Some mathematical methods of physics

Title page
Preface
References
PART ONE. SYSTEMS WITH A FINITE NUMBER OF DEGREES OF FREEDOM
Chapter 1 Formulation of the Problem and Development of Notation
1.1 Introduction
1.2 Standardization of Notation
1.3 Matrices
1.4 Elementary Arithmetic Operations with Matrices
1.5 The Row-Column Rule
1.6 Wamings
1.7 Some Properties of Determinants
1.8 Inverses
1.9 Linear Independence
Chapter 2 Solution for Diagonalizable Matrices
2.1 Solution by Taylor Series
2.2 Eigenvalues and Eigencolumns
2.3 Superposition
2.4 Completeness
2.5 Diagonalization of Nondegenerate Matrices
2.6 Outline of Computation Procedure with Examples
2.7 Change of Variable
2.8 The Steady-state Solution
2.9 The Inhomogeneous Problem
Chapter 3 The Evaluation of a Fonction of a Matrix for an Arbitrary Matrix
3.1 Introduction
3.2 The Cauchy-integral Formula
3.3 Application to Matrices
3.4 Evaluation of f(A) with Illustrations
3.5 The Inversion Formula
3.6 Laplace Transforms
3.7 Inhomogeneous Equations
3.8 The Convolution Theorem
Chapter 4 Vector Spaces and Linear Operators
4.1 Introduction
4.2 Base Vectors and Basis
4.3 Change of Basis
4.4 Linear Operators
4.5 The Representation of Linear Operators by Matrices
4.6 The Operator in the Dual Space
4.7 Effect of Change of Basis on the Representation of an Operator
4.8 The Spectral Representation of an Operator
4.9 The Formation of a Basis by Eigenvectors of a Linear Operator
4.10 The Diagonalization of Normal Matrices
Chapter 5 The Dirac Notation
5.1 Introduction
5.2 The Change of Basis
5.3 Linear Operators in the Dirac Notation
5.4 Eigenvectors and Eigenvalues
5.5 The Spectral Representation of an Operator
5.6 Theorems on Hermitian Operators
Chapter 6 Periodic Structures
6.1 Motivation
6.2 The RC Line
6.3 Diagonalizing M
6.4 The Loaded String
6.5 Difference Operators
PART TWO. SYSTEMS WITH AN INFINITE NUMBER OF DEGREES OF FREEDOM
Chapter 7 The Transition to Continuous Systems
7.1 Introduction
7.2 The RC Line-Change of Notation
7.3 The RC Line-Transition to the Continuous Case
7.4 Solution of the Discrete Problem
7.5 Solution in the Limit (Continuous Problem)
7.6 The Fourier Transform
Chapter 8 Operators in Continuons Systems
8.1 Introduction
8.2 Operators on Functions
8.3 The Dirac Function
8.4 Coordinate Transformations
8.5 Adjoints
8.6 Orthogonality of Eigenfunctions
8.7 Functions of Operators
8.8 Three-dimensional Continuous Systems
8.9 DifferentiaI Operators
Chapter 9 The Laplacian (∇²) in One Dimension
9.1 Introduction
9.2 The Infinite Domain, -∞ < x < +∞
9.3 The Semi-infinite Domain, 0 < x < +∞
9.4 The Finite Domain, 0 < x < L
9.5 The Circular Domain
9.6 The Method of Images
Chapter 10 The Laplacian (∇²) in Two Dimensions
10.1 Introduction
10.2 Conduction of Heat in an Infinite Insulated Plate. Cartesian Coordinates
10.3 The Vibrating Rectangular Membrane
10.4 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates
10.5 Theorems on Cylindrical Functions (of Integral Order n)
10.6 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates (Concluded)
10.7 The Circular Membrane
10.8 The Vibrating Circular Ring and Circular Sector
Chapter 11 The Laplacian (∇²) in Three Dimensions
11.1 Introduction
11.2 The Wave Equation in Three Dimensions
11.3 The Eigenvalues of L² and L_z
11.4 The Simultaneous Eigenfunctions of L² and L_z
11.5 Solution of ∇²ψ = 0
11.6 Solution of (∇² + k²)ψ = 0
11.7 Recurrence Relations for the Spherical Harmonies
11.8 Some Expansion Theorems
11.9 Solution of the Wave Equation
11.10 Heat Conduction in an Infinite Solid
Chapter 12 Green's Functions
12.1 Definition
12.2 The Necessary and Sufficient Condition for a Green's Function
12.3 The Operator -α²d²/dx² + 1 in an Infinite Domain
12.4 The Operator -α²d²/dx² + 1 in a Finite Domain
12.5 The Operator -α²∇² + 1 in Spherical Coordinates
Chapter 13 Radiation and Scattering Problems
13.1 The Outgoing Wave Condition
13.2 The Green's Function Solution
13.3 The Multipole Expansion
13.4 The Radiation Far from the Source
13.5 Radiation from an Infinitely Long Cylinder
13.6 The Scattering Problem
13.7 The Scattering Cross Section
13.8 The Method of Partial Waves
13.9 The Born Approximation
13.10 Gratings
PART THREE. APPROXIMATE METHODS
Chapter 14 Perturbation of Eigenvalues
14.1 Introduction
14.2 Formulation of the Problem
14.3 A Simple Solution
14.4 Nondegenerate Eigenvalues
14.5 Change of Notation and an Extension
14.6 An Application - The Vibrating String
14.7 Degenerate Eigenvalues
Chapter 15 Variational Estimates
15.1 Introduction
15.2 The Rayleigh Variational Principle
15.3 A Lower Bound
15.4 The Ritz Method
15.5 Higher Eigenvalues by the Ritz Method
15.6 Example of the Ritz Method
Chapter 16 Iteration Procedures
16.1 Introduction
16.2 Eigenvalue Problems
16.3 Inverses by Iteration
Chapter 17 Construction of Eigenvalue Problems
17.1 Introduction
17.2 The Method
17.3 Application to the Scattering Problem
Chapter 18 Numerical Procedures
18.1 Introduction
18.2 Simplification of the Model
18.3 Difference Equations from the Variational Principle
Appendix lA Determinants
Appendix 1B Convergence of Matrix Power Series
Appendix 1C Remarks on Theory of FunctioDS of Complex Variables
1C.l Analytic Functions
1C.2 The Cauchy Integral Theorem and Corollary
1C.3 Singularities
1C.4 Cauchy's Integral Formula
1C.5 The Theorem of Residues
Appendix 2A Evaluation of Integrais of the Form \int_{-∞}^{+∞}'F(x)e^{itx}dx)
Appendix 2B Fourier Transforms, Integrals, and Series
2B.l Introduction
2B.2 Transforms
2B.3 Infinite One-dimensional Transforms
2B.4 Infinite Multidimensional Transforms - Cartesian Coordinates
2B.5 Finite One-dimensional Transforms
2B.6 The Fourier-Bessel Integral
2B.7 The Fourier-Bessel Expansion
Appendix 2C The Cylindrical Functions
2C.l Introduction
2C.2 The Integral Representation of J_n(ρ)
2C.3 The Integral Representations of the General Cylindrical Functions
2C.4 The Integral Representation of the Bessel Function J_ν
2C.5 The Hankel Functions
2C.6 Series Expansions at the Origin
2C.7 The Asymptotic Expansions
2C.8 The Asymptotic Series of Debye
2C.9 The Addition Theorems for Bessel Functions
Index