Mathematical Physics _ Applications and Problems

Preface
Contents
About the Author
1 Warming Up: Functions of a Real Variable
	1.1 Sketching Functions
		1.1.1 Features of Interest in a Function
		1.1.2 Powers of x
		1.1.3 A Family of Ovals
		1.1.4 A Family of Spirals
	1.2 Maps of the Unit Interval
2 Gaussian Integrals, Stirling's Formula, and Some Integrals
	2.1 Gaussian Integrals
		2.1.1 The Basic Gaussian Integral
		2.1.2 A Couple of Higher Dimensional Examples
	2.2 Stirling's Formula
	2.3 The Dirichlet Integral and Its Descendants
	2.4 Solutions
3 Some More Functions
	3.1 Functions Represented by Integrals
		3.1.1 Differentiation Under the Integral Sign
		3.1.2 The Error Function
		3.1.3 Fresnel Integrals
		3.1.4 The Gamma Function
		3.1.5 Connection to Gaussian Integrals
	3.2 Interchange of the Order of Integration
	3.3 Solutions
4 Generalized Functions
	4.1 The Step Function
	4.2 The Dirac Delta Function
		4.2.1 Defining Relations
		4.2.2 Sequences of Functions Tending to the δ-Function
		4.2.3 Relation Between δ(x) and θ(x)
		4.2.4 Fourier Representation of the δ-Function
		4.2.5 Properties of the δ-Function
		4.2.6 The Occurrence of the δ-Function in Physical Problems
		4.2.7 The δ-Function in Polar Coordinates
	4.3 Solutions
5 Vectors and Tensors
	5.1 Cartesian Tensors
		5.1.1 What Are Scalars and Vectors?
		5.1.2 Rotations and the Index Notation
		5.1.3 Isotropic Tensors
		5.1.4 Dot and Cross Products in Three Dimensions
		5.1.5 The Gram Determinant
		5.1.6 Levi-Civita Symbol in d Dimensions
	5.2 Rotations in Three Dimensions
		5.2.1 Proper and Improper Rotations
		5.2.2 Scalars and Pseudoscalars; Polar and Axial Vectors
		5.2.3 Transformation Properties of Physical Quantities
	5.3 Invariant Decomposition of a 2nd Rank Tensor
		5.3.1 Spherical or Irreducible Tensors
		5.3.2 Stress, Strain, and Stiffness Tensors
		5.3.3 Moment of Inertia
		5.3.4 The Euler Top
		5.3.5 Multipole Expansion; Quadrupole Moment
		5.3.6 The Octupole Moment
	5.4 Solutions
6 Vector Calculus
	6.1 Orthogonal Curvilinear Coordinates
		6.1.1 Cylindrical and Spherical Polar Coordinates
		6.1.2 Elliptic and Parabolic Coordinates
		6.1.3 Polar Coordinates in d Dimensions
	6.2 Scalar and Vector Fields and Their Derivatives
		6.2.1 The Gradient of a Scalar Field
		6.2.2 The Flux and Divergence of a Vector Field
		6.2.3 The Circulation and Curl of a Vector Field
		6.2.4 Some Physical Aspects of the Curl of a Vector Field
		6.2.5 Any Vector Field is the Sum of a Curl and a Gradient
		6.2.6 The Laplacian Operator
		6.2.7 Why Do div, curl, and del-Squared Occur so Frequently?
		6.2.8 The Standard Identities of Vector Calculus
	6.3 Solutions
7 A Bit of Fluid Dynamics
	7.1 Equation of Motion of a Fluid Element
		7.1.1 Hydrodynamic Variables
		7.1.2 Equation of Motion
	7.2 Flow When Viscosity Is Neglected
		7.2.1 Euler's Equation
		7.2.2 Barotropic Flow
		7.2.3 Bernoulli's Principle in Steady Flow
		7.2.4 Irrotational Flow and the Velocity Potential
	7.3 Vorticity
		7.3.1 Vortex Lines
		7.3.2 Equations in Terms of v Alone
	7.4 Flow of a Viscous Fluid
		7.4.1 The Viscous Force in a Fluid
		7.4.2 The Navier–Stokes Equation
	7.5 Solutions
8 Some More Vector Calculus
	8.1 Integral Theorems of Vector Calculus
		8.1.1 The Fundamental Theorem of Calculus
		8.1.2 Stokes' Theorem
		8.1.3 Green's Theorem
