Herbert S Wilf Mathematics for the Physical Sciences MATHEMATICS FOR THE PHYSICAL SCIENCES by Herbert S Wilf, Ph.D Professor of Mathematics University of Pennsylvania DOVER PUBLICATIONS, INC NEW YORK Copyright © 1962 by Herbert S Wilf All rights reserved under Pan American and International Copyright Conventions Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC2H 7EG This Dover edition, first published in 1978, is an unabridged republication of the work originally published by John Wiley & Sons, Inc., New York, in 1962 Corrections have been made on pages 27, 219 and 228 of the present edition International Standard Book Number: 0-486-68635-6 Library of Congress Catalog Card Number: '17-94182 Manufactured in the United States of America Dover Publications, Inc 180 Varick Street New York, N.Y 10014 To my mother and father Preface This book is based on a two-semester course in "The Mathematical Methods of Physics" which I have given in the mathematics department of the University of Illinois in recent years The audience has consisted primarily of physicists, engineers, and other natural scientists in their first or second year of graduate study Knowledge of the theory of functions of real and complex variables is assumed The subject matter has been shaped by the needs of the students and by my own experience In many cases students who not major in mathematics have room in their schedules for only one or two mathematics courses The purpose of this book, therefore, is to provide the student with some heavy artillery in several fields of mathematics, in confidence that targets for these weapons will be amply provided by the student's own special field of interest Naturally, in such an attempt, something must be sacrificed, and I have regarded as most expendable discussions of physical applications of the material being presented Again, in the short space allotted to each subject there is little chance to develop the theory beyond fundamentals Thus in each chapter I have gone straight to (what I regard as) the heart of the matter, developing a subject just far enough so that applications can easily be made by the student himself The exercises at the end of each chapter, along with their solutions at the back of the book, afford some further opportunities for using the theoretical apparatus The material herein is, for the most part, classical The bibliographical references, particularly to journal articles, are given not so much to provide a jumping-off point for further research as to give the reader a vii viii PREFACE feeling for the chronological development of these subjects and for the names of the men who created them Finally, I have, where possible, tried to say something about numerical methods for computing the solutions of various kinds of problems These discussions, while brief, are oriented toward electronic computers and are intended to help bridge the gap between the "there exists" of a pure mathematician and the "find it to three decimal places" of an engineer I am indebted to Professor L A Rubel for permission to publish Theorem of Chapter here for the first time and to Professor R P Jerrard for some of the exercises in Chapter To the welI-known volume of Courant and Hilbert lowe the intriguing notion that, even in an age of specialization, it may be possible for physicists and mathematicians to understand each other HERBERT S WILF Philadelphia, Pennsylvania March, 1962 Contents Chapter Vector Spaces and Matrices 1.1 1.2 ! 