Springer Undergraduate Mathematics Series Advisory Board M.A.J Chaplain University of Dundee K Erdmann Oxford University A MacIntyre Queen Mary, University of London L.C.G Rogers University of Cambridge E Süli Oxford University J.F Toland University of Bath Other books in this series A First Course in Discrete Mathematics I Anderson Analytic Methods for Partial Differential Equations G Evans, J Blackledge, P Yardley Applied Geometry for Computer Graphics and CAD, Second Edition D Marsh Basic Linear Algebra, Second Edition T.S Blyth and E.F Robertson Basic Stochastic Processes Z Brzeź niak and T Zastawniak Calculus of One Variable K.E Hirst Complex Analysis J.M Howie Elementary Differential Geometry A Pressley Elementary Number Theory G.A Jones and J.M Jones Elements of Abstract Analysis M Ó Searcóid Elements of Logic via Numbers and Sets D.L Johnson Essential Mathematical Biology N.F Britton Essential Topology M.D Crossley Fields and Galois Theory J.M Howie Fields, Flows and Waves: An Introduction to Continuum Models D.F Parker Further Linear Algebra T.S Blyth and E.F Robertson Game Theory: Decisions, Interaction and Evolution J.N Webb General Relativity N.M.J Woodhouse Geometry R Fenn Groups, Rings and Fields D.A.R Wallace Hyperbolic Geometry, Second Edition J.W Anderson Information and Coding Theory G.A Jones and J.M Jones Introduction to Laplace Transforms and Fourier Series P.P.G Dyke Introduction to Lie Algebras K Erdmann and M.J Wildon Introduction to Ring Theory P.M Cohn Introductory Mathematics: Algebra and Analysis G Smith Linear Functional Analysis, Second Edition B.P Rynne and M.A Youngson Mathematics for Finance: An Introduction to Financial Engineering M Capiń ski and T Zastawniak Matrix Groups: An Introduction to Lie Group Theory A Baker Measure, Integral and Probability, Second Edition M Capiń ski and E Kopp Metric Spaces M Ĩ Searcóid Multivariate Calculus and Geometry, Second Edition S Dineen Numerical Methods for Partial Differential Equations G Evans, J Blackledge, P Yardley Probability Models J Haigh Real Analysis J.M Howie Sets, Logic and Categories P Cameron Special Relativity N.M.J Woodhouse Sturm-Liouville Theory and its Applications: M.A Al-Gwaiz Symmetries D.L Johnson Topics in Group Theory G Smith and O Tabachnikova Vector Calculus P.C Matthews Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century J Gray M.A Al-Gwaiz Sturm-Liouville Theory and its Applications ABC M.A Al-Gwaiz Department of Mathematics King Saud University Riyadh, Saudi Arabia malgwaiz@ksu.edu.sa Cover illustration elements reproduced by kind permission of: Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E Kent-Kangley Road, Maple Valley, WA 98038, USA Tel: (206) 432 - 7855 Fax (206) 432 - 7832 email: info@aptech.com URL: www.aptech.com American Statistical Association: Chance Vol No 1, 1995 article by KS and KW Heiner ‘Tree Rings of the Northern Shawangunks’ page 32 fig Springer-Verlag: Mathematica in Education and Research Vol Issue 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor ‘Illustrated Mathematics: Visualization of Mathematical Objects’ page fig 11, originally published as a CD ROM ‘Illustrated Mathematics’ by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4 Mathematica in Education and Research Vol Issue 1995 article by Richard J Gaylord and Kazume Nishidate ‘Traffic Engineering with Cellular Automata’ page 35 fig Mathematica in Education and Research Vol Issue 1996 article by