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Functional Equations and How to Solve Them (Problem Books in Mathematics)by Christopher G. Small
Synopses & Reviews
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.
Many books have been written on the theory of functional equations, but very few help readers solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap, offering explanatory text and illustrative problems of varied difficulty.
This book covers topics in the theory and practice of functional equations. Special emphasis is given to methods for solving functional equations that appear in mathematics contests, such as the Putnam competition and the International Mathematical Olympiad. This book will be of particular interest to university students studying for the Putnam competition, and to high school students working to improve their skills on mathematics competitions at the national and international level. Mathematics educators who train students for these competitions will find a wealth of material for training on functional equations problems. The book also provides a number of brief biographical sketches of some of the mathematicians who pioneered the theory of functional equations. The work of Oresme, Cauchy, Babbage, and others, is explained within the context of the mathematical problems of interest at the time. Christopher Small is a Professor in the Department of Statistics and Actuarial Science at the University of Waterloo. He has served as the co-coach on the Canadian team at the IMO (1997, 1998, 2000, 2001, and 2004), as well as the Waterloo Putnam team for the William Lowell Putnam Competition (1986-2004). His previous books include Numerical Methods for Nonlinear Estimating Equations (Oxford 2003), The Statistical Theory of Shape (Springer 1996), Hilbert Space Methods in Probability and Statistical Inference (Wiley 1994). From the reviews: Functional Equations and How to Solve Them fills a need and is a valuable contribution to the literature of problem solving. - Henry Ricardo, MAA Reviews The main purpose and merits of the book...are the many solved, unsolved, partially solved problems and hints about several particular functional equations. - Janos Aczel, Zentralblatt
Many books have been written on the theory of functional equations, but very few help readers solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap. Each chapter includes a list of problems associated with the covered material. These vary in difficulty, with the easiest being accessible to any high school student who has read the chapter carefully. The most difficult will challenge students studying for the International Mathematical Olympiad or the Putnam Competition. An appendix provides a springboard for further investigation of the concepts of limits, infinite series and continuity.
Table of Contents
Preface.- An historical introduction.- Functional equations with two variables.- Functional equations with one variable.- Miscellaneous methods for functional equations.- Some closing heuristics.- Appendix: Hamel bases.- Hints and partial solutions to problems.- Bibliography.- Index.
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