		8.1.4 A Topological Restriction; ``Exact'' Versus ``Closed''
		8.1.5 Gauss's Theorem
		8.1.6 Green's Identities and Reciprocity Relation
		8.1.7 Comment on the Generalized Stokes' Theorem
	8.2 Harmonic Functions
		8.2.1 Mean Value Property
		8.2.2 Harmonic Functions Have No Absolute Maxima or Minima
		8.2.3 What Is the Significance of the Laplacian?
	8.3 Singularities of Planar Vector Fields
		8.3.1 Critical Points and the Poincaré Index
		8.3.2 Degenerate Critical Points and Unfolding Singularities
		8.3.3 Singularities of Three-Vector Fields
	8.4 Solutions
9 A Bit of Electromagnetism and Special Relativity
	9.1 Classical Electromagnetism
		9.1.1 Maxwell's Field Equations
		9.1.2 The Scalar and Vector Potentials
		9.1.3 Gauge Invariance and Choice of Gauge
		9.1.4 The Coulomb Gauge
		9.1.5 Electrostatics
		9.1.6 Magnetostatics
		9.1.7 The Lorenz Gauge
	9.2 Special Relativity
		9.2.1 The Principle and the Postulate of Relativity
		9.2.2 Boost Formulas
		9.2.3 Collinear Boosts: Velocity Addition Rule
		9.2.4 Rapidity
		9.2.5 Lorentz Scalars and Four-Vectors
		9.2.6 Matrices Representing Lorentz Transformations
	9.3 Relativistic Invariance of Electromagnetism
		9.3.1 Covariant Form of the Field Equations
		9.3.2 The Electromagnetic Field Tensor
		9.3.3 Transformation Properties of E and B
		9.3.4 Lorentz Invariants of the Electromagnetic Field
		9.3.5 Energy Density and the Poynting Vector
	9.4 Solutions
10 Linear Vector Spaces
	10.1 Definitions and Basic Properties
		10.1.1 Definition of a Linear Vector Space
		10.1.2 The Dual of a Linear Space
		10.1.3 The Inner Product of Two Vectors
		10.1.4 Basis Sets and Dimensionality
	10.2 Orthonormal Basis Sets
		10.2.1 Gram–Schmidt Orthonormalization
		10.2.2 Expansion of an Arbitrary Vector
		10.2.3 Basis Independence of the Inner Product
	10.3 Some Important Inequalities
		10.3.1 The Cauchy–Schwarz Inequality
		10.3.2 The Triangle Inequality
		10.3.3 The Gram Determinant Inequality
	10.4 Solutions
11 A Look at Matrices
	11.1 Pauli Matrices
		11.1.1 Expansion of a (2times2) Matrix
		11.1.2 Basic Properties of the Pauli Matrices
	11.2 The Exponential of a Matrix
		11.2.1 Occurrence and Definition
		11.2.2 The Exponential of an Arbitrary (2times2) Matrix
	11.3 Rotation Matrices in Three Dimensions
		11.3.1 Generators of Infinitesimal Rotations and Their Algebra
		11.3.2 The General Rotation Matrix
		11.3.3 The Finite Rotation Formula for a Vector
	11.4 The Eigenvalue Spectrum of a Matrix
		11.4.1 The Characteristic Equation
		11.4.2 Gershgorin's Circle Theorem
		11.4.3 The Cayley–Hamilton Theorem
		11.4.4 The Resolvent of a Matrix
	11.5 A Generalization of the Gaussian Integral
	11.6 Inner Product in the Linear Space of Matrices
	11.7 Solutions
12 More About Matrices
	12.1 Matrices as Operators in a Linear Space
		12.1.1 Representation of Operators
		12.1.2 Projection Operators
	12.2 Hermitian, Unitary, and Positive Definite Matrices
		12.2.1 Definitions and Eigenvalues
		12.2.2 The Eigenvalues of a Rotation Matrix in d Dimensions
		12.2.3 The General Form of a (2times2) Unitary Matrix
	12.3 Diagonalization of a Matrix and all That
		12.3.1 Eigenvectors, Nullspace, and Nullity
		12.3.2 The Rank of a Matrix and the Rank-Nullity Theorem
		12.3.3 Degenerate Eigenvalues and Defective Matrices
		12.3.4 When Can a Matrix Be Diagonalized?
		12.3.5 The Minimal Polynomial of a Matrix