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 U6 1.17 U8 1.19 1.20 1.21 1.22 1.23 Vector Spaces, Schwarz Inequality and Orthogonal Sets, Linear Dependence and Independence, Linear Operators on a Vector Space, Eigenvalues and Hermitian Operators, Unitary Operators, 10 Projection Operators, 11 Euclidean n-space and Matrices, 12 Matrix Algebra, 14 The Adjoint Matrix, 15 The Inverse Matrix, 16 Eigenvalues of Matrices, 18 Diagonalization of Matrices, 21 Functions of Matrices, 23 The Companion Matrix, 25 Bordering Hermitian Matrices, 26 Definite Matrices, 28 Rank and Nullity, 30 Simultaneous Diagonalization and Commutativity, 33 The Numerical Calculation of Eigenvalues, 34 Application to Differential Equations, 36 Bounds for the Eigenvalues, 38 Matrices with Nonnegative Elements, 39 Bibliography, 44 Exercises, 45 ix x CONTENTS Chapter 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 Orthogonal Functions Introduction, 48 Orthogonal Polynomials, 49 Zeros, 51 The Recurrence Formula, 52 The Christoffel-Darboux Identity, 55 Modifying the Weight Function, 57 Rodrigues' Formula, 58 Location of the Zeros, 59 Gauss Quadrature, 61 The Classical Polynomials, 64 Special Polynomials, 67 The Convergence of Orthogonal Expansions, 69 Trigonometric Series, 72 Fejer Summability, 75 Bibliography, 79 Exercises, 79 Chapter The Roots of Polynomial Equations 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 82 Introduction, 82 The Gauss-Lucas Theorem, 83 Bounds for the Moduli of the Zeros, 85 Sturm Sequences, 90 Zeros in a Half-Plane, 95 Zeros in a Sector; Erdos-Turan's Theorem, 98 Newton's Sums, 100 Other Numerical Methods, 104 Bibliography, 106 Exercises, 106 Chapter Asymptotic Expansions 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 48 Introduction; the 0, 0, , symbols, 108 Sums, 114 Stirling's Formula, 120 Sums of Powers, 122 The Functional Equation of Us), 124 The Method of Laplace for Integrals, 127 The Method of Stationary Phase, 131 Recurrence Relations, 136 Bibliography, 139 Exercises, 140 108 Books referred to in the text Bernstein, S., Leryons sur les Proprietes Extremales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Reele, Gauthier-Villars, Paris, 1926 Bliss, G A., Lectures on the Calculus of Variations, University of Chicago Press, 1946 de Brnijn, N G., Asymptotic Method$ in Analysis, Interscience Publishers, New York, 1958 Caratheodory, C., Conformal Representation, Cambridge Tracts, 28, Cambridge University Press, New York, 1932 Churchill, R V., Introduction to Complex Variables and ApR'ications, McGraw-Hill Book Co., New York, 1948 Coddington, E A and N Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Co., New York, 1955 Collatz, L., The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin, 1960 Courant, R., and D Hilbert, Method$ of Mathematical PhYSiCS, Interscience Publishers, New York, 1953 Dickson, L E., A First Course in the Theory of Equations, John Wiley and Sons, New York, 1922 to Dieudonne, J., La Theorie Analytique des Polynomes d'une Variable, Memorial des Sciences Mathematiques, vol 93, Paris, 1938 11 Erdelyi, A., Asymptotic Expansions, Dover Publications, New York, 1956 12 Erdelyi, A., et al., Bateman Manuscript Project, McGraw-Hill Book Co., New York, 1954 13 Fox, L., The Numerical Solution of Two-Point Bounr:kzry Problems, Clarendon Press, Oxford, 1957 14 Gantmacher, F R., Applications of the Theory of Matrices, Chelsea Press, New York 1960 277 Original works cited in the text S Bernstein Sur les formules de quadrature de Cotes et de Tschebycheff, C R de L'Academie des Sciences de L'URSS, 14 (1937), 323~326 N Bourbaki Elements de Mathematique, xir, livre IV, chap IV, Equations l)ifferentielles, Hermann and Cie, Paris, 1951 A Brauer Limits for the Characteristic Roots of a Matrix, Duke Math J., 13 (1946), 387-395 A Cauchy J Ecole Poly., 25 (1837), 176 E B Christoffel Ober die Gaussische Quadratur und eine Verallgemeinerung derselben, J.fur Math., 55 (1858), 61-62 G Dantzig The Maximization of a Linear Function of Variables Subject to Linear Inequalities, Activity Analysis of Production and Allocation, Cowles Commission Monograph 13, John Wiley and Sons, New York, 1951 G Darboux Memoire sur l'approximation des fonctions de tres grands nombres, J de MatM- matiques, (1878), 5-56 279 280 MATIIEMATICS FOR THE PHYSICAL SCIENCES P ErdOs and P Turan On the Distribution of Roots of Polynomials, Annals of Math., 51 (1950), 105-119 L Euler Commentarii Acad Sci Imp