Michael Trott ‘The Implicitization of a Trefoil Knot’ page 14 Mathematica in Education and Research Vol Issue 1996 article by Lee de Cola ‘Coins, Trees, Bars and Bells: Simulation of the Binomial Process’ page 19 fig Mathematica in Education and Research Vol Issue 1996 article by Richard Gaylord and Kazume Nishidate ‘Contagious Spreading’ page 33 fig Mathematica in Education and Research Vol Issue 1996 article by Joe Buhler and Stan Wagon ‘Secrets of the Madelung Constant’ page 50 fig Mathematics Subject Classification (2000): 34B24, 34L10 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2007938910 Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 978-1-84628-971-2 e-ISBN 978-1-84628-972-9 © Springer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency Enquiries concerning reproduction outside those terms should be sent to the publishers The use of registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made Printed on acid-free paper 987654321 springer.com Preface This book is based on lecture notes which I have used over a number of years to teach a course on mathematical methods to senior undergraduate students of mathematics at King Saud University The course is offered here as a prerequisite for taking partial differential equations in the final (fourth) year of the undergraduate program It was initially designed to cover three main topics: special functions, Fourier series and integrals, and a brief sketch of the Sturm– Liouville problem and its solutions Using separation of variables to solve a boundary-value problem for a second-order partial differential equation often leads to a Sturm–Liouville eigenvalue problem, and the solution set is likely to be a sequence of special functions, hence the relevance of these topics Typically, the solution of the partial differential equation can then be represented (pointwise) by a Fourier series or a Fourier integral, depending on whether the domain is finite or infinite But it soon became clear that these “mathematical methods” could be developed into a more coherent and substantial course by presenting them within the more general Sturm–Liouville theory in L2 According to this theory, a linear second-order differential operator which is self-adjoint has an orthogonal sequence of eigenfunctions that spans L2 This immediately leads to the fundamental theorem of Fourier series in L2 as a special case in which the operator is simply d2 /dx2 The other orthogonal functions of mathematical physics, such as the Legendre and Hermite polynomials or the Bessel functions, are similarly generated as eigenfunctions of particular differential operators The result is a generalized version of the classical theory of Fourier series, which ties up the topics of the course mentioned above and provides a common theme for the book vi Preface In Chapter the stage is set by defining the inner product space of square integrable functions L2 , and the basic analytical tools needed in the chapters to follow These include the convergence properties of sequences and series of functions and the important notion of completeness of L2 , which is defined through Cauchy sequences The difficulty with building Fourier analysis on the Sturm–Liouville theory is that the latter is deeply rooted in functional analysis, in particular the spectral theory of compact operators, which is beyond the scope of an undergraduate treatment such as this We need a simpler proof of the existence