**MSRI Mathematical Circles Library**

Volume: 6;
2011;
363 pp;
Softcover

MSC: Primary 39; 00; 26;

**Print ISBN: 978-0-8218-5314-6
Product Code: MCL/6**

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

**Electronic ISBN: 978-1-4704-1617-1
Product Code: MCL/6.E**

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

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# Introduction to Functional Equations: Theory and problem-solving strategies for mathematical competitions and beyond

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*Costas Efthimiou*

A co-publication of the AMS and the Mathematical Sciences Research Institute

Functions and their properties have been part of the rigorous
precollege curriculum for decades. And functional equations have been a
favorite topic of the leading national and international mathematical
competitions. Yet the subject has not received equal attention by
authors at an introductory level. The majority of the books on the
topic remain unreachable to the curious and intelligent precollege
student. The present book is an attempt to eliminate this disparity.

The book opens with a review chapter on functions, which collects the
relevant foundational information on functions, plus some material
potentially new to the reader. The next chapter presents a working
definition of functional equations and explains the difficulties in
trying to systematize the theory. With each new chapter, the author
presents methods for the solution of a particular group of equations.
Each chapter is complemented with many solved examples, the majority of
which are taken from mathematical competitions and professional
journals. The book ends with a chapter of unsolved problems and some
other auxiliary material.

The book is an invaluable resource for precollege and college students
who want to deepen their knowledge of functions and their properties,
for teachers and instructors who wish to enrich their curricula, and for
any lover of mathematical problem-solving techniques.

In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics
profession.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

#### Readership

High school teachers and students, undergraduate students interested in functional equations; anyone interested in tips, tricks, and techniques for mathematical competitions and math circles.

#### Table of Contents

# Table of Contents

## Introduction to Functional Equations: Theory and problem-solving strategies for mathematical competitions and beyond

- Cover Cover11
- Title page iii4
- Contents vii8
- Introduction xi12
- Acknolwedgments xv16
- Part I. Background 118
- Functions 320
- Part II. Basic equations 3956
- A primer on functional relations 4158
- Equations for arithmetic functions 5774
- Equations reducing to algebraic systems 7794
- Cauchy’s equations 91108
- Cauchy’s ℕℚℝ method 107124
- Equations for trigonometric functions 117134
- Part III. Generalizations 135152
- The Pexider, Vincze and Wilson equations 137154
- Vector and matrix variables 153170
- Systems of equations 161178
- Part IV. Changing the rules 167184
- Less than continuity 169186
- More than continuity 185202
- Functional equations for polynomials 199216
- Conditional functional equations 221238
- Functional inequalities 225242
- Part V. Equations with no parameters 231248
- Iterations 233250
- Solving by invariants and linearization 265282
- More on fixed points 277294
- Part VI. Getting additional experience 281298
- Miscellaneous problems 283300
- Additional problems 303320
- Part VII. Auxiliary material 343360
- Acronyms and abbreviations 345362
- Set conventions 347364
- Named equations 349366
- Bibliography 351368
- Index 355372
- Back Cover Back Cover1381