Petropolitanae, vol 11 (1739) L Fejer Untersuchungen uber die Fourierschen Reihen, Math Ann., 58 (1903), 51 Nombre des changements de signe d'une fonction dans un intervalle et ses moments, C R Acad Sci., 158 (1914), 1328-1331 M Fekete Sur une limite inferieure des changements de signe d'une fonction dans un intervalle, C R Acad Sci., 158 (1914), 1256-1258 G Frobenius Vber Matrizen aus positiven Elementen, Sitz Akad Wiss Berlin, Phys Math Klasse, 1908, pp 471-476; 1909, pp 514-.158 Vber Matrizen aus nicht negativen Elementen, Sitz Akad Wiss Berlin, Phys Math Klasse, 1912, pp 456-477 M Fujiwara Vber die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung, Tahoku Math J., 10 (1916), 167-171 C F Gauss Collected Works, vol 10, part II, pp 189-191 Methodus nova integralium valores per approximationem inveniendi, Werke, vol 3, pp 163-196 S Gerschgorin Bull Acad Sci de I'URSS, Leningrad, Classe Math., 7e serie, 1931, pp 749-754 T H Gronwall Proc National Academy of Sciences, U.s.A., (1920),300 302 J Hadamard Lerons sur la Propagation des Ondes, COllege de France, Paris (1903) pp 13-14 W K Hayman A Generalisation of Stirling's Formula, Jour fiir Math., 196 (1956), 67-95 M R Hestenes and W Karush A Method of Gradients for the Calculation of the Characteristic Roots and Vectors of a Real Symmetric Matrix, J Res Nat'l Bur of Stds., vol 47, pp 45-61 A Hurwitz Vber die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reelen Theilen besitzt, Math Ann., 46 (1895),273-284 ORIGINAL WORKS CITED IN THE TEXT 281 C J G Jacobi Uber Gauss' neue Methode, die Werthe der Integrale niiherungsweise zu tinden, I fur Math., (1826), pp 30F 308 P Koebe I.fur die Reine und Angewandte Mathematik, 145 (1915), 177-225 T Kojima On the limits of the roots of an algebraic equation, Tohoku Math J., 11 (1917), 119-127 E Landau Abschiitzung der Koeffizientensumme einer Potenzreihe, Arch fiir Math., 21 42-50 P S Laplace Oeuvres, vol 7, Gauthier-Villars, Paris, 1886, p 89 L Levy C R Acad Sci., 93 (1881), 707-708 K Lowner Untersuchungen tiber die Verzerrung bei konformen Abbildungen des Einheitskreises Izl 1, die durch Funktionen mit nieht versehwindender Ableitung geliefert werden, Leipzig Ber., 69 (1917),89-106 F.Lucas Geometrie des Polynomes, J Ecole Poly., 46 (1879), 1-33 C MacLaurin Treatise of Fluxions, Edinburgh, 1742 P Montel Le~ons sur les fonetions univalentes ou multivalentes, Paris, 1933 M A Pellet Sur un mode de separation des racines des equations et la formule de Lagrange, Bull des Sciences Math., (1881), 393-395 O Perron Grundlagen fUr eine Theorie des Jaeobisehen Kettenbruchalgorithmus, Math Ann., 64, (1907), 1-76 Zur Theorie der Matrizen, Math Ann 64 (1907), 248-263 W W Rogosinski Uber positive harmonische Sinusentwicklungen, Jahrsber deutsch Math Ver., 40 (1931), 33-35 E J Routh Dynamics of a System of Rigid Bodies, London, 1905 282 MATHEMATICS FOR THE PHYSICAL SCIENCES T Sherman and W • T Morrison Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column or a Given Row of the Original Matrix, Ann Math Stat., 20 (1949), 621 L Throumo)opou)os On the modulus of the roots of polynomials, Bu/! Greek Math Soc., 23 (1947), 18-20 .T L Ullman The Tschebycheff Method of Approximate Integration, Abstract, Notices of the Amer Math Soc., (1961), 49 H Wielandt Unzer1egbare, nicht negative Matrizen, Math Z., 52 (1950), 642-648 H S Wilr Perron-Frobenius Theory and the Zeros of Polynomials, Proc Amer Math Soc., 12 (1961),247-250 The Possibility of Tschebycheff Quadrature on Infinite Intervals, Froc Nat 'I Acad Sci U.S.A., 47 (1961),209-213 J E Wilkins Jr The Average of the Re.:iprocal of a Function, Pmc Amer Math Soc., (1955), 806-815 A Wintner On the Process of Successive Approximation in Initial Value Problems, Annali di Matematica Pura ed Applicata, 41 (1956), 343-357 M Woodbury Inverting Modified Matrices, Memo Rpt 42, Stat Res Group, Princeton, 1950 Index differential equations, matrix methods, 36 numerical solution, 153 ff regular points of, 166 