		12.3.6 Simple Illustrative Examples
		12.3.7 Jordan Normal Form
		12.3.8 Other Matrix Decompositions
		12.3.9 Circulant Matrices
		12.3.10 A Simple Illustration: A 3-state Random Walk
	12.4 Commutators of Matrices
		12.4.1 Mutually Commuting Matrices in Quantum Mechanics
		12.4.2 The Lie Algebra of (n timesn) Matrices
	12.5 Spectral Representation of a Matrix
		12.5.1 Right and Left Eigenvectors of a Matrix
		12.5.2 An Illustration
	12.6 Solutions
13 Infinite-Dimensional Vector Spaces
	13.1 The Space ell2 of Square-Summable Sequences
	13.2 The Space mathcalL2 of Square-Integrable Functions
		13.2.1 Definition of mathcalL2
		13.2.2 Continuous Basis
		13.2.3 Weight Functions: A Generalization of mathcalL2
		13.2.4 mathcalL2(-infty,infty) Functions and Fourier Transforms
		13.2.5 The Wave Function of a Particle
	13.3 Hilbert Space and Subspaces
		13.3.1 Hilbert Space
		13.3.2 Linear Manifolds and Subspaces
	13.4 Solutions
14 Linear Operators on a Vector Space
	14.1 Some Basic Notions
		14.1.1 Domain, Range, and Inverse
		14.1.2 Linear Operators, Norm, and Bounded Operators
	14.2 The Adjoint of an Operator
		14.2.1 Densely Defined Operators
		14.2.2 Definition of the Adjoint Operator
		14.2.3 Symmetric, Hermitian, and Self-adjoint Operators
	14.3 The Derivative Operator in mathcalL2
		14.3.1 The Momentum Operator of a Quantum Particle
		14.3.2 The Adjoint of the Derivative Operator in mathcalL2(-infty,infty)
		14.3.3 When Is -i(d/dx) Self-adjoint in mathcalL2[a,b]?
		14.3.4 Self-adjoint Extensions of Operators
		14.3.5 Deficiency Indices
		14.3.6 The Radial Momentum Operator in d 2 Dimensions
	14.4 Nonsymmetric Operators
		14.4.1 The Operators xpmip
		14.4.2 Oscillator Ladder Operators and Coherent States
		14.4.3 Eigenvalues and Non-normalizable Eigenstates of  x and p
		14.4.4 Matrix Representations for Unbounded Operators
	14.5 Solutions
15 Operator Algebras and Identities
	15.1 Operator Algebras
		15.1.1 The Heisenberg Algebra
		15.1.2 Some Other Basic Operator Algebras
	15.2 Useful Operator Identities
		15.2.1 Perturbation Series for an Inverse Operator
		15.2.2 Hadamard's Lemma
		15.2.3 Weyl Form of the Canonical Commutation Relation
		15.2.4 The Zassenhaus Formula
		15.2.5 The Baker–Campbell–Hausdorff Formula
	15.3 Some Physical Applications
		15.3.1 Angular Momentum Operators
		15.3.2 Representation of Rotations by SU(2) Matrices
		15.3.3 Connection Between the Groups SO(3) and SU(2)
		15.3.4 The Parameter Space of SU(2)
		15.3.5 The Parameter Space of SO(3)
		15.3.6 The Parameter Space of SO(2)
	15.4 Some More Physical Applications
		15.4.1 The Displacement Operator and Coherent States
		15.4.2 The Squeezing Operator and the Squeezed Vacuum
		15.4.3 Values of z That Produce Squeezing in x or p
		15.4.4 The Squeezing Operator and the Group SU(1,1)
		15.4.5 SU(1,1) Generators in Terms of Pauli Matrices
	15.5 Solutions
16 Orthogonal Polynomials
	16.1 General Formalism
		16.1.1 Introduction
		16.1.2 Orthogonality and Completeness
		16.1.3 Expansion and Inversion Formulas
		16.1.4 Uniqueness and Explicit Representation
		16.1.5 Recursion Relation
	16.2 The Classical Orthogonal Polynomials
		16.2.1 Polynomials of the Hypergeometric Type
		16.2.2 The Hypergeometric Differential Equation
		16.2.3 Rodrigues Formula and Generating Function
		16.2.4 Class I.Hermite Polynomials
