Functional Equations and How to Solve Them / Edition 1

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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.

About the Author:
Christopher Small is a Professor in the Department of Statistics and Actuarial Science at the University of Waterloo

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Editorial Reviews

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"This book is devoted to functional equations of a special type, namely to those appearing in competitions … . The book contains many solved examples and problems at the end of each chapter. … The book has 130 pages, 5 chapters and an appendix, a Hints/Solutions section, a short bibliography and an index. … The book will be valuable for instructors working with young gifted students in problem solving seminars." (EMS Newsletter, June, 2008)

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Product Details

  • ISBN-13: 9780387345390
  • Publisher: Springer New York
  • Publication date: 8/28/2007
  • Series: Problem Books in Mathematics Series
  • Edition description: 2007
  • Edition number: 1
  • Pages: 131
  • Product dimensions: 5.90 (w) x 9.10 (h) x 0.40 (d)

Table of Contents

Preface     vii
An historical introduction     1
Preliminary remarks     1
Nicole Oresme     1
Gregory of Saint-Vincent     4
Augustin-Louis Cauchy     6
What about calculus?     8
Jean d'Alembert     9
Charles Babbage     10
Mathematics competitions and recreational mathematics     16
A contribution from Ramanujan     21
Simultaneous functional equations     24
A clarification of terminology     25
Existence and uniqueness of solutions     26
Problems     26
Functional equations with two variables     31
Cauchy's equation     31
Applications of Cauchy's equation     35
Jensen's equation     37
Linear functional equation     38
Cauchy's exponential equation     38
Pexider's equation     39
Vincze's equation     40
Cauchy's inequality     42
Equations involving functions of two variables     43
Euler's equation     44
D'Alembert's equation     45
Problems     49
Functional equations with one variable     55
Introduction     55
Linearization     55
Some basic families of equations     57
A menagerie of conjugacy equations     62
Finding solutions for conjugacy equations     64
The Koenigs algorithm for Schroder's equation     64
The Levy algorithm for Abel's equation     66
An algorithm for Bottcher's equation     66
Solving commutativity equations     67
Generalizations of Abel's and Schroder's equations     67
General properties of iterative roots     69
Functional equations and nested radicals     72
Problems     75
Miscellaneous methods for functional equations     79
Polynomial equations     79
Power series methods     81
Equations involving arithmetic functions     82
An equation using special groups     87
Problems     89
Some closing heuristics     91
Appendix: Hamel bases     93
Hints and partial solutions to problems     97
A warning to the reader     97
Hints for Chapter 1     97
Hints for Chapter 2     102
Hints for Chapter 3     107
Hints for Chapter 4      113
Bibliography     123
Index     125
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