and completeness of the eigenfunctions In the case of the regular Sturm–Liouville problem, this is achieved in Chapter by invoking the existence theorem for linear differential equations to construct Green’s function for the Sturm– Liouville operator, and then using the Ascoli–Arzela theorem to arrive at the desired conclusions This is covered in Sections 2.4.1 and 2.4.2 along the lines of Coddington and Levinson in [6] Chapters through present special applications of the Sturm–Liouville theory Chapter 3, which is on Fourier series, provides the prime example of a regular Sturm–Liouville problem In this chapter the pointwise theory of Fourier series is also covered, and the classical theorem (Theorem 3.9) in this context is proved The advantage of the L2 theory is already evident from the simple statement of Theorem 3.2, that a function can be represented by a Fourier series if and only if it lies in L2 , as compared to the statement of Theorem 3.9 In Chapters and we discuss some of the more important examples of a singular Sturm–Liouville problem These lead to the orthogonal polynomials and Bessel functions which are familiar to students of science and engineering Each chapter concludes with applications to some well-known equations of mathematical physics, including Laplace’s equation, the heat equation, and the wave equation Chapters and on the Fourier and Laplace transformations are not really part of the Sturm–Liouville theory, but are included here as extensions of the Fourier series method for representing functions These have important applications in heat transfer and signal transmission They also allow us to solve nonhomogeneous differential equations, a subject which is not discussed in the previous chapters where the emphasis is mainly on the eigenfunctions The reader is assumed to be familiar with the convergence properties of sequences and series of functions, which are usually presented in advanced calculus, and with elementary ordinary differential equations In addition, we have used some standard results of real analysis, such as the density of continuous functions in L2 and the Ascoli–Arzela theorem These are used to prove the existence of eigenfunctions for the Sturm–Liouville operator in Chapter 2, and they Preface vii have the advantage of avoiding any need for Lebesgue measure and integration It is for that reason that smoothness conditions are imposed on the coefficients of the Sturm–Liouville operator, for otherwise integrability conditions would have sufficed The only exception is the dominated convergence theorem, which is invoked in Chapter to establish the continuity of the Fourier transform This is a marginal result which lies outside the context of the Sturm–Liouville theory and could have been handled differently, but the temptation to use that powerful theorem as a shortcut was irresistible This book follows a strict mathematical style of presentation, but the subject is important for students of science and engineering In these disciplines, Fourier analysis and special functions are used quite extensively for solving linear differential equations, but it is only through the Sturm–Liouville theory in L2 that one discovers the underlying principles which clarify why the procedure works The theoretical treatment in Chapter need not hinder students outside mathematics who may have some difficulty with the analysis Proof of the existence and completeness of the eigenfunctions (Sections 2.4.1 and 2.4.2) may be skipped by those who are mainly interested in the results of the theory But the operator-theoretic approach to differential equations in Hilbert space has proved extremely convenient and fruitful in quantum mechanics, where it is introduced at the undergraduate level, and it should not be avoided where it seems to brings clarity and coherence in other disciplines I have occasionally used the symbols ⇒ (for “implies”) and ⇔ (for “if and only if”) to connect mathematical statements This is done mainly for the sake of typographical convenience and economy of expression, especially where displayed relations are involved A first draft of this book was written in the summer of 2005 while I was on vacation in Lebanon I should like to thank the librarian of the American University of Beirut for allowing me to use the facilities of their library during my stay there A number of colleagues in our department were kind enough to check the manuscript for errors and misprints, and to comment on parts of it I am grateful to them all Professor Saleh Elsanousi prepared the figures for the book, and my former student Mohammed Balfageh helped me to set up the software used in the SUMS Springer series I would not have been able to complete these tasks without their help Finally, I wish to express my deep appreciation to Karen Borthwick at Springer-Verlag for her gracious handling of all the communications leading to publication M.A Al-Gwaiz Riyadh, March 2007 Contents Preface v Inner Product Space 1.1 Vector Space 1.2 Inner Product Space 1.3 The Space L2 1.4 Sequences of Functions 1.5 Convergence in L2 1.6 Orthogonal Functions 1 14 20 31 36 The Sturm–Liouville Theory 2.1 Linear Second-Order Equations 2.2 Zeros of Solutions 2.3 Self-Adjoint Differential Operator 2.4 The Sturm–Liouville Problem 2.4.1 Existence of Eigenfunctions 2.4.2 Completeness of the Eigenfunctions 2.4.3 The Singular SL Problem 41 41 49 55 67 68 79 88 Fourier Series 93 3.1 Fourier Series in L2 93 3.2 Pointwise Convergence of Fourier Series 102 3.3 Boundary-Value Problems 117 3.3.1 The Heat Equation 118 3.3.2 The Wave Equation 123 x Contents Orthogonal Polynomials 129 4.1 Legendre Polynomials 130 4.2 Properties of the Legendre Polynomials 135 4.3 Hermite and Laguerre Polynomials 141 4.3.1 Hermite Polynomials 141 4.3.2 Laguerre Polynomials 145 4.4 Physical Applications 148 4.4.1 Laplace’s Equation 148 4.4.2 Harmonic Oscillator 153 Bessel Functions 157 5.1 The Gamma Function 157 5.2 Bessel Functions of the First Kind 160 5.3 Bessel Functions of the Second Kind 168 5.4 Integral Forms of the Bessel Function Jn 171 5.5 Orthogonality Properties 174 The Fourier Transformation 185 6.1 The Fourier Transform 185 6.2 The Fourier Integral 193 6.3 Properties and Applications 206 6.3.1 Heat Transfer in an Infinite Bar 208 6.3.2 Non-Homogeneous Equations 214 The Laplace Transformation 221 7.1 The Laplace Transform 221 7.2 Properties and Applications 227 7.2.1 Applications to Ordinary Differential Equations 230 7.2.2 The Telegraph Equation 236 Solutions to Selected Exercises 245 References 259 Notation 261 Index 263 250 Solutions to Selected Exercises 4.7 Differentiate Rodrigues’ formula for Pn and replace n by n + to obtain Pn+1 = 2n n! dn dxn (2n + 1)x2 − (x2 − 1)n−1 , then differentiate Pn−1 and subtract The first integral formula follows directly from (4.14) and the equality Pn (±1) = (±1)n The second results from setting x = 4.11 (a) − x3 = P0 (x) − 35 P1 (x) − 25 P3 (x) (b) |x| = 12 P0 (x) + 58 P2 (x) − 4.13 f (x) = ∞ n=0 16 P4 (x) + ··· cn Pn (x), where cn = (2n + 1)/2 −1 f (x)Pn (x)dx Because f is odd, cn = for all even values of n For n = 2k + 1, c2k+1 = (4k + 3) P2k+1 (x)dx [P2k (0) − P2k+2 (0)] 4k + (2k + 2)! (2k)! + 2k+2 = (−1)k 2k k!k! (k + 1)!