Contents

• Introduction
• Infinite Dimensional Vector Spaces
  • Inner Product Spaces
  • Normed Linear Spaces
  • Metric Spaces
  • Complete Metric Spaces
    • Limit of a Sequence
    • Cauchy Sequence
    • Cauchy Completeness: Complete Metric Space, Banach Space, and Hilbert Space
  • Hilbert Spaces
    • Two Prototypical Examples
    • The Riesz-Fischer Theorem
    • Orthogonal Basis and Orthogonalization
    • Approximation via Subspaces: Analysis and Geometry
    • Subspace Approximation vs. the Riesz-Fischer Theorem
    • Recapitulation
    • Isomorphic Hilbert Spaces
  
  
• Fourier Theory
  • The Dirichlet Kernel
    • Basic Properties
    • Three Applications
  • The Dirac Delta Function
  • The Fourier Integral
    • Transition from Fourier Series to Fourier Integral
    • The Fourier Integral Theorem
    • The Fourier Transform as a Unitary Transformation
    • Fourier Transform via Parseval's Relation
    • Efficient Calculation: Fourier Transform via Convolution
  • Orthonormal Wave Packet Representation
    • Orthonormal Wave Packets: General Construction
    • Orthonormal Wave Packets: Definition and Properties
    • Phase Space Representation
  • Orthonormal Wavelet Representation
    • Construction and Properties
  • Multiresolution Analysis
    • Chirped Signals and the Principle of Unit-Economy
    • Irregular Signals and Variable Resolution Analysis
    • Multiresolution Analysis as Hierarchical
    • Unit-Economy via the Two Parent Wavelets
    • Multiscale Analysis as a Method of Measurement
    • Multiscale Analysis vs Multiresolution Analysis: MSA or MRA?
    • The Pyramid Algorithm
    • The Requirement of Commensurability
    • Wavelet Analysis
  
  
• Sturm-Liouville Theory
  • Three Archetypical Linear Problems
  • The Homogeneous Problem
  • Sturm-Liouville Systems
    • Sturm-Liouville Differential Equation
    • Homogeneous Boundary Conditions
    • Basic Properties of a Sturm-Liouville Eigenvalue Problem
  • Phase Analysis of a Linear Second Order O.D.E.
    • The Prüfer System
  • Qualitative Results
  • Phase Analysis of a Sturm-Liouville System
    • The Boundary Conditions
    • The Boundary Value Problem
    • The Behavior of the Phase: The Oscillation Theorem
    • Discrete Unbounded Sequence of Eigenvalues
  • Completeness of the Set of Eigenfunctions via Rayleigh's Quotient
  
  
• Green's Function Theory
  • The Adjoint of an Operator
    • Adjoint Boundary Conditions
    • Second Order Operator and the Bilinear Concomitant
  • Green's Function and Its Adjoint
    • Translation Invariant Systems
  • Pictorial Definition of a Green's Function
    • The Simple String and Poisson's Equation
    • Point Force Applied to the System
  • Properties and Utility of a Green's Function
  • Construction of the Green's Function
  • Unit Impulse Response: General Homogeneous Boundary Conditions
  • The Totally Inhomogeneous Boundary Value Problem
  • Spectral Representation
    • Spectral Resolution of the Resolvent of a Matrix
    • Spectral Resolution of the Green's Function
    • Green's Function as the Fountainhead of the Eigenvalues and Eigenvectors of a System
    • String with Free Ends: Green's Function, Spectrum, and Completeness
  • Boundary Value Problem via Green's Function: Integral Equation
    • One-dimensional Scattering Problem: Exterior Boundary Value Problem
    • One-dimensional Cavity Problem: Interior Boundary Value Problem
    • Eigenfunctions via Integral Equations
    • Types of Integral Equations
  • Singular Boundary Value Problem: Infinite Domain
    • Review: Branches, Branch Cuts, and Riemann Sheets
    • Square Integrability
    • Infinite String
    • Infinite String as the Limit of a Finite String
  • Spectral Representation of the Dirac Delta Function
    • Coalescence of Poles into a Branch Cut
    • Contour Integration Around the Branch Cut
    • Fourier Sine Theorem
  
  
• Special Function Theory
  • The Helmholtz Equation
    • Cartesian versus Polar Coordinates
    • Degenerate Eigenvalues
    • Complete Set of Commuting Operators
    • Translations and Rotations in the Euclidean Plane
    • Symmetries of the Helmholtz Equations
    • Wanted: Rotation Invariant Solutions to the Helmholtz Equation
  • Properties of Hankel and Bessel Functions
  • Applications of Hankel and Bessel Functions
    • Exterior Boundary Value Problem: Scattering
    • Finite Interior Boundary Value Problem: Cavity Vibrations
    • Infinite Interior Boundary Value Problem: Waves Propagating in a Cylindrical Pipe
  • More Properties of Hankel and Bessel Functions
  • The Method of Steepest Descent and Stationary Phase
  • Boundary Value Problems in Two Dimensions
    • Solution via Green's Function
    • Green's Function via Dimensional Reduction
    • Green's Function: 2-D Laplace vs. (Limit of) 2-D Helmholtz
  • Wave Equation for Spherically Symmetric Systems
    • Spherically Symmetric Solutions
    • Factorization Method for Solving a Partial Differential Equation: Spherical Harmonics
  • Static Solutions
    • Static Multipole Field
    • Addition Theorem for Spherical Harmonics
    • The Bessel-Legendre Connection
  
  
• Partial Differential Equations
  • Single Partial Differential Equations: Their Origin
    • Boundary Conditions of a Typical Partial Differential Equation in Two Dimensions
    • Cauchy Problem and Characteristics
    • Hyperbolic Equations
    • Riemann's Method for Integrating the Most General 2nd Order Linear Hyperbolic Equation
  • System of Partial Differential Equations: How to Solve Maxwell's Equations Using Linear Algebra
    • Maxwell Wave Equation
    • The Overdetermined System $A\vec u = \vec b$
    • Maxwell Wave Equation (continued)
    • Cylindrical Coordinates
    • Spherical Coordinates
    • Electromagnetic Fields in a Spherical Coordinate System
  
  
• Index