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    Mathematical Constants

    Famous mathematical constants include the ratio of circular circumference to diameter, π = 3.14 . . . , and the natural logarithmic base, e = 2.178 . . . . Students and professionals usually can name at most a few others, but there are many more buried in the literature and awaiting discovery.

    How do such constants arise, and why are they important? Here Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. Topics covered include the statistics of continued fractions, chaos in nonlinear systems, prime numbers, sum-free sets, isoperimetric problems, approxi- mation theory, self-avoiding walks and the Ising model (from statistical physics), binary and digital search trees (from theoretical computer science), the Prouhet–Thue–Morse sequence, complex analysis, geometric probability, and the traveling salesman problem. This book will be helpful both to readers seeking information about a specific constant and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This is an outstanding scholarly attempt to bring together all significant mathematical constants in one place.

    Steven R. Finch studied at Oberlin College and the University of Illinois in Urbana- Champaign, and held positions at TASC, MIT Lincoln Laboratory, and MathSoft Inc. He is presently a freelance mathematician in the Boston area. He is also a composer and has released a CD entitled “An Apple Gathering” devoted to his vocal and choral music.

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    ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

    FOUNDING EDITOR GIAN-CARLO ROTA Editorial Board R. Doran, P. Flajolet, M. Ismail, T.-Y. Lam, E. Lutwak, Volume 94

    27 N. H. Bingham, C. M. Goldie, and J. L. Teugels Regular Variation 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski, and R. Ger Iterative Functional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional

    Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 38 N. Korneichuk Exact Constants in Approximation Theory 39 R. Brualdi and H. Ryser Combinatorial Matrix Theory 40 N. White (ed.) Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Basic Model Theory 43 H. Stahl and V. Totik General Orthogonal Polynomials 45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions 46 A. Björner et al. Oriented Matroids 47 G. Edgar and L. Sucheston Stopping Times and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 49 T. Palmer Banach Algebras and the General Theory of *-Algebras I 50 F. Borceux Handbook of Categorical Algebra I 51 F. Borceux Handbook of Categorical Algebra II 52 F. Borceux Handbook of Categorical Algebra III 53 V. F. Kolchin Random Graphs 54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Tomography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants, 2ed 60 J. Krajicek Bounded Arithmetic, Propositional Logic and Complexity Theory 61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics 62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63 A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65 K. Engel Sperner Theory 66 D. Cvetkovic, P. Rowlinson, and S. Simic Eigenspaces of Graphs 67 F. Bergeron, G. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures 68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups 69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2ed 70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry 71 G. E. Andrews, R. Askey, and R. Roy Special Functions 72 R. Ticciati Quantum Field Theory for Mathematicians 73 M. Stern Semimodular Lattices 74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I 75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II 76 A. A. Ivanov Geometry of Sporadic Groups 1 77 A. Schinzel Polynomials with Special Regard to Reducibility 78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2ed 79 T. Palmer Banach Algebras and the General Theory of *-Algebras II 80 O. Stormark Lie’s Structural Approach to PDE Systems 81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables 82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets 83 C. Foias et al. Navier–Stokes Equations and Turbulence 84 B. Polster and G. Steinke Geometries on Surfaces

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    85 R. B. Paris and D. Karninski Asymptotics and Mellin–Barnes Integrals 86 R. McEliece The Theory of Information and Coding, 2ed 87 B. Magurn Algebraic Introduction to K-Theory 88 T. Mora Systems of Polynomial Equations I 89 K. Bichteler Stochastic Integration with Jumps 90 M. Lothaire Algebraic Combinatorics on Words 91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II 92 P. McMullen and E. Schulte Abstract Regular Polytopes 93 G. Gierz et al. Continuous Lattices and Domains

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    EN C Y C LO PED IA O F MATHEMATICS AND ITS APPLICATIONS

    Mathematical Constants

    STEVEN R. FINCH

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    CAMBRIDGE UNIVERSITY PRESS

    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

    Cambridge University Press 40 West 20th Street, New York, NY 10011-4211, USA

    www.cambridge.org Information on this title: www.cambridge.org/9780521818056

    C© Steven R. Finch 2003

    This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements,

    no reproduction of any part may take place without the written permission of Cambridge University Press.

    First published 2003

    Printed in the United States of America

    A catalog record for this book is available from the British Library.

    Library of Congress Cataloging in Publication Data

    Finch, Steven R., 1959–

    Mathematical constants / Steven R. Finch.

    p. cm. – (Encyclopedia of mathematics and its applications; v. 94)

    Includes bibliographical references and index.

    ISBN 0-521-81805-2

    I. Mathematical constants. I. Title. II. Series.

    QA41 .F54 2003 513 – dc21 2002074058

    ISBN-13 978-0-521-81805-6 hardback ISBN-10 0-521-81805-2 hardback

    Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or

    third-party Internet Web sites referred to in this book and does not guarantee that any content on such

    Web sites is, or will remain, accurate or appropriate.

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    For Nancy Armstrong, the one constant

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    Contents

    Preface page xvii Notation xix

    1 Well-Known Constants 1 1.1 Pythagoras’ Constant,

    √ 2 1

    1.1.1 Generalized Continued Fractions 3 1.1.2 Radical Denestings 4

    1.2 The Golden Mean, ϕ 5 1.2.1 Analysis of a Radical Expansion 8 1.2.2 Cubic Variations of the Golden Mean 8 1.2.3 Generalized Continued Fractions 9 1.2.4 Random Fibonacci Sequences 10 1.2.5 Fibonacci Factorials 10

    1.3 The Natural Logarithmic Base, e 12 1.3.1 Analysis of a Limit 14 1.3.2 Continued Fractions 15 1.3.3 The Logarithm of Two 15

    1.4 Archimedes’ Constant, π 17 1.4.1 Infinite Series 20 1.4.2 Infinite Products 21 1.4.3 Definite Integrals 22 1.4.4 Continued Fractions 23 1.4.5 Infinite Radical 23 1.4.6 Elliptic Functions 24 1.4.7 Unexpected Appearances 24

    1.5 Euler–Mascheroni Constant, γ 28 1.5.1 Series and Products 30 1.5.2 Integrals 31 1.5.3 Generalized Euler Constants 32 1.5.4 Gamma Function 33

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    x Contents

    1.6 Apéry’s Constant, ζ (3) 40 1.6.1 Bernoulli Numbers 41 1.6.2 The Riemann Hypothesis 41 1.6.3 Series 42 1.6.4 Products 45 1.6.5 Integrals 45 1.6.6 Continued Fractions 46 1.6.7 Stirling Cycle Numbers 47 1.6.8 Polylogarithms 47

    1.7 Catalan’s Constant, G 53 1.7.1 Euler Numbers 54 1.7.2 Series 55 1.7.3 Products 56 1.7.4 Integrals 56 1.7.5 Continued Fractions 57 1.7.6 Inverse Tangent Integral 57

    1.8 Khintchine–Lévy Constants 59 1.8.1 Alternative Representations