Mathematical Methods for Physics: 45th Anniversary Edition

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Cover Half Title Title Page Copyright Page Table of Contents List of Figures List of Tables Editor’s Preface to the 45th Anniversary Edition Preface to the First Edition Section I Homogeneous Boundary Value Problems and Special Functions Chapter 1 • The Partial Differential Equations of Mathematical Physics 1.1 Introduction 1.2 Heat Conduction and Diffusion 1.3 Quantum Mechanics 1.4 Waves on Strings and Membranes 1.5 Hydrodynamics and Aerodynamics 1.6 Acoustic Waves in a Compressible Fluid 1.7 Irrotational Flow in an Incompressible Fluid 1.8 Electrodynamics 1.8.1 Time Independent Phenomena 1.8.2 Vacuum Equations 1.8.3 General Case 1.9 Summary Problems Chapter 2 • Separation of Variables and Ordinary Differential Equations 2.1 Introduction 2.2 Separation of Variables 2.3 Rectangular Coordinates (x, y, z) 2.4 Cylindrical Coordinates (r, θ, z) 2.5 Spherical Coordinates (r, θ, ?) 2.6 Series Solutions of Ordinary Differential Equations: Preliminaries 2.7 Expansion About a Regular Singular Point 2.8 Sturm-Liouville Eigenvalue Problem 2.9 Fourier Series and Integrals 2.10 Numerical Solution of Ordinary Differential Equations Problems Chapter 3 • Spherical Harmonics and Applications 3.1 Introduction 3.2 Series Solution of Legendre’s Equation-Legendre Polynomials 3.3 Properties of Legendre Polynomials 3.4 The Second Solution Ql(x) of Legendre’s Equation 3.5 Associated Legendre Polynomials 3.6 Spherical Harmonics 3.7 The Spherical Harmonics Addition Theorem 3.8 Multipole Expansions 3.9 Laplace’s Equation in Spherical Coordinates 3.9.1 Interior Problem I, r ≤ a with ψ(a, θ, ϕ)= u(θ, ϕ) given 3.9.2 Interior Problem II, r ≤ a with ∂ψ/∂rr=a = v(θ, ϕ) given 3.9.3 Exterior Problem, r ≥ a with ψ(a, θ, ϕ)= u(θ, ϕ) given 3.9.4 Exterior Problem, r ≥ a with ∂ψ/∂rr=a = v(θ, ϕ) given 3.9.5 Region Between Two Spheres, a ≤ r ≤ b with ψ(a, θ, ϕ)=u(θ, ϕ) and ψ(b, θ, ϕ)= v(θ, ϕ) given 3.9.6 Notes on Solving Other Boundary Conditions on Regions Between Two Spheres 3.10 Conducting Sphere in a Uniform External Electric Field 3.11 Flow of an Incompressible Fluid Around a Spherical Obstacle Problems Chapter 4 • Bessel Functions and Applications 4.1 Introduction 4.2 Series Solutions of Bessel’s Equation; Bessel Functions 4.3 Neumann Functions 4.4 Small Argument and Asymptotic Expansions 4.5 Bessel Functions of Imaginary Argument 4.6 Laplace’s Equation in Cylindrical Coordinates 4.7 Interior of a Cylinder of Finite Length 4.8 The Sturm-Liouville Eigenvalue Problem and Application of The Expansion Theorem 4.9 Interior of a Cylinder of Finite Length - Continued 4.10 Exterior of an Infinitely Long Cylinder 4.11 Cylinder in an External Field 4.12 Space between Two Infinite Planes 4.13 Fourier Bessel Transforms 4.14 Space between Two Infinite Planes - Continued Problems Chapter 5 • Normal Mode Eigenvalue Problems 5.1 Introduction 5.2 Reduction of the Diffusion Equation and Wave Equation to an Eigenvalue Problem 5.3 The Vibrating String 5.4 The Vibrating Drumhead 5.5 Heat Conduction in a Cylinder of Finite Length 5.6 Particle in a Cylindrical Box (Quantum Mechanics) 5.7 Normal Modes of an Acoustic Resonant Cavity 5.8 Acoustic Wave Guide Problems Chapter 6 • Spherical Bessel Functions and Applications 6.1 Introduction 6.2 Formulas for Spherical Bessel Functions in Terms of Elementary Functions 6.3 Eigenvalue Problem and Application of the Expansion Theorem 6.4 Expansion of Plane and Spherical Waves in Spherical Coordinates 6.5 The Emission of Spherical Waves 6.6 Scattering of Waves by a Sphere Problems Summary of Part I Section II Inhomogeneous Problems, Green’s Functions, and Integral Equations Chapter 7 • Dielectric and Magnetic Media 7.1 Introduction 7.2 Macroscopic Electrostatics in the Presence of Dielectrics 7.3 Boundary Value Problems