Equations of Mathematical Physics

Description

The differential equations of mathematical physics have a twofold character: their physical content and their mathematical solutions. This book discusses the basic tools of theoretical physicists, applied mathematicians, and engineers, providing detailed insights into linear algebra, Fourier transforms, special functions, Laplace and Poisson, diffusion and vector equations. These basic tools are a set of methods and techniques, known as the equations of mathematical physics. At first sight, they look like a collection of disparate things. Many students in theoretical physics perceive them as strange, autonomous, inflexible, and ultimately unknown objects, whose sole use resides in their being applied to solving usually standard physical problems. While mathematicians are oriented towards empty generalizations and the so-called mathematical rigour, theoretical physicists often limit themselves to giving a set of recipes and examples. Both succeed in producing large, heavy tomes, which are, to a large extent, useless. The only exception seems to be Sommerfeld’s Partielle Differentialgleichungen der Physik, which, however, is rather limited to a restricted list of subjects. The physical nature and origin of the equations of mathematical physics is emphasized in this book, and their various elements and great flexibility are described. The book reveals the indissoluble connection between physical ideas and mathematical concepts, and how these visions can be transcribed into accurate mathematics. Contents 1 Preface 2 Introductory Elements 2.1 Linear Algebra 2.1.1 Vectors 2.1.2 Matrices 2.1.3 Quadratic forms. Diagonalization 2.1.4 Bessel inequality 2.2 Integral Equations 2.2.1 Fredholm equations 2.2.2 Degenerate kernels 2.2.3 Volterra equation 2.3 Calculus of Variations 2.3.1 Extrema points 2.3.2 Variational problems 2.4 Fourier Transform 2.4.1 Delta function 2.4.2 Fourier transform 2.4.3 Fourier series 2.4.4 Periodic functions 2.4.5 Particular orthogonal circular functions 2.5 Cauchy Integral 2.5.1 Cauchy integral 2.5.2 Integrals. Laplace and Mellin transforms 2.6 Series Expansions 2.6.1 Taylor series 2.6.2 Laurent series 2.6.3 A series of Darboux 2.6.4 Bernoulli numbers and polynomials 2.6.5 Euler-Maclaurin formula 2.6.6 Expansion in rational fractions. Infinite products 2.6.7 Asymptotic series 2.6.8 Steepest descent 2.6.9 Numerical series and series of functions 2.7 Curvilinear Coordinates 2.7.1 Laplacian 2.7.2 Divergence and curl 2.8 Coulomb Potential 2.8.1 Basic equation 2.8.2 Fourier transform 2.8.3 2+1 dimensions 2.9 Bessel Functions 2.9.1 Definition 2.9.2 m − th order 2.9.3 Completeness, orthogonality and addition theorem 2.9.4 Other Bessel functions 2.9.5 A few recurrence relations 2.9.6 Bessel functions of half-integer order 2.10 Legendre Polynomials 2.10.1 Definition 2.10.2 Generating function and recurrence relations 2.10.3 Legendre’s equation 2.11 Spherical Harmonics 2.11.1 Associated Legendre functions 2.11.2 Spherical harmonics 2.11.3 Poisson’s integral 2.11.4 Laplace equation 2.11.5 Spherical Bessel functions 2.12 Spherical Waves 2.12.1 Wave equation 2.12.2 2+1 dimensions 2.12.3 Spherical wave at infinity, Hankel function 2.12.4 Two dimensions, cylindrical waves 2.12.5 Helmholtz equation, addition theorem 2.13 Physical Equations 2.13.1 Physical equations 2.13.2 Laplace equation 2.13.3 Associated Legendre functions 2.13.4 Bessel functions 2.13.5 Wave equation 2.13.6 Heat equation 2.14 Poisson Equation 2.14.1 Generalized Poisson equation 2.14.2 Planar geometry 2.14.3 Cylindrical geometry 2.14.4 Spherical geometry 2.15 Transcendental Functions 2.15.1 Differential equations. Hermite polynomials 2.15.2 Airy function 2.15.3 Hypergeometric function 2.15.4 Laguerre polynomials and other orthogonal polynomials 2.15.5 Gamma function 2.15.6 Zeta function 2.15.7 Mathieu functions 2.15.8 Elliptic functions 3 Differential Equations. Generalities 4 The Equation of the Harmonic Oscillator 4.1 Homogeneous equation (free equation) 4.2 Inhomogeneous equation. Fundamental solution 4.3 Green function 4.4 Another representation of the solution. The Green theorem 4.5 Image sources 4.6 Generalized equation 4.7 Another Green function 4.8 Damped harmonic oscillator 4.9 Resonance 5 Laplace and Poisson Equations 5.1 Green functions 5.2 Green theorem (formulae) 5.3 Boundary values 5.4 Image Green functions 5.5 Generalized equation 5.6 Fourier transform of the Coulomb potential 5.7 Poisson formula for sphere and circle 5.8 Eigenvalues and eigenfunctions 5.9 Plane waves. Fourier eigenfunctions 5.10 Solution by series expansion 5.11 Spherical coordinates. Legendre polynomials 5.12 Spherical harmonics 5.13 Cylindrical coordinates. Bessel functions 5.14 Some properties of Jm(z) 5.15 Other Bessel functions 5.16 Bessel functions of half-integer 5.17 Spherical Bessel functions 5.18 Generalized Poisson equation in spherical geometry 5.19 Generalized Poisson equation in cylindrical geometry 6 Wave Equation 6.1 Green function 6.2 Spherical waves. Helmholtz equation 6.3 Cylindrical waves in 2D. The Hankel function 6.4 Waves in 1D 6.5 Decomposition in spherical Bessel functions 6.6 Initial conditions 6.7 The wave of a uniformly moving source 6.8 Vibrations. 1D str 6.9 The circular drum and cymbal 6.10 The vibrations of the sphere 6.11 A special example 6.12 Radiation 6.13 Scattering 6.14 Weyl-Sommerfeld integrals 7 Vector Equations 7.1 Elastic waves. Helmholtz potentials 7.2 Hertz potentials 7.3 Vibrations of the elastic sphere 7.4 Static elasticity 7.5 Kelvin’s problem 7.6 Fluids 7.7 Maxwell equations 7.8 Hansen vectors 8 Quasiclassical Approximation 8.1 JWKB method 8.2 Sphere 8.3 Spherical Bessel functions 8.4 Legendre functions 9 Ordinary Differential Equations 9.1 Introduction 9.2 Constant coefficients 9.3 First-order differential equations 9.4 Second-order differential e 9.5 Sturm-Liouville theory Index