		16.2.5 Linear Harmonic Oscillator Eigenfunctions
		16.2.6 Oscillator Coherent State Wave Functions
		16.2.7 Class II.Generalized Laguerre Polynomials
		16.2.8 Class III.Jacobi Polynomials
	16.3 Gegenbauer Polynomials
		16.3.1 Ultraspherical Harmonics
		16.3.2 Chebyshev Polynomials of the 1st Kind
		16.3.3 Chebyshev Polynomials of the Second Kind
	16.4 Legendre Polynomials
		16.4.1 Basic Properties
		16.4.2 Pn(x) by Gram–Schmidt Orthonormalization
		16.4.3 Expansion in Legendre Polynomials
		16.4.4 Expansion of xn in Legendre Polynomials
		16.4.5 Legendre Function of the Second Kind
		16.4.6 Associated Legendre Functions
		16.4.7 Spherical Harmonics
		16.4.8 Expansion of the Coulomb Kernel
	16.5 Solutions
17 Fourier Series
	17.1 Series Expansion of Periodic Functions
		17.1.1 Dirichlet Conditions
		17.1.2 Orthonormal Basis
		17.1.3 Fourier Series Expansion and Inversion Formula
		17.1.4 Parseval's Formula for Fourier Series
		17.1.5 Simplified Formulas When (a,b) = (-π,π)
	17.2 Asymptotic Behavior and Convergence
		17.2.1 Uniform Convergence of Fourier Series
		17.2.2 Large-n Behavior of Fourier Coefficients
		17.2.3 Periodic Array of δ-Functions: The Dirac Comb
	17.3 Summation of Series
		17.3.1 Some Examples
		17.3.2 The Riemann Zeta Function ζ(2k)
		17.3.3 Fourier Series Expansions of cosαx and sinαx
	17.4 Solutions
18 Fourier Integrals
	18.1 Expansion of Nonperiodic Functions
		18.1.1 Fourier Transform and Inverse Fourier Transform
		18.1.2 Parseval's Formula for Fourier Transforms
		18.1.3 Fourier Transform of the δ-Function
		18.1.4 Examples of Fourier Transforms
		18.1.5 Relative ``Spreads'' of a Fourier Transform Pair
		18.1.6 The Convolution Theorem
		18.1.7 Generalized Parseval Formula
	18.2 The Fourier Transform Operator in mathcalL2
		18.2.1 Iterates of the Fourier Transform Operator
		18.2.2 Eigenvalues and Eigenfunctions of mathcalF
		18.2.3 The Adjoint of an Integral Operator
		18.2.4 Unitarity of the Fourier Transformation
	18.3 Generalization to Several Dimensions
	18.4 The Poisson Summation Formula
		18.4.1 Derivation of the Formula
		18.4.2 Some Illustrative Examples
		18.4.3 Generalization to Higher Dimensions
	18.5 Solutions
19 Discrete Probability Distributions
	19.1 Some Elementary Distributions
		19.1.1 Mean and Variance
		19.1.2 Bernoulli Trials and the Binomial Distribution
		19.1.3 Number Fluctuations in a Classical Ideal Gas
		19.1.4 The Geometric Distribution
		19.1.5 Photon Number Distribution in Blackbody Radiation
	19.2 The Poisson Distribution
		19.2.1 From the Binomial to the Poisson Distribution
		19.2.2 Photon Number Distribution in Coherent Radiation
		19.2.3 Photon Number Distribution in the Squeezed Vacuum State
		19.2.4 The Sum of Poisson-Distributed Random Variables
		19.2.5 The Difference of Two Poisson-Distributed Random Variables
	19.3 The Negative Binomial Distribution
	19.4 The Simple Random Walk
		19.4.1 Random Walk on a Linear Lattice
		19.4.2 Some Generalizations of the Simple Random Walk
	19.5 Solutions
20 Continuous Probability Distributions
	20.1 Continuous Random Variables
		20.1.1 Probability Density and Cumulative Distribution
		20.1.2 The Moment-Generating Function
		20.1.3 The Cumulant-Generating Function
		20.1.4 Application to the Discrete Distributions
		20.1.5 The Characteristic Function
		20.1.6 The Additivity of Cumulants
	20.2 The Gaussian Distribution
