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Numbers and Functions from a Classical- Experimental Mathematician’S Point of View STUDENT MATHEMATICAL LIBRARY Volume 65 Numbers and Functions From a classical- experimental mathematician’s point of view Victor H. Moll http://dx.doi.org/10.1090/stml/065 Numbers and Functions From a classical- experimental mathematician’s point of view STUDENT MATHEMATICAL LIBRARY Volume 65 Numbers and Functions From a classical- experimental mathematician’s point of view Victor H. Moll American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05A05, 05C05, 11A07, 11A41, 11A55, 11B39, 11B65, 33B10, 33B15, 33F10. For additional information and updates on this book, visit www.ams.org/bookpages/stml-65 Library of Congress Cataloging-in-Publication Data Moll, Victor H., 1956– Numbers and functions : from a classical-experimental mathematician’s point of view / Victor H. Moll. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-8795-0 (alk. paper) 1. Functions—Textbooks. 2. Calculus—Textbooks. I. Title. QA331.M72 2012 515.25–dc23 2012014772 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Dedicated to Olivier Espinosa Contents Preface xiii Chapter 1. The Number Systems 1 §1.1. The natural numbers 1 §1.2. An automatic approach to finite sums 3 §1.3. Elementary counting 8 §1.4. The integers and divisibility 11 §1.5. The Euclidean algorithm 12 §1.6. Modular arithmetic 19 §1.7. Prime numbers 20 §1.8. The rational numbers 29 §1.9. The set of real numbers 40 §1.10. Fundamental sequences and completions 52 §1.11. Complex numbers 54 Chapter 2. Factorials and Binomial Coefficients 57 §2.1. The definitions 57 §2.2. A counting argument 60 §2.3. The generating function of binomial coefficients 61 §2.4. An extension of the binomial theorem to noninteger exponents 63 vii viii Contents §2.5. Congruences for factorials and binomial coefficients 66 §2.6. The prime factorization of n!76 §2.7. The central binomial coefficients 82 §2.8. Bertrand’s postulate 88 §2.9. Some generating functions involving valuations 91 §2.10. The asymptotics of factorials: Stirling’s formula 94 §2.11. The trinomial coefficients 96 Chapter 3. The Fibonacci Numbers 105 §3.1. Introduction 105 §3.2. What do they count? 106 §3.3. The generating function 108 §3.4. A family of related numbers 110 §3.5. Some arithmetical properties 112 §3.6. Modular properties of Fibonacci numbers 120 §3.7. Continued fractions of powers of Fibonacci quotients 123 §3.8. Fibonacci polynomials 124 §3.9. Series involving Fibonacci numbers 129 Chapter 4. Polynomials 133 §4.1. Introduction 133 §4.2. Examples of polynomials 134 §4.3. The division algorithm 139 §4.4. Roots of polynomials 141 §4.5. The fundamental theorem of algebra 144 §4.6. The solution of polynomial equations 146 §4.7. Cubic polynomials 151 §4.8. Quartic polynomials 155 Chapter 5. Binomial Sums 159 §5.1. Introduction 159 §5.2. Power sums 160 §5.3. Moment sums 167 Contents ix §5.4. Recurrences for powers of binomials 171 §5.5. Calkin’s identity 173 Chapter 6. Catalan Numbers 179 §6.1. The placing of parentheses 179 §6.2. A recurrence 179 §6.3. The generating function 180 §6.4. Arithmetical properties 186 §6.5. An integral expression 190 Chapter 7. The Stirling Numbers of the Second Kind 191 §7.1. Introduction 191 §7.2. A recurrence 192 §7.3. An explicit formula 194 §7.4. The valuations of Stirling numbers 197 Chapter 8. Rational Functions 211 §8.1. Introduction 211 §8.2. The method of partial fractions 213 §8.3. Rational generating functions 217 §8.4. The operator point of view 219 §8.5. A dynamical system 221 §8.6. Sums of four squares 225 §8.7. The integration of rational functions 229 §8.8. Symbolic integration. The methods of Hermite and Rothstein-Trager 237 Chapter 9. Wallis’ Formula 245 §9.1. An experimental approach 245 §9.2. A proof based on recurrences 247 §9.3. A proof based on generating functions 248 §9.4. A trigonometric version 249 §9.5. An automatic proof 252 x Contents Chapter 10. Farey Fractions 255 §10.1. Introduction 255 §10.2. Farey fractions and the Stern-Brocot tree 255 §10.3. The distribution of denominators 262 Chapter 11. The Exponential Function 269 §11.1. Introduction 269 §11.2. Elementary properties of the exponential function 272 §11.3. The constant e 274 §11.4. The