Equations of Mathematical Physics Equations of Mathematical Physics By Marian Apostol Equations of Mathematical Physics By Marian Apostol This book first published 2018 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2018 by Marian Apostol All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-1616-4 ISBN (13): 978-1-5275-1616-8 Contents 1 Preface 1 2 Introductory Elements 5 2.1 LinearAlgebra ...................... 5 2.1.1 Vectors ...................... 5 2.1.2 Matrices...................... 6 2.1.3 Quadratic forms. Diagonalization . 8 2.1.4 Besselinequality . 12 2.2 IntegralEquations . 13 2.2.1 Fredholmequations . 13 2.2.2 Degeneratekernels . 14 2.2.3 Volterraequation. 15 2.3 CalculusofVariations . 16 2.3.1 Extremapoints. 16 2.3.2 Variationalproblems . 16 2.4 FourierTransform .. .. .. .. .. .. .. .. 18 2.4.1 Deltafunction . .. .. .. .. .. .. .. 18 2.4.2 Fouriertransform. 20 2.4.3 Fourierseries . .. .. .. .. .. .. .. 21 2.4.4 Periodicfunctions . 23 2.4.5 Particular orthogonal circular functions . 24 2.5 CauchyIntegral...................... 26 2.5.1 Cauchyintegral. 26 2.5.2 Integrals. Laplace and Mellin transforms . 28 2.6 SeriesExpansions. 29 2.6.1 Taylorseries ................... 29 2.6.2 Laurentseries. 30 2.6.3 AseriesofDarboux . 30 2.6.4 Bernoulli numbers and polynomials . 31 2.6.5 Euler-Maclaurinformula. 32 v Contents 2.6.6 Expansion in rational fractions. Infinite products 33 2.6.7 Asymptoticseries. 34 2.6.8 Steepestdescent . 35 2.6.9 Numerical series and series of functions . 36 2.7 CurvilinearCoordinates . 38 2.7.1 Laplacian ..................... 38 2.7.2 Divergenceandcurl . 39 2.8 CoulombPotential . 41 2.8.1 Basicequation . .. .. .. .. .. .. .. 41 2.8.2 Fouriertransform. 42 2.8.3 2+1dimensions. 44 2.9 BesselFunctions ..................... 45 2.9.1 Definition ..................... 45 2.9.2 m th order ................... 47 2.9.3 Completeness,− orthogonality and addition the- orem........................ 48 2.9.4 OtherBesselfunctions . 50 2.9.5 Afewrecurrencerelations . 51 2.9.6 Bessel functions of half-integer order . 52 2.10 LegendrePolynomials . 53 2.10.1 Definition ..................... 53 2.10.2 Generating function and recurrence relations . 55 2.10.3 Legendre’sequation . 55 2.11 SphericalHarmonics . 56 2.11.1 AssociatedLegendrefunctions . 56 2.11.2 Sphericalharmonics . 57 2.11.3 Poisson’sintegral . 59 2.11.4 Laplaceequation . 60 2.11.5 SphericalBesselfunctions . 62 2.12 SphericalWaves . .. .. .. .. .. .. .. .. 64 2.12.1 Waveequation . 64 2.12.2 2+1dimensions. 65 2.12.3 Spherical wave at infinity, Hankel function . 67 2.12.4 Two dimensions, cylindrical waves . 68 2.12.5 Helmholtz equation, addition theorem . 70 2.13 PhysicalEquations . 71 2.13.1 Physicalequations . 71 2.13.2 Laplaceequation . 72 vi Contents 2.13.3 AssociatedLegendrefunctions . 74 2.13.4 Besselfunctions. 75 2.13.5 Waveequation . 76 2.13.6 Heatequation. 77 2.14 PoissonEquation . 78 2.14.1 GeneralizedPoissonequation . 78 2.14.2 Planargeometry . 79 2.14.3 Cylindricalgeometry . 80 2.14.4 Sphericalgeometry . 82 2.15 TranscendentalFunctions . 84 2.15.1 Differential equations. Hermite polynomials . 84 2.15.2 Airyfunction . 86 2.15.3 Hypergeometricfunction. 87 2.15.4 Laguerre polynomials and other orthogonal poly- nomials ...................... 89 2.15.5 Gammafunction . 90 2.15.6 Zetafunction . 91 2.15.7 Mathieufunctions . 92 2.15.8 Elliptic functions . 94 3 Differential Equations. Generalities 97 4 TheEquationoftheHarmonicOscillator 101 4.1 Homogeneous equation (free equation) . 101 4.2 Inhomogeneous equation. Fundamental solution . 102 4.3 Greenfunction ...................... 103 4.4 Another representation of the solution. The Green the- orem............................ 104 4.5 Imagesources....................... 106 4.6 Generalizedequation . 107 4.7 AnotherGreenfunction . 108 4.8 Dampedharmonicoscillator . 109 4.9 Resonance......................... 110 5 LaplaceandPoissonEquations 113 5.1 Greenfunctions. 113 5.2 Greentheorem(formulae) . 115 5.3 Boundaryvalues . .. .. .. .. .. .. .. .. 117 vii Contents 5.4 ImageGreenfunctions . 119 5.5 Generalizedequation . 120 5.6 Fourier transform of the Coulomb potential . 123 5.7 Poissonformulaforsphereand circle . 125 5.8 Eigenvalues and eigenfunctions . 126 5.9 Plane waves. Fourier eigenfunctions . 129 5.10 Solutionbyseriesexpansion . 132 5.11 Spherical coordinates. Legendre polynomials . 134 5.12 Sphericalharmonics . 138 5.13 Cylindrical coordinates. Bessel functions . 143 5.14 Some properties of Jm(z) ................ 148 5.15 OtherBesselfunctions . 