		20.2.1 The Normal Density and Distribution
		20.2.2 Moments and Cumulants of a Gaussian Distribution
		20.2.3 Simple Functions of a Gaussian Random Variable
		20.2.4 Mean Collision Rate in a Dilute Gas
	20.3 The Gaussian as a Limit Law
		20.3.1 Linear Combinations of Gaussian Random Variables
		20.3.2 The Central Limit Theorem
		20.3.3 An Explicit Illustration of the Central Limit Theorem
	20.4 Random Flights
		20.4.1 From Random Flights to Diffusion
		20.4.2 The Probability Density for Short Random Flights
	20.5 The Family of Stable Distributions
		20.5.1 What Is a Stable Distribution?
		20.5.2 The Characteristic Function of Stable Distributions
		20.5.3 Three Important Cases: Gaussian, Cauchy, and Lévy
		20.5.4 Some Connections Between the Three Cases
	20.6 Infinitely Divisible Distributions
		20.6.1 Divisibility of a Random Variable
		20.6.2 Infinite Divisibility Does Not Imply Stability
	20.7 Solutions
21 Stochastic Processes
	21.1 Multiple-Time Joint Probabilities
	21.2 Discrete Markov Processes
		21.2.1 The Two-Time Conditional Probability
		21.2.2 The Master Equation
		21.2.3 Formal Solution of the Master Equation
		21.2.4 The Stationary Distribution
		21.2.5 Detailed Balance
	21.3 The Autocorrelation Function
	21.4 The Dichotomous Markov Process
		21.4.1 The Stationary Distribution
		21.4.2 Solution of the Master Equation
	21.5 Birth-and-Death Processes
		21.5.1 The Poisson Pulse Process and Radioactive Decay
		21.5.2 Biased Random Walk on a Linear Lattice
		21.5.3 Connection with the Skellam Distribution
		21.5.4 Asymptotic Behavior of the Probability
	21.6 Continuous Markov Processes
		21.6.1 Master Equation for the Conditional density
		21.6.2 The Fokker–Planck Equation
		21.6.3 The Autocorrelation Function for a Continuous Process
	21.7 The Stationary Gaussian Markov Process
		21.7.1 The Ornstein–Uhlenbeck Process
		21.7.2 The Ornstein–Uhlenbeck Distribution
		21.7.3 Velocity Distribution in a Classical Ideal Gas
		21.7.4 Solution for an Arbitrary Initial Velocity Distribution
		21.7.5 Diffusion of a Harmonically Bound Particle
	21.8 Solutions
22 Analytic Functions of a Complex Variable
	22.1 Some Preliminaries
		22.1.1 Complex Numbers
		22.1.2 Equations to Curves in the Plane in Terms of z
	22.2 The Riemann Sphere
		22.2.1 Stereographic Projection
		22.2.2 Maps of Circles on the Riemann Sphere
		22.2.3 A Metric on the Extended Complex Plane
	22.3 Analytic Functions of z
		22.3.1 The Cauchy–Riemann Conditions
		22.3.2 The Real and Imaginary Parts of an Analytic Function
	22.4 The Derivative of an Analytic Function
	22.5 Power Series as Analytic Functions
		22.5.1 Radius and Circle of Convergence
		22.5.2 An Instructive Example
		22.5.3 Behavior on the Circle of Convergence
		22.5.4 Lacunary Series
	22.6 Entire Functions
		22.6.1 Representation of Entire Functions
		22.6.2 The Order of an Entire Function
	22.7 Solutions
23 More on Analytic Functions
	23.1 Cauchy's Integral Theorem
	23.2 Singularities
		23.2.1 Simple Pole; Residue at a Pole
		23.2.2 Multiple pole
		23.2.3 Essential Singularity
		23.2.4 Laurent Series
		23.2.5 Singularity at Infinity
		23.2.6 Accumulation Points
		23.2.7 Meromorphic Functions
	23.3 Contour Integration
		23.3.1 A Basic Formula
		23.3.2 Cauchy's Residue Theorem
		23.3.3 The Dirichlet Integral; Cauchy Principal Value
		23.3.4 The ``iε-Prescription'' for a Singular Integral