in Dielectrics 7.3.1 Free Charge Distribution ρF Embedded in an Infinite Uniform Dielectric with a Constant Dielectric Constant ε 7.3.2 Point Charge in Front of a Semi-infinite Dielectric 7.3.3 Dielectric Sphere in a Uniform External Electric Field 7.4 Magnetostatics and the Multipole Expansion for the Vector Potential 7.5 Magnetic Media 7.6 Boundary Value Problems in Magnetic Media 7.6.1 Uniformly Magnetized Sphere, M Given 7.6.2 Magnetic Sphere in a Uniform External Magnetic Field 7.6.3 Long Straight Wire Carrying Current I Parallel to a Semi-infinite Slab of Material of Permeability μ Problems Chapter 8 • Green’s Functions: Part One 8.1 Introduction 8.2 Ordinary Differential Equations 8.3 General Theory, Various Boundary Conditions 8.4 The Bowed Stretched String 8.5 Expansion of Green’s Function in Eigenfunctions 8.6 Poisson’s Equation 8.7 Poisson’s Equation for All Space 8.8 Electrostatics with Boundary Conditions on Surfaces at Finite Distances – The Image Method 8.9 Expansion of the Green’s Function for the Interior of a Sphere in Series 8.10 The Helmholtz Equation – The Forced Drumhead 8.11 Eigenfunction Expansion of the Green’s Function for the Helmholtz Equation Problems Chapter 9 • Green’s Functions: Part Two 9.1 Introduction 9.2 The Helmholtz Equation for Infinite Regions, Radiation, and the Wave Equation; Sinusoidal Time Dependence 9.3 General Time Dependence 9.4 The Wave Equation 9.5 The Wave Equation for All Space, No Boundaries at Finite Distances 9.6 Field Due to a Point Source 9.6.1 Point Source Moving with Constant Velocity, v c 9.7 The Diffusion Equation 9.8 The Diffusion Equation for All Space, No Boundaries at Finite Distances Problems Chapter 10 • Integral Equations 10.1 Introduction 10.2 Quantum Theory of Scattering 10.3 Types of Integral Equations 10.3.1 First Kind 10.3.2 Second Kind 10.3.3 Volterra 10.3.4 Eigenvalue Problem 10.4 Integral Equations with Separable Kernels 10.5 Convolution Integral Equations 10.6 Iteration – Liouville-Neumann Series 10.7 Numerical Solution 10.8 Fredholm’s Formulas 10.9 Conditions for Validity of Fredholm’s Formulas 10.10 Hilbert-Schmidt Theory Problems Section III Complex Variable Techniques Chapter 11 • Complex Variables; Basic Theory 11.1 Introduction 11.2 Analytic Functions; The Cauchy-Riemann Equations 11.3 Power Series 11.4 Multivalued Functions; Cuts; Riemann Sheets 11.5 Contour Integrals; Cauchy’s Theorem 11.6 Cauchy’s Integral Formula 11.7 Taylor and Laurent Expansions 11.8 Analytic Continuation Problems Chapter 12 • Evaluation of Integrals 12.1 Introduction 12.2 The Residue Theorem 12.3 Rational Functions (−∞, ∞) 12.4 Exponential Factors; Jordan’s Lemma 12.5 Integrals on the Range (0, ∞) 12.6 Angular Integrals 12.7 Transforming the Contour 12.8 Partial Fraction and Product Expansions Problems Chapter 13 • Dispersion Relations 13.1 Introduction 13.2 Plemelj Formulas; Dirac’s Formula 13.3 Discontinuity Problem 13.4 Dispersion Relations; Spectral Representations 13.5 Examples 13.6 Integral Equations with Cauchy Kernels Problems Chapter 14 • Special Functions 14.1 Introduction 14.2 The Gamma Function 14.3 Asymptotic Expansions; Stirling’s Formula 14.4 The Hypergeometric Function 14.5 Legendre Functions 14.6 Bessel Functions 14.7 Asymptotic Expansions for Bessel Functions Problems Chapter 15 • Integral Transforms in the Complex Plane 15.1 Introduction 15.2 The Calculation of Green’s Functions by Fourier Transform Methods 15.2.1 The Helmholtz Equation 15.2.2 The Wave Equation 15.2.3 The Klein-Gordon Equation 15.3 One-Sided Fourier Transforms; Laplace Transforms 15.4 Linear Differential Equations with Constant Coefficients 15.5 Integral Equations of Convolution Type 15.6 Mellin Transforms 15.7 Partial Differential Equations 15.8 The Wiener-Hopf Method 15.8.1 Potential Given on Semi-Infinite Plate 15.8.2 Diffraction by a Knife Edge Problems Bibliography Index