Functional Equations and How to Solve Them (Problem Books in Mathematics)

143 Pages·2006·1.03 MB·English

by  Christopher G. Small

Problem Books in Mathematics
Edited by P. Winkler
Problem Books in Mathematics
Series Editor:Peter Winkler
Pell’s Equation
by Edward J.Barbeau
Polynomials
by Edward J.Barbeau
Problems in Geometry
by Marcel Berger,Pierre Pansu,Jean-Pic Berry,and Xavier Saint-Raymond
Problem Book for First Year Calculus
by George W.Bluman
Exercises in Probability
by T.Cacoullos
Probability Through Problems
by Marek Capin´ski and Tomasz Zastawniak
An Introduction to Hilbert Space and Quantum Logic
by David W.Cohen
Unsolved Problems in Geometry
by Hallard T.Croft,Kenneth J.Falconer,and Richard K.Guy
Berkeley Problems in Mathematics,(Third Edition)
by Paulo Ney de Souza and Jorge-Nuno Silva
The IMO Compendium:A Collection of Problems Suggested for the
International Mathematical Olympiads:1959-2004
by Dusˇan Djukic´,Vladimir Z.Jankovic´,Ivan Matic´,and Nikola Petrovic´
Problem-Solving Strategies
by Arthur Engel
Problems in Analysis
by Bernard R.Gelbaum
Problems in Real and Complex Analysis
by Bernard R.Gelbaum
(continued after index)
Christopher G. Small
Functional Equations
and How to Solve Them
Christopher G.Small
Department of Statistics & Actuarial Science
University of Waterloo
200 University Avenue West
Waterloo N2L 3G1
Canada
[email protected]
Series Editor:
Peter Winkler
Department of Mathematics
Dartmouth College
Hanover,NH 03755
USA
[email protected]
Mathematics Subject Classification (2000):39-xx
Library ofCongress Control Number:2006929872
ISBN-10:0-387-34534-5 e-ISBN-10:0-387-48901-0
ISBN-13:978-0-387-34534-5 e-ISBN-13:978-0-387-48901-8
Printed on acid-free paper.
© 2007 Springer Science+Business Media,LLC
All rights reserved.This work may not be translated or copied in whole or in part without the
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2
1.5
f(x)
1
0.5
0
-2 -1 0 1 2
x
-0.5
-1
-1.5
f(x)+f(2x)+f(3x)=0
for all real x.
This functional equation is satisfied by the function f(x) ≡ 0, and also by
thestrangeexamplegraphedabove.Tofindoutmoreaboutthisfunction,see
Chapter 3.
4
f(x)
2
0
-4 -2 0 x 2 4
-2
-4
f(f(f(x)))=x
Can you discover a function f(x) which satisfies this functional equation?
Contents
Preface ........................................................ ix
1 An historical introduction ................................. 1
1.1 Preliminary remarks ..................................... 1
1.2 Nicole Oresme .......................................... 1
1.3 Gregory of Saint-Vincent ................................. 4
1.4 Augustin-Louis Cauchy................................... 6
1.5 What about calculus?.................................... 8
1.6 Jean d’Alembert ........................................ 9
1.7 Charles Babbage ........................................ 10
1.8 Mathematics competitions and recreational mathematics ..... 16
1.9 A contribution from Ramanujan........................... 21
1.10 Simultaneous functional equations ......................... 24
1.11 A clarification of terminology ............................. 25
1.12 Existence and uniqueness of solutions ...................... 26
1.13 Problems............................................... 26
2 Functional equations with two variables.................... 31
2.1 Cauchy’s equation ....................................... 31
2.2 Applications of Cauchy’s equation ......................... 35
2.3 Jensen’s equation........................................ 37
2.4 Linear functional equation................................ 38
2.5 Cauchy’s exponential equation ............................ 38
2.6 Pexider’s equation ....................................... 39
2.7 Vincze’s equation........................................ 40
2.8 Cauchy’s inequality...................................... 42
2.9 Equations involving functions of two variables............... 43
2.10 Euler’s equation......................................... 44
2.11 D’Alembert’s equation ................................... 45
2.12 Problems............................................... 49
viii Contents
3 Functional equations with one variable..................... 55
3.1 Introduction ............................................ 55
3.2 Linearization............................................ 55
3.3 Some basic families of equations........................... 57
3.4 A menagerie of conjugacy equations ....................... 62
3.5 Finding solutions for conjugacy equations................... 64
3.5.1 The Koenigs algorithm for Schro¨der’s equation........ 64
3.5.2 The L´evy algorithm for Abel’s equation .............. 66
3.5.3 An algorithm for Bo¨ttcher’s equation ................ 66
3.5.4 Solving commutativity equations .................... 67
3.6 Generalizations of Abel’s and Schro¨der’s equations........... 67
3.7 General properties of iterative roots........................ 69
3.8 Functional equations and nested radicals ................... 72
3.9 Problems............................................... 75
4 Miscellaneous methods for functional equations............ 79
4.1 Polynomial equations .................................... 79
4.2 Power series methods .................................... 81
4.3 Equations involving arithmetic functions ................... 82
4.4 An equation using special groups .......................... 87
4.5 Problems............................................... 89
5 Some closing heuristics .................................... 91
6 Appendix: Hamel bases.................................... 93
7 Hints and partial solutions to problems.................... 97
7.1 A warning to the reader.................................. 97
7.2 Hints for Chapter 1...................................... 97
7.3 Hints for Chapter 2......................................102
7.4 Hints for Chapter 3......................................107
7.5 Hints for Chapter 4......................................113
8 Bibliography...............................................123
Index..........................................................125
Preface
Over the years, a number of books have been written on the theory of func-
tional equations. However, few books have been published on solving func-
tional equations which arise in mathematics competitions and mathematical
problem solving. The intention of this book is to go some distance towards
filling this gap.
This work began life some years ago as a set of training notes for
mathematics competitions such as the William Lowell Putnam Competition
for undergraduate university students, and the International Mathematical
Olympiad for high school students. As part of the training for these competi-
tions,Itriedtoputtogethersomesystematicmaterialonfunctionalequations,
which have formed a part of the International Mathematical Olympiad and a
small component of the Putnam Competition. As I became more involved
in coaching students for the Putnam and the International Mathematical
Olympiad, I started to understand why there is not much training mate-
rial available in systematic form. Many would argue that there is no theory
attachedtofunctionalequationsthatareencounteredinmathematicscompe-
titions.Eachsuchequationrequiresdifferenttechniquestosolveit.Functional
equations are often the most difficult problems to be found on mathematics
competitions because they require a minimal amount of background theory
andamaximalamountofingenuity.Thegreatadvantageofaprobleminvolv-
ing functional equations is that you can construct problems that students at
all levels can understand and play with. The great disadvantage is that, for
manyproblems,fewstudentscanmakemuchprogressinfindingsolutionseven
if the required techniques are essentially elementary in nature. It is perhaps
this view of functional equations which explains why most problem-solving
textshavelittlesystematicmaterialonthesubject.Problembooksinmathe-
maticsusuallyincludesomefunctionalequationsintheirchaptersonalgebra.
Butbyincludingfunctionalequationsamongtheproblemsonpolynomialsor
inequalities the essential character of the methodology is often lost.
Asmytrainingnotesgrew,sogrewmyconvictionthatweoftendonotdo
full justice to the role of theory in the solution of functional equations. The