		23.3.5 Residue at Infinity
	23.4 Summation of Series Using Contour Integration
	23.5 Linear Recursion Relations with Constant Coefficients
		23.5.1 The Generating Function
		23.5.2 Hemachandra-Fibonacci Numbers
		23.5.3 Catalan Numbers
		23.5.4 Connection with Wigner's Semicircular Distribution
	23.6 Mittag-Leffler Expansion of Meromorphic Functions
	23.7 Solutions
24 Linear Response and Analyticity
	24.1 The Dynamic Susceptibility
		24.1.1 Linear, Causal, Retarded Response
		24.1.2 Frequency-Dependent Response
		24.1.3 Symmetry Properties of the Dynamic Susceptibility
	24.2 Dispersion Relations
		24.2.1 Derivation of the Relations
		24.2.2 Complex Admittance of an LCR Circuit
		24.2.3 Subtracted Dispersion Relations
		24.2.4 Hilbert Transform Pairs
		24.2.5 Discrete and Continuous Relaxation Spectra
	24.3 Solutions
25 Analytic Continuation and the Gamma Function
	25.1 Analytic Continuation
		25.1.1 What Is Analytic Continuation?
		25.1.2 The Permanence of Functional Relations
	25.2 The Gamma Function for Complex Argument
		25.2.1 Stripwise Analytic Continuation of Γ(z)
		25.2.2 Mittag-Leffler Expansion of Γ(z)
		25.2.3 Logarithmic Derivative of Γ(z)
		25.2.4 Infinite Product Representation of Γ(z)
		25.2.5 Connection with the Riemann Zeta Function
		25.2.6 The Beta Function
		25.2.7 Reflection Formula for Γ(z)
		25.2.8 Legendre's Doubling Formula
	25.3 Solutions
26 Multivalued Functions and Integral Representations
	26.1 Multivalued Functions
		26.1.1 Branch Points and Branch Cuts
		26.1.2 Types of Branch Points
		26.1.3 Contour Integrals in the Presence of Branch Points
	26.2 Contour Integral Representations
		26.2.1 The Gamma Function
		26.2.2 The Beta Function
		26.2.3 The Riemann Zeta Function
		26.2.4 Connection with Bernoulli Numbers
		26.2.5 The Legendre Functions Pν(z) and Qν(z)
	26.3 Singularities of Functions Defined by Integrals
		26.3.1 End Point and Pinch Singularities
		26.3.2 Singularities of the Legendre Functions
	26.4 Solutions
27 Möbius Transformations
	27.1 Conformal Mapping
	27.2 Möbius (or Fractional Linear) Transformations
		27.2.1 Definition
		27.2.2 Fixed Points
		27.2.3 The Cross-Ratio and Its Invariance
	27.3 Normal Form of a Möbius Transformation
		27.3.1 Normal Forms in Different Cases
		27.3.2 Iterates of a Möbius Transformation
		27.3.3 Classification of Möbius Transformations
		27.3.4 The Isometric Circle
	27.4 Group Properties
		27.4.1 The Möbius Group
		27.4.2 The Möbius Group Over the Reals
		27.4.3 The Invariance Group of the Unit Circle
		27.4.4 The Group of Cross-Ratios
	27.5 Solutions
28 Laplace Transforms
	28.1 Definition and Properties
		28.1.1 Definition of the Laplace Transform
		28.1.2 Transforms of Some Simple Functions
		28.1.3 The Convolution Theorem
		28.1.4 Laplace Transforms of Derivatives
	28.2 The Inverse Laplace Transform
		28.2.1 The Mellin Formula
		28.2.2 LCR Circuit Under a Sinusoidal Applied Voltage
	28.3 Bessel Functions and Laplace Transforms
		28.3.1 Differential Equations and Power Series Representations
		28.3.2 Generating Functions and Integral Representations
		28.3.3 Spherical Bessel Functions
		28.3.4 Laplace Transforms of Bessel Functions
	28.4 Laplace Transforms and Random Walks
		28.4.1 Random Walk in d Dimensions
		28.4.2 The First-Passage-Time Distribution
	28.5 Solutions
