Mathematical Methods for Physics

✍ Scribed by H.W. Wyld, Gary Powell

• Publisher: CRC
• Year: 2021
• Tongue: English
• Leaves: 476
• Category: Library

✦ Synopsis

From classical mechanics and classical electrodynamics to modern quantum mechanics many physical phenomena are formulated in terms of similar partial differential equations while boundary conditions determine the specifics of the problem. This 45th anniversary edition of the advanced book classic Mathematical Methods for Physics demonstrates how many physics problems resolve into similar inhomogeneous partial differential equations and the mathematical techniques for solving them. The text has three parts: Part I establishes solving the homogenous Laplace and Helmholtz equations in the three main coordinate systems, rectilinear, cylindrical, and spherical and develops the solution space for series solutions to the Sturm-Liouville equation, indicial relations, and the expansion of orthogonal functions including spherical harmonics and Fourier series, Bessel, and Spherical Bessel functions. Many examples with figures are provided including electrostatics, wave guides and resonant cavities, vibrations of membranes, heat flow, potential flow in fluids, and plane and spherical waves. In Part II the inhomogeneous equations are addressed where source terms are included for Poisson's equation, the wave equation, and the diffusion equation. Coverage includes many examples from averaging approaches for electrostatics and magnetostatics, from Green function solutions for time independent and time dependent problems, and from integral equation methods. In Part III complex variable techniques are presented for solving integral equations involving Cauchy Residue theory, contour methods, analytic continuation, and transforming the contour; for addressing dispersion relations; for revisiting special functions in the complex plane; and for transforms in the complex plane including Green’s functions and Laplace transforms.

Key Features:

· Mathematical Methods for Physics creates a strong, solid anchor of learning and is useful for reference.

· Lecture note style suitable for advanced undergraduate and graduate students to learn many techniques for solving partial differential equations with boundary conditions

· Many examples across various subjects of physics in classical mechanics, classical electrodynamics, and quantum mechanics

· Updated typesetting and layout for improved clarity

This book, in lecture note style with updated layout and typesetting, is suitable for advanced undergraduate, graduate students, and as a reference for researchers. It has been edited and carefully updated by Gary Powell.

✦ Table of Contents

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
9.6.2 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