INTRODUCTION TO FUNCTIONAL EQUATIONS - MSRI ?· INTRODUCTION TO FUNCTIONAL EQUATIONS ... (e.g. functional equations, inequalities, finite and infinite sums, etc.) instead. x

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<ul><li><p>INTRODUCTION TO FUNCTIONAL EQUATIONStheory and problem-solving strategies for mathematical competitions and beyond</p><p>COSTAS EFTHIMIOU</p><p>Department of PhysicsUNIVERSITY OF CENTRAL FLORIDA</p><p>VERSION: 2.00September 12, 2010</p></li><li><p>Cover picture: Domenico Fettis Archimedes Thoughtful, Oil on canvas, 1620. State ArtCollections Dresden.</p></li><li><p> To My Mother who taught me that love is notmeasured by how many things</p><p>one can offer to a child butby how many things</p><p>a parent sacrificesfor the child</p></li><li><p>Contents</p><p>INTRODUCTION ix</p><p>ACKNOWLEDGEMENTS xiii</p><p>I BACKGROUND 1</p><p>1 Functions 3</p><p>1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3</p><p>1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6</p><p>1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8</p><p>1.4 Limits &amp; Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13</p><p>1.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16</p><p>1.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22</p><p>II BASIC EQUATIONS 33</p><p>2 Functional Relations Primer 35</p><p>2.1 The Notion of Functional Relations . . . . . . . . . . . . . . . . . . . . . . . . 35</p><p>2.2 Beginning Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>3 Equations for Arithmetic Functions 51</p><p>3.1 The Notion of Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 51</p><p>3.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54</p><p>3.3 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55</p><p>3.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63</p><p>4 Equations Reducing to Algebraic Systems 69</p><p>4.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69</p><p>4.2 Group Theory in Functional Equations . . . . . . . . . . . . . . . . . . . . . . 74</p><p>v</p></li><li><p>vi CONTENTS</p><p>5 Cauchys Equations 81</p><p>5.1 First Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81</p><p>5.2 Second Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83</p><p>5.3 Third Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84</p><p>5.4 Fourth Cauchy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86</p><p>5.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89</p><p>6 CauchysNQR-Method 95</p><p>6.1 TheNQR-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96</p><p>6.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98</p><p>7 Equations for Trigonometric Functions 105</p><p>7.1 Characterization of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 105</p><p>7.2 DAlembert-Poisson I Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 113</p><p>7.3 DAlembert-Poisson II Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 115</p><p>7.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118</p><p>III GENERALIZATIONS 121</p><p>8 Pexider, Vincze &amp; Wilson Equations 123</p><p>8.1 First Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123</p><p>8.2 Second Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125</p><p>8.3 Third Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126</p><p>8.4 Fourth Pexider Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127</p><p>8.5 Vincze Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128</p><p>8.6 Wilson Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 133</p><p>8.7 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134</p><p>9 Vector and Matrix Variables 137</p><p>9.1 Cauchy &amp; Pexider Type Equations . . . . . . . . . . . . . . . . . . . . . . . . 138</p><p>9.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140</p><p>10 Systems of Equations 145</p><p>10.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147</p><p>IV CHANGING THE RULES 151</p><p>11 Less Than Continuity 153</p><p>11.1 Imposing Weaker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 153</p><p>11.2 Non-Continuous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155</p><p>11.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161</p></li><li><p>CONTENTS vii</p><p>12 More Than Continuity 169</p><p>12.1 Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169</p><p>12.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176</p><p>12.3 Stronger Conditions as a Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 179</p><p>13 Functional Equations for Polynomials 183</p><p>13.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183</p><p>13.2 Symmetric Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185</p><p>13.3 More on the Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 188</p><p>13.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197</p><p>14 Conditional Functional Equations 205</p><p>14.1 The Notion of Conditional Equations . . . . . . . . . . . . . . . . . . . . . . . 205</p><p>14.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206</p><p>15 Functional Inequalities 209</p><p>15.1 Useful Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209</p><p>15.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212</p><p>V EQUATIONS WITH NO PARAMETERS 215</p><p>16 Iterations 217</p><p>16.1 The Need for New Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217</p><p>16.2 Iterates, Orbits, Fixed Points, and Cycles . . . . . . . . . . . . . . . . . . . . . 220</p><p>16.3 Fixed Points: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223</p><p>16.4 Cycles: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227</p><p>16.5 From Iterations to Difference Equations . . . . . . . . . . . . . . . . . . . . . 230</p><p>16.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231</p><p>16.7 A Taste of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237</p><p>17 Solving by Invariants &amp; Linearization 247</p><p>17.1 Constructing Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247</p><p>17.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249</p><p>17.3 The Abel and Schroder Equations . . . . . . . . . . . . . . . . . . . . . . . . . 250</p><p>17.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251</p><p>17.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253</p><p>18 More on Fixed Points 259</p><p>18.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259</p></li><li><p>viii CONTENTS</p><p>VI GETTING ADDITIONAL EXPERIENCE 263</p><p>19 Miscellaneous Problems 265</p><p>19.1 Integral Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26519.2 Problems Solved by Functional Relations . . . . . . . . . . . . . . . . . . . . 