dimension of a vector space, Dirac delta function, 81 Dirichlet's integral, 73 Abel's lemma, 132 Adams-Bashforth formulas, 165 asymptotic sequences, 112 asymptotic series, 112 basis, Bernoulli's method, 102 Bernoulli's numbers, 115 Bessel functions, 141, 179 ff Bessel's inequality, bordering Hermitian matrices, 26 bounds for eigenvalues, 38 brachistochrone problem, 223 Budan's rule, 94 eigenvalues and eigenvectors, bounds for, 38 definition, numerical calculation of, 34-35 of matrices, 18 equicontinuity, 193 equiconvergence, 78 equivalence relation, 46 Erdos-Turiin's theorem, 100 Euler equations, 222 Euler-MacLaurin sum formula, extended form, 119 first form, 117 extreme points, 234 Cauchy integral formula, 47 Cayley-Hamilton theorem, 24 characteristic polynomial, 19, 24 Christoffel-Darbou" identity, 55, 70 classical orthogonal polynomials, 64 commutator, 8, 46 companion matrix, 25, 40, 86 complete orthonormal set, conformal mapping, applications, 205 ff definition, 190 of polygons, 203 convergence of orthogonal expansions, Hjer summabiIity, 75 Fibonacci numbers, 136 Fourier coefficients, 4, 5, 70 Fourier series, 2, 70 69 convex hull, 85 convex set, 84 degeneracy, 19 Descartes' rule of signs, 94 differential equations, existence theorems, 143 ff Gamma function, 173 ff Gauss-Lucas theorem, 83 ff., 89 Gauss quadrature, 61 generating functions, 137 Graeffe's process, 105 Gram-Schmidt process, 283 284 Heine-Borel theorem, 195 INDEX Koebe image domains, 202 Perron-Frobenius theorem, 42, 60 Perron root, 43, 61, 86 Picard's theorem, 145 Poisson's equation, 37 positive definite matrices, 28 predictor-corrector formulas, 158 Lagrange multipliers, 217 Lagrangian formulas, 155 Landau's numbers, 141 Laplace's method, 127 linear independence, linear pn.gramming, 233 ff_ Lipschitz' condition, 75 local uniform boundedness, 193 rank,30 recurrence formula, 52, 136 ff relaxation, 38 Riemann-Lebesgue lemma, 6, 75 Riemann mapping theorem, 197 Riemann zeta function, 114, 124, 125 Rodrigues' formula, 58 Rolle's theorem, 83, 95 matrix, adjoint, 15 adjugate, 17 algebra, 14, 15 companion, 25, 40 diagonalization of, 21 Hermitian, 16 inverse, 16 modal,21 of rank one, 31, 32 polar, 21 positive definite, 28 reducible, 40 skew-Hermitian, 45 square root of, 46 symmetric, 16 modified weight function, 57 moduli of zeros, bounds for, 85 moments, 49 monodromy theorem, 199 Montel family of functions, 193 Mantel's theorem, 196 saddle point, 217 schlicht domain, 192 Schwarz-Christoffel mapping, 203 Schwarz' inequality, Schwarz' lemma, 209 second mean value theorem, 132 similar matrices, 21 Simplex method, 235 Simpson's rule, 61 stability, 161 If st?tionary phase, 131 ff Stirling's formula, 120 Sturm sequences, 90 subordination, 209 sums of powers, 122 inner product, Jacobi matrix, 55 Newton-Cotes formula, 61 Newton-Raphson iteration, 104 Newton's sums, 100 normalization constants, 54 nullity, 30 operators, linear, ff orthogonal, functions, 48 If polynomials, 49 ff vectors, Parseval's identity, partial differential equations, 37 trace, 20 trapezoidal rule, 61 triangle inequality, 45 trigonometrit series, 72 truncation error, 156 Tschebychelf-Bernstein quadrature, 80 unitary space, univalent function, 192 variational notation, 230 variational principle, 233 vector space, Weierstrass approximation theorem, 240 weight function, 49 Wintner's method, 149 zeros of orthogonal polynomials, 59 ... many cases students who not major in mathematics have room in their schedules for only one or two mathematics courses The purpose of this book, therefore, is to provide the student with some heavy... MATHEMATICS FOR THE PHYSICAL SCIENCES by Herbert S Wilf, Ph.D Professor of Mathematics University of Pennsylvania DOVER PUBLICATIONS, INC NEW YORK Copyright © 1962 by Herbert S Wilf All... by the student himself The exercises at the end of each chapter, along with their solutions at the back of the book, afford some further opportunities for using the theoretical apparatus The