149 5.16 Bessel functions of half-integer order . 150 5.17 SphericalBesselfunctions . 151 5.18 Generalized Poisson equation in spherical geometry . 154 5.19 Generalized Poisson equation in cylindrical geometry . 156 6 Wave Equation 159 6.1 Greenfunction ...................... 159 6.2 Spherical waves. Helmholtz equation . 161 6.3 Cylindrical waves in 2D. The Hankel function . 163 6.4 Waves in 1D ....................... 165 6.5 Decomposition in spherical Bessel functions . 167 6.6 Initialconditions . 168 6.7 The wave of a uniformly moving source . 170 6.8 Vibrations. 1D string .................. 171 6.9 Thecirculardrumandcymbal . 176 6.10 Thevibrationsofthesphere. 180 6.11 Aspecialexample . 184 6.12Radiation ......................... 188 6.13Scattering ......................... 192 6.14 Weyl-Sommerfeldintegrals. 194 7 Vector Equations 199 7.1 Elastic waves. Helmholtz potentials . 199 7.2 Hertzpotentials. 200 7.3 Vibrationsoftheelasticsphere . 201 7.4 Staticelasticity . 208 viii Contents 7.5 Kelvin’sproblem . 209 7.6 Fluids ........................... 210 7.7 Maxwellequations . 216 7.8 Hansenvectors ...................... 219 8 QuasiclassicalApproximation 223 8.1 JWKBmethod ...................... 223 8.2 Sphere........................... 225 8.3 SphericalBesselfunctions . 227 8.4 Legendrefunctions . 228 9 OrdinaryDifferentialEquations 231 9.1 Introduction........................ 231 9.2 Constantcoefficients . 231 9.3 First-orderdifferentialequations . 232 9.4 Second-orderdifferentialequations . 233 9.5 Sturm-Liouvilletheory . 236 Index 239 ix 1 Preface Mathematics are the Equations of Mathematical Physics. They are based on numbers and mental constructs which we feel to be satisfy- ing and helpful in our endeavour to survive in relation with the world, nature and humans (Planck). The equations are the tools of Theoret- ical Physics. Both Mathematics and Physics have each their own halo of pseudo-science, a lot of nonsense which goes under the same name as the main scientific core. This is so, because both these sciences are successful and nobody can say what they are in fact. As if they came as strange living beings from an alien, superior world (God’s world?). Wigner would have said, approximately: "The miracle of the appro- priateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve"; many believe it is God’s gift. But nothing could impose itself upon us, and we would not have accepted such a wonderful gift, without understanding it; except for one thing: our own subjectivity. Mathematics and Physics are our own subjectivity; the difference is made by the quality of our subjectivity. The objectivity is the sub- jectivity of the "great" men. Mathematics, as well as Physics, contains a few things which impose themselves upon our mind with necessity, with force, authority, and beauty; or, perhaps, simply, we just recognize them as being quite fa- miliar to us. We do not know, in fact, what they are, nor where they come from. It is in the human soul the perversion to assert our arro- gance in matters which we do not understand. Out of this perversion, an immense body of useless, fake, ungracious and, ultimately, perni- cious things arise, which go, falsely, under the name of Mathematics or Physics. The most important thing in Mathematics and Physics is to be lucky enough to avoid the nonsense, the monstrous, pathological constructions which bring pain (Landau); such constructions are not even wrong (Pauli). I chose here the basic elements of Mathematics 1 1 Preface which I felt are healthy. I have never seen a textbook on Mathematics which was not long, though titles like Methods, Introduction, Course, etc, seem to be meant to convey that only important things were in- cluded; nothing more misleading. Mathematics or Physics authors, for their great majority, seem to not compare themselves at all with mathematicians or physicists. I hope this booklet is different, at least by its brevity. Mathematics has been made by numerous people along many years. I give here a list: Isaac Newton (1642-1727), Gottfried Wilhelm Leibniz (1646-1714), Jacob Bernoulli (1654-1705), Johann Bernoulli (1667-1748), Brook Taylor (1685-1731), Nicolas Bernoulli (1687-1759), Daniel Bernoulli (1700-1782), Leonhard Euler (1707-1783), Jean le Rond d’Alembert (1717-1783), Joseph-Louis Lagrange (1736-1813), Pierre-Simon Laplace (1749-1827), Adrien-Marie Legendre (1752-1833), Jean Baptiste Joseph Fourier (1768-1830), Carl Friedrich Gauss (1777-1855), Simeon De- nis Poisson (1781-1840), Friedrich Bessel (1784-1846), Augustin-Louis Cauchy (1789-1857),