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Functional Equations and How to Solve Them
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.
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.