(k + 1)! (2k)! (4k + 3) , k ∈ N0 = (−1)k 2k k!k! (2k + 2) = (4k + 3) Hence f (x) = 32 P1 (x) − 78 P3 (x) + 11 16 P5 (x) ∞ cn Pn (0) = = n=0 4.15 ∞ −x2 e dx −∞ =4 + · · · At x = 0, [f (0+ ) + f (0− )] ∞ ∞ −(x2 +y ) e dxdy 0 =4 ∞ π/2 r e r 0 drdθ = π 4.16 Replace t by −t in Equation (4.25) to obtain ∞ ∞ 1 Hn (x)(−t)n = e−2xt−t = Hn (−x)tn , n! n! n=0 n=0 which implies Hn (−x) = (−1)n Hn (x) 4.17 Setting x = in (4.25) yields ∞ k=0 ∞ 1 Hk (0)tk = e−t = (−1)n t2n k! n! n=0 By equating corresponding coefficients we obtain the desired formulas Solutions to Selected Exercises 251 4.19 If m = 2n, (2n)! 22n x2n = n k=0 H2k (x) (2k)!(n − k)! If m = 2n + 1, x2n+1 = (2n + 1)! 22n+1 n k=0 H2k+1 (x) , (2k + 1)!(n − k)! x ∈ R, n ∈ N0 4.23 Use Leibnitz’ rule for the derivative of a product, ∞ (f g)(n) = k=0 n (n−k) (k) f g , k with f (x) = xn and g(x) = e−x 4.25 xm = m n=0 ∞ e−x xm Ln (x)dx = (−1)n cn Ln (x), where cn = x 4.28 u(x) = c1 + c2 x2 e dx = c1 + c2 log x + x + + ··· x 2! m!m! n!(m − n)! 4.29 The surface ϕ = π/2, corresponding to the xy-plane 4.31 The solution of Laplace’s equation in the spherical coordinates (r, ϕ) is given by Equation (4.42) Using the given boundary condition in Equation (4.43), an = 2n + 2Rn π/2 10Pn (cos ϕ)sin ϕ dϕ 5(2n + 1) Pn (x)dx Rn n ∈ N, = n [Pn−1 (0) − Pn+1 (0)], R where the result of Exercise 4.7 is used in the last equality We therefore arrive at the solution = ∞ r R Pn−1 (0) − Pn+1 (0)] u(r, ϕ) = + n=1 =5 1+ 3r P1 (cos ϕ) − 2R r R n Pn (cos ϕ) P3 (cos ϕ) + · · · Note that u(R, ϕ) − is an odd function of ϕ, hence the summation (starting with n = 1) is over odd values of n 4.33 In view of the boundary condition uϕ (r, π/2) = 0, f may be extended as an even function of ϕ from [0, π/2] to [0, π] By symmetry the solution is even about ϕ = π/2, hence the summation is over even orders of the Legendre polynomials 252 Solutions to Selected Exercises Chapter n 5.1 For all n ∈ N, the integral In (x) = e−t tx−1 dt is a continuous function of x ∈ [a, b], where < a < b < ∞ Because ∞ 0≤ e−t tx−1 dt ≤ n ∞ e−t tb−1 dt → 0, u n it follows that In converges uniformly to Γ (x) Therefore Γ (x) is continuous on [a, b] for any < a < b < ∞, and hence on (0, ∞) By a similar pro∞ cedure we can also show that its derivatives Γ (x) = e−t tx−1 log tdt, ∞ −t x−1 (log t)2 dt, · · · are all continuous on (0, ∞) Γ (x) = n e t 5.3 Γ n + = n− know that Γ ··· √ = π 2 2 Γ = (2n)! Γ n!22n From Exercise 5.2 we 5.5 Use the integral definition of the gamma function to obtain 22x−1 Γ (x)Γ x+ ∞ ∞ =4 e−(α +β ) (2αβ)2x−1 (α + β)dαdβ, then change the variables of integration to ξ = α2 + β , η = 2αβ to arrive at the desired formula 5.7 Apply the ratio test 5.9 Differentiate Equation (5.12) and multiply by x 5.11 Substitute ν = −1/2 into the identity and use Exercise 5.8 5.17 Substitute directly into Bessel’s equation Note that, whereas Jn (x) is bounded at x = 0, yn (x) is not Hence the two functions cannot be linearly dependent 5.25 The definition of Iν , as given in Exercise 5.22, extends to negative values of ν Equation (5.18) is invariant under a change of sign of ν, hence it is satisfied by both Iν and I−ν 