Power series methods 81Equations involving arithmetic functions 82An equation using special groups 87Problems 89Some closing heuristics 91Appendix: Hamel bases 93Hints and partial solutions to problems 97A warning to the reader 97Hints for Chapter 1 97Hints for Chapter 2 102Hints for Chapter 3 107Hints for Chapter 4 113Bibliography 123Index 125