series representation of e 275 §11.5. Arithmetical properties of e 278 §11.6. Continued fractions connected to e 284 §11.7. Derangements: The presence of e in combinatorics 288 §11.8. The natural logarithm 293 §11.9. The binary expansion of ln 2 294 §11.10. The irrationality of ln 2 295 §11.11. Harmonic numbers 302 Chapter 12. Trigonometric Functions 309 §12.1. Introduction 309 §12.2. The notion of angle 309 §12.3. Sine and cosine 311 §12.4. The additional trigonometric functions 314 §12.5. The addition theorem 320 §12.6. Stirling’s formula and π 323 §12.7. The continued fraction of π 325 §12.8. The digits of π in base 16 330 §12.9. Special values of trigonometric functions 331 §12.10. The roots of a cubic polynomial 335 §12.11. A special trigonometric integral 339 §12.12. The infinite product for sin x 343 Contents xi §12.13. The irrationality of π 346 §12.14. Arctangent sums and a dynamical system 349 Chapter 13. Bernoulli Polynomials 355 §13.1. Introduction 355 §13.2. The exponential generating function 356 §13.3. Elementary properties of Bernoulli numbers 357 §13.4. Integrals involving Bernoulli polynomials 366 §13.5. A relation to Stirling numbers 369 §13.6. Arithmetic properties of Bernoulli numbers 371 §13.7. The Euler-MacLaurin summation formula 376 §13.8. Bernoulli numbers and solitons 383 §13.9. The Giuga-Agoh conjectured criterion for primality 384 Chapter 14. A Sample of Classical Polynomials: Legendre, Chebyshev, and Hermite 387 §14.1. Introduction 387 §14.2. Legendre polynomials 387 §14.3. Chebyshev polynomials 400 §14.4. Hermite polynomials 403 Chapter 15. Landen Transformations 411 §15.1. Introduction 411 §15.2. An elementary example 413 §15.3. The case of rational integrands 415 §15.4. The evaluation of a quartic integral 417 §15.5. An integrand of degree six 435 §15.6. The original elliptic case 438 Chapter 16. Three Special Functions: Γ,ψ,and ζ 445 §16.1. Introduction 445 §16.2. The gamma function 446 xii Contents §16.3. Elementary properties of the gamma function 447 §16.4. Special values of the gamma function 449 §16.5. The infinite product for the gamma function 450 §16.6. The beta function 455 §16.7. The digamma function 458 §16.8. The Riemann zeta function 460 §16.9. The values of ζ(2n) 461 §16.10. Ap´ery’s constant ζ(3) 463 Bibliography 473 Index 493 Preface In the process of writing a mathematics book, an author has to make a variety of decisions. The central theme of the book must be followed by deciding on a potential audience to whom the book is directed, and then a choice of style for presenting the material must be made. This book began as a collection of additional notes given to stu- dents participating in the courses taught by the author at Tulane University. These courses included: • The calculus sequence. This is the typical two-semester course on differential and integral calculus. The standard book used at Tulane is J. Stewart [281]. The author has used M. Spivak [277]for the Honor section, which is slightly more advanced than the regular one. • Discrete mathematics. This course introduces students to mathematical induction and provides a glimpse of number theory. This is the first time where the students are exposed to proofs. The books used in the past include M. Aigner [4]andK.Rosen[256]. • Combinatorics. This is a one-semester course that includes basic counting techniques, recurrences, combinatorial identities, and the ideas behind bijective proofs. The author has used a selection of texts, including T. Andreescu and Z. Feng [17], A. Benjamin and J. Quinn [46], M. Bona [58], and R. Brualdi [82]. xiii xiv Preface • Number theory. This is also a one-semester course covering the basics of the subject: primality and factorization, congruences, diophantine equations, continued fractions, primitive roots, and qua- dratic reciprocity. The texts used by the author include G. H. Hardy and E. M. Wright [160], K. Ireland and M. Rosen [178], K. Rosen [257], and J. H. Silverman [274]. • Real analysis. This is one of the few required courses for a mathematics major. It introduces the student to the real line and all of its properties. Sequences and completeness, the study of the real line, continuity, and compactness form the bulk of the course.
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