5.27 Follows from the bounds on the sine and cosine functions 5.29 Applying Parseval’s relation to Equations (5.22) and (5.23), we obtain ∞ π −π cos2 (x sin θ)dθ = 2πJ02 (x) + 4π m=1 ∞ π sin2 (x sin θ)dθ = 4π −π J2m (x) J2m−1 (x) m=1 By adding these two equations we arrive at the desired identity Solutions to Selected Exercises 253 5.31 Apply Lemma 3.7 to Equations (5.24) and (5.25) b 5.33 (a) 1, J0 (µk x) x = J0 (µk x)xdx = Therefore 1= (c) x2 , J0 (µk x) x ∞ b k=1 x = b ∞ x = k=1 b J12 (µk b) µ2k b2 − J0 (µk x) µ3k J1 (µk b) J0 (µk x)xdx = f (x) = J1 (µk b) Hence b/2 (e) f, J0 (µk x) b x= J0 (µk x) µk J1 (µk b) b3 4b − µk µk = b J1 (µk b), J0 (µk x) µk ∞ k=1 b J1 (µk b/2) Hence 2µk J1 (µk b/2) J0 (µk x) µk J12 (µk b) 5.35 From Exercises 5.13 and 5.14(a) we have x, J1 (µk x) x = J1 (µk x)x2 dx = −J0 (µk )/µk = J2 (µk )/µk , and, from Equation (5.34), J1 (µk x) x = 2 J2 (µk ) Therefore ∞ x=2 k=1 J1 (µk x), µk J2 (µk ) < x < 5.37 Using the results of Exercises 5.13 and 5.14(a), f, J1 (µk x) x x2 J1 (µk x)dx = = 2µk J1 (µk ) − µ2k J0 (µk ) µk3 J2 (µk ) = µk Bessel’s equation also implies J1 (µk x) x = 2[J1 (2µk )]2 + 4µ2k − (4µ2k − 1)J12 (2µk ) = J1 (2µk ) 2µk 2µ2k Consequently, ∞ f (x) = µk J2 (µk ) J1 (µk x), − 1)J12 (2µk ) (4µ2k k=1 < x < This representation is not pointwise At x = 1, f (1) = whereas the right-hand side is 12 [f (1+ ) + f (1− )] = 12 254 Solutions to Selected Exercises 5.39 Assuming u(r, t) = v(r)w(t) leads to w = kw v v + v r = −µ2 Solve these two equations and apply the boundary condition to obtain the desired representation for u 5.41 Use separation of variables to conclude that ∞ J0 (µk r)[ak cos µk ct + bk sin µk ct), u(r, t) = k=1 ak = bk = R R2 J12 (µk R) f (r)J0 (µk r)rdr, R cµk R2 J12 (µk R) g(r)J0 (µk r)rdr Chapter 6.1 (a) fˆ(ξ) = (1 ξ2 − cos ξ) (c) fˆ(ξ) = − e−iξ iξ 6.3 For any fixed point ξ ∈ J, let ξ n be a sequence in J which converges to ξ Because |F (ξ n ) − F (ξ)| ≤ |ϕ(x, ξ n ) − ϕ(x, ξ)| dx, I and |ϕ(x, ξ n ) − ϕ(x, ξ)| ≤ 2g(x) ∈ L1 (I), we can apply Theorem 6.4 to the sequence of functions ϕn (x) = ϕ(x, ξ n ) − ϕ(x, ξ) to conclude that lim |F (ξ n ) − F (ξ)| ≤ lim n→∞ n→∞ = |ϕ(x, ξ n ) − ϕ(x, ξ)| dx I lim |ϕ(x, ξ n ) − ϕ(x, ξ)| dx = I n→∞ 6.5 Suppose ξ ∈ J, and let ξ n → ξ Define ψ n (x, ξ) = ϕ(x, ξ n ) − ϕ(x, ξ) , ξn − ξ then ψ n (x, ξ) → ϕξ (x, ξ) pointwise ψ n is integrable on I and, by the mean value theorem, ψ n (x, ξ) = ϕn (x, η n ) for some η n between ξ n and ξ Solutions to Selected Exercises 255 Therefore |ψ n (x, ξ)| ≤ h(x) on I × J Now use the dominated convergence theorem to conclude that I ψ n (x, ξ)dx → I ϕξ (x, ξ)dx This proves F (ξ n ) − F (ξ) → ξn − ξ ϕξ (x, ξ)dx I The continuity of F follows from Exercise 6.3 6.8 (a) 1, (b) 1/2, (c) 6.9 Express the integral over (a, b) as a sum of integrals over the subintervals (a, x1 ), , (xn , b) Because both f and g are smooth over each subinterval, the formula for integration by parts applies to each integral in the sum π + cos πξ 6.10 (a) f is even, hence B(ξ) = 0, A(ξ) = sin x cos ξx dx = , − ξ2 ∞ + cos xξ cos xξ dξ and f (x) = π − ξ2 (c) f (x) = π ∞ ξ − sin ξ sin xξ dξ ξ2 6.13 Define x>0 e−x cos x, x < −ex cos x, Because f is odd its cosine transform is zero and f (x) = ∞ 2ξ ξ4 + Now f (x) may be represented on (−∞, ∞) by the inversion formula (6.28), B(ξ) = e−x cos x sin ξx dx = ∞ ξ3 sin xξ dξ ξ4 + Because f is not continuous at x = 0, this integral is not uniformly convergent e−x cos x = π 6.15 Extend 1, 0