29 Green Function for the Laplacian Operator
	29.1 The Partial Differential Equations of Physics
	29.2 Green Functions
		29.2.1 Green Function for an Ordinary Differential Operator
		29.2.2 An Illustrative Example
	29.3 The Fundamental Green Function for 2
		29.3.1 Poisson's Equation in Three Dimensions
		29.3.2 The Solution for G(3)(r, r')
		29.3.3 Solution of Poisson's Equation
		29.3.4 Connection with the Coulomb Potential
	29.4 The Coulomb Potential in d > 3 Dimensions
		29.4.1 Simplification of the Fundamental Green Function
		29.4.2 Power Counting and a Divergence Problem
		29.4.3 Dimensional Regularization
		29.4.4 A Direct Derivation
	29.5 The Coulomb Potential in d=2 Dimensions
		29.5.1 Dimensional Regularization
		29.5.2 Direct Derivation
		29.5.3 An Alternative Regularization
	29.6 Solutions
30 The Diffusion Equation
	30.1 The Fundamental Gaussian Solution
		30.1.1 Fick's Laws of Diffusion
		30.1.2 Further Remarks on Linear Response
		30.1.3 The Fundamental Solution in d Dimensions
		30.1.4 Solution for an Arbitrary Initial Distribution
		30.1.5 Moments of the Distance Travelled in Time t
	30.2 Diffusion in One Dimension
		30.2.1 Continuum Limit of a Biased Random Walk
		30.2.2 Free Diffusion on an Infinite Line
		30.2.3 Absorbing and Reflecting Boundary Conditions
		30.2.4 Finite Boundaries: Solution by the Method of Images
		30.2.5 Finite Boundaries: Solution by Separation  of Variables
		30.2.6 Survival Probability and Escape-Time Distribution
		30.2.7 Equivalence of the Solutions
	30.3 Diffusion with Drift: Sedimentation
		30.3.1 The Smoluchowski Equation
		30.3.2 Equilibrium Barometric Distribution
		30.3.3 The Time-Dependent Solution
	30.4 The Schrödinger Equation for a Free Particle
		30.4.1 Connection with the Free-Particle Propagator
		30.4.2 Spreading of a Quantum Mechanical Wave Packet
		30.4.3 The Wave Packet in Momentum Space
	30.5 Solutions
31 The Wave Equation
	31.1 Causal Green Function of the Wave Operator
		31.1.1 Formal Solution as a Fourier Transform
		31.1.2 Simplification of the Formal Solution
	31.2 Explicit Solutions for d =1, 2 and 3
		31.2.1 The Green Function in (1+1) Dimensions
		31.2.2 The Green Function in (2+1) Dimensions
		31.2.3 The Green Function in (3+1) Dimensions
		31.2.4 Retarded Solution of the Wave Equation
	31.3 Remarks on Propagation in Dimensions d > 3
	31.4 Solutions
32 Integral Equations
	32.1 Fredholm Integral Equations
		32.1.1 Equation of the First Kind
		32.1.2 Equation of the Second Kind
		32.1.3 Degenerate Kernels
		32.1.4 The Eigenvalues of a Degenerate Kernel
		32.1.5 Iterative Solution: Neumann Series
	32.2 Nonrelativistic Potential Scattering
		32.2.1 The Scattering Amplitude
		32.2.2 Integral Equation for Scattering
		32.2.3 Green Function for the Helmholtz Operator
		32.2.4 Formula for the Scattering Amplitude
		32.2.5 The Born Approximation
		32.2.6 Yukawa and Coulomb Potentials; Rutherford's Formula
	32.3 Partial Wave Analysis
		32.3.1 The Physical Idea Behind Partial Wave Analysis
		32.3.2 Expansion of a Plane Wave in Spherical Harmonics
		32.3.3 Partial Wave Scattering Amplitude and Phase Shift
		32.3.4 The Optical Theorem
	32.4 The Fredholm Solution
		32.4.1 The Fredholm Formulas
		32.4.2 Remark on the Application to the Scattering Problem
	32.5 Volterra Integral Equations
	32.6 Solutions
Appendix  Bibliography and Further Reading
		
Index