26919.3 Assortment of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280</p><p>20 Unsolved Problems 285</p><p>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</p><p>VII AUXILIARY MATERIAL 329</p><p>ACRONYMS &amp; ABBREVIATIONS 331</p><p>SET CONVENTIONS 333</p><p>NAMED EQUATIONS 335</p><p>BIBLIOGRAPHY 337</p><p>INDEX 341</p></li><li><p>Introduction</p><p>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</p><p>f (x + y) = f (x) + f (y) .</p><p>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 mathematics. For example, the definition of an evenfunction is a functional equation,</p><p>f (x) = f (x) ,that is encountered as soon as the notion of a function is introduced. At the other ex-treme, in the forefront of research, during the last two to three decades, the celebratedYoung-Baxter functional equation has been at the heart of many different areas of math-ematics and theoretical physics such as lattice integrable systems, factorized scattering inquantum field theory, braid and knot theory, and quantum groups to name a few. TheYoung-Baxter equation is a system of N6 functional equations for the (N2 N2)-matrix R(x)whose entries are functions of x:</p><p>N</p><p>,,=1</p><p>Rjk</p><p>(x y) Ri</p><p>(x) Rmn (y) =</p><p>N</p><p>,,=1</p><p>Ri j</p><p>(y) Rnk</p><p>(x) Rm(x y) ,</p><p>where i, j, k, ,m, n take the values 1, 2, ,N. Amazingly, although the Yang-Baxter equa-tion appears to impose more constraints than the number of unknowns (N6 equations forN4 unknowns), it has a rich set of solutions. The study of this equation is well beyondthe goals of the book; however, the reader will hopefully come to appreciate the role thatfunctional equations play in mathematics and science over the course of reading this book.</p><p>Despite being such a rich subject, this topic does not fit perfectly into any of the conven-tional areas of mathematics and thus a systematic way of studying functional equations isnot found in any traditional book used in the traditional education of students. Severalgood books on the topic exist but, unfortunately, they are either relatively hard to find [39]or quite advanced [7, 8, 27]. While this book was in the making, a nice, simple book [38]appeared which has considerable overlap with, and is complementary to, this book.</p><p>We should point out in advance that there is no universal technique for solving func-tional equations, as will become obvious once the reader solves the problems throughoutthis book. At the current time, thorough knowledge of many areas of mathematics, lots ofimagination, and, perhaps, a bit of luck are the best methods for solving such equations.</p><p>The core of the book is the result of a series of lectures I presented to the UCF Putnamteam after my arrival at UCF. My personal belief is that the training of a math teamshould be based on a systematic approach for each subject of interest for the mathematicalcompetitions (e.g. functional equations, inequalities, finite and infinite sums, etc.) instead</p></li><li><p>x More on How to Approach This Book</p><p>of randomly solving hard problems to get experience. Therefore, this book is an attemptto present the fundamentals of the topic at hand in a pedagogical manner, accessibleto students who have a background on the theory of functions up to differentiability.Although there are some parts of the book that use additional ideas from calculus, suchparts may be omitted at first reading. In particular, the book should be useful to highschool students who participate in math competitions and have an interest in InternationalMath Olympiads (IMO). Many of the problems I have included are problems I solved asa high school student1 in preparation for the national math exam of my home country,Greece. I still enjoy them as much as I enjoyed them then. In fact, I now appreciate theirbeauty much more, as I have grown to have a better understanding of the subject andits importance. College students with an interest in the Putnam Competition should alsofind the book useful, as standard texts do not provide a thorough coverage of functionalequations. Finally, we hope that any person with interest in mathematics will find in thisbook something to his or her interest. However, a piece of advice is in order here for allreaders: The majority of the problems discussed in this book are related to mathematicalcompetitions which target ingenuity and insight, so they are not easy. Approach them withcaution. They should serve as a source of enjoyment, not despair!</p><p>And a final comment: The interested reader should, perhaps, read the book simultane-ously with Smtals [39] and Smalls [38] books, which are at the same level. I cover morefunctional equations but Smtal presents beautifully the topic of iterations and functionalequations of one variable2. Similarly, Smalls book [38] is a very enjoyable, well writtenbook and focuses on the most essential aspects of functional equations. Once the readeris done with these three books, he may read Aczels and Kuczmas authoritative books[7, 8, 27], which contain a vast amount of information. Beyond that level, the reader will beready to immerse himself in the current literature on the subject, and perhaps even conducthis own research in the area.</p><p>More on How to Approach This Book</p><p>Mathematics has a reputation of being a very stiff subject that only follows a pre-determinedpattern: definitions, axioms, theorems, proofs. Any deviations from this pre-establishedpattern are neither welcome nor wise. In this respect, mathematics appears to be verydifferent from sciences and, in particular, theoretical physics, which is the closest discipline.For, mathematics is using demonstrative reasoning and sciences use plausible reasoning.3</p><p>As was mentioned previously, the majority of the problems in this book are taken frommathematical competitions. As such, these problems are precious not only for their origi-</p><p>1Yes, I still have many of my notes! As a result, some of the problems I present might be taken from nationalor local competitions without reference. I would appreciate a note from readers who discover such omissionsor other typos and mistakes.</p><p>2One can detect some influence from Smtals book on my presentation in Chapters 16 and 17. I highlyrecommended this book to any serious student.</p><p>3The terms are discussed in G. Polyas classic two-volume work [31] which teaches students the...</p></li></ul>