# INTRODUCTION TO FUNCTIONAL EQUATIONS - MSRI .INTRODUCTION TO FUNCTIONAL EQUATIONS ... (e.g. functional equations, inequalities, ï¬nite and inï¬nite sums, etc.) instead. x

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• INTRODUCTION TO FUNCTIONAL EQUATIONStheory and problem-solving strategies for mathematical competitions and beyond

COSTAS EFTHIMIOU

Department of PhysicsUNIVERSITY OF CENTRAL FLORIDA

VERSION: 2.00September 12, 2010

• Cover picture: Domenico Fettis Archimedes Thoughtful, Oil on canvas, 1620. State ArtCollections Dresden.

• To My Mother who taught me that love is notmeasured by how many things

one can offer to a child butby how many things

a parent sacrificesfor the child

• Contents

INTRODUCTION ix

ACKNOWLEDGEMENTS xiii

I BACKGROUND 1

1 Functions 3

1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Limits & Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

II BASIC EQUATIONS 33

2 Functional Relations Primer 35

2.1 The Notion of Functional Relations . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Beginning Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Equations for Arithmetic Functions 51

3.1 The Notion of Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Equations Reducing to Algebraic Systems 69

4.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Group Theory in Functional Equations . . . . . . . . . . . . . . . . . . . . . . 74

v

• vi CONTENTS

5 Cauchys Equations 81

5.1 First Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Second Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Third Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Fourth Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 CauchysNQR-Method 95

6.1 TheNQR-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7 Equations for Trigonometric Functions 105

7.1 Characterization of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 DAlembert-Poisson I Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.3 DAlembert-Poisson II Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

III GENERALIZATIONS 121

8 Pexider, Vincze & Wilson Equations 123

8.1 First Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2 Second Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.3 Third Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.4 Fourth Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.5 Vincze Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.6 Wilson Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.7 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9 Vector and Matrix Variables 137

9.1 Cauchy & Pexider Type Equations . . . . . . . . . . . . . . . . . . . . . . . . 138

9.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

10 Systems of Equations 145

10.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

IV CHANGING THE RULES 151

11 Less Than Continuity 153

11.1 Imposing Weaker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11.2 Non-Continuous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

11.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

• CONTENTS vii

12 More Than Continuity 169

12.1 Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

12.3 Stronger Conditions as a Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

13 Functional Equations for Polynomials 183

13.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

13.2 Symmetric Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

13.3 More on the Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 188

13.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

14 Conditional Functional Equations 205

14.1 The Notion of Conditional Equations . . . . . . . . . . . . . . . . . . . . . . . 205

14.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

15 Functional Inequalities 209

15.1 Useful Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

15.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

V EQUATIONS WITH NO PARAMETERS 215

16 Iterations 217

16.1 The Need for New Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

16.2 Iterates, Orbits, Fixed Points, and Cycles . . . . . . . . . . . . . . . . . . . . . 220

16.3 Fixed Points: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

16.4 Cycles: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

16.5 From Iterations to Difference Equations . . . . . . . . . . . . . . . . . . . . . 230

16.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

16.7 A Taste of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

17 Solving by Invariants & Linearization 247

17.1 Constructing Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

17.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

17.3 The Abel and Schroder Equations . . . . . . . . . . . . . . . . . . . . . . . . . 250

17.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

17.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

18 More on Fixed Points 259

18.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

• viii CONTENTS

19 Miscellaneous Problems 265

19.1 Integral Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26519.2 Problems Solved by Functional Relations . . . . . . . . . . . . . . . . . . . . 26919.3 Assortment of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

20 Unsolved Problems 285

20.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28520.2 Problems That Can Be Solved Using Functions . . . . . . . . . . . . . . . . . 29120.3 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29220.4 Functional Equations With Parameters . . . . . . . . . . . . . . . . . . . . . . 29720.5 Functional Equations with No Parameters . . . . . . . . . . . . . . . . . . . . 30520.6 Fixed Points and Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30820.7 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31120.8 Systems of Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31420.9 Conditional Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31520.10Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31620.11Functional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32120.12Functional Equations Containing Derivatives . . . . . . . . . . . . . . . . . . 32420.13Functional Relations Containing Integrals . . . . . . . . . . . . . . . . . . . . 325

VII AUXILIARY MATERIAL 329

ACRONYMS & ABBREVIATIONS 331

SET CONVENTIONS 333

NAMED EQUATIONS 335

BIBLIOGRAPHY 337

INDEX 341

• Introduction

One of the most beautiful mathematical topics I encountered as a student was the topicof functional equations that is, the topic that deals with the search of functions whichsatisfy given equations, such as

f (x + y) = f (x) + f (y) .

This topic is not only remarkable for its beauty but also impressive for the fact that functionalequations arise in all areas of mathematics and, even more, science, engineering, and socialsciences. They appear at all levels of mathema

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