Small C.G. Functional Equations and How to Solve Them

Springer, 2007. — xii, 130 p. — (Problem books in mathematics). — ISBN: 0-387-34534-5, 0-387-48901-0, 978-0-387-34534-5, 978-0-387-48901-8.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.An historical introduction
Preliminary remarks
Nicole Oresme
Gregory of Saint-Vincent
Augustin-Louis Cauchy
What about calculus?
Jean d’Alembert
Charles Babbage
Mathematics competitions and recreational mathematics
A contribution from Ramanujan
Simultaneous functional equations
A clarification of terminology
Existence and uniqueness of solutions
Problems
Functional equations with two variables
Cauchy’s equation
Applications of Cauchy’s equation
Jensen’s equation
Linear functional equation
Cauchy’s exponential equation
Pexider’s equation
Vincze’s equation
Cauchy’s inequality
Equations involving functions of two variables
Euler’s equation
D’Alembert’s equation
Problems
Functional equations with one variableLinearization
Some basic families of equations
A menagerie of conjugacy equations
Finding solutions for conjugacy equations
The Koenigs algorithm for Schröder’s equation
The Lévy algorithm for Abel’s equation
An algorithm for B¨ottcher’s equation
Solving commutativity equations
Generalizations of Abel’s and Schröder’s equations
General properties of iterative roots
Functional equations and nested radicals
Problems
Miscellaneous methods for functional equations
Polynomial equations
Power series methods
Equations involving arithmetic functions
An equation using special groups
Problems
Some closing heuristics
Appendix: Hamel bases
Hints and partial solutions to problems
A warning to the reader
Hints for Chapter 1
Hints for Chapter 2
Hints for Chapter 3
Hints for Chapter 4
Bibliography
Index