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Functional Equations and How to Solve Them : Problem Books in Mathematics - Christopher G. Small

Functional Equations and How to Solve Them

Problem Books in Mathematics


Published: 8th August 2007
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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.

From the reviews:

"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)

Prefacep. vii
An historical introductionp. 1
Preliminary remarksp. 1
Nicole Oresmep. 1
Gregory of Saint-Vincentp. 4
Augustin-Louis Cauchyp. 6
What about calculus?p. 8
Jean d'Alembertp. 9
Charles Babbagep. 10
Mathematics competitions and recreational mathematicsp. 16
A contribution from Ramanujanp. 21
Simultaneous functional equationsp. 24
A clarification of terminologyp. 25
Existence and uniqueness of solutionsp. 26
Problemsp. 26
Functional equations with two variablesp. 31
Cauchy's equationp. 31
Applications of Cauchy's equationp. 35
Jensen's equationp. 37
Linear functional equationp. 38
Cauchy's exponential equationp. 38
Pexider's equationp. 39
Vincze's equationp. 40
Cauchy's inequalityp. 42
Equations involving functions of two variablesp. 43
Euler's equationp. 44
D'Alembert's equationp. 45
Problemsp. 49
Functional equations with one variablep. 55
Introductionp. 55
Linearizationp. 55
Some basic families of equationsp. 57
A menagerie of conjugacy equationsp. 62
Finding solutions for conjugacy equationsp. 64
The Koenigs algorithm for Schroder's equationp. 64
The Levy algorithm for Abel's equationp. 66
An algorithm for Bottcher's equationp. 66
Solving commutativity equationsp. 67
Generalizations of Abel's and Schroder's equationsp. 67
General properties of iterative rootsp. 69
Functional equations and nested radicalsp. 72
Problemsp. 75
Miscellaneous methods for functional equationsp. 79
Polynomial equationsp. 79
Power series methodsp. 81
Equations involving arithmetic functionsp. 82
An equation using special groupsp. 87
Problemsp. 89
Some closing heuristicsp. 91
Appendix: Hamel basesp. 93
Hints and partial solutions to problemsp. 97
A warning to the readerp. 97
Hints for Chapter 1p. 97
Hints for Chapter 2p. 102
Hints for Chapter 3p. 107
Hints for Chapter 4p. 113
Bibliographyp. 123
Indexp. 125
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387345390
ISBN-10: 0387345396
Series: Problem Books in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 131
Published: 8th August 2007
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.4 x 15.6  x 0.7
Weight (kg): 0.21