Reading Assignments
(Sections and chapters refer to "Linear Mathematics in Infinite Dimensions", March 2009 Beta Edition)
  1. Monday 3/30/09: Section 1.5.7  "Isomorphic Hilbert Spaces". You should already know Section 1.5.6 and Theorem 1.5.2          
  2. Wednesday 4/1/09:  Chapter 2 "Why Fourier Theory?", Section 2.1 "The Dirichelet Kernel", Section 2.1.2 "Fraunhofer-Kirchoff  diffraction theorem".
  3. Friday 4/3/09:  Section 2.1.2 "Sampling theorem", "Fourier series theorem".
  4. Monday 4/6/09: Section 2.2 "The Dirac Delta Function", Section 2.3 "The Fourier Integral" (Transition from Fourier Series to Fourier Integral)
  5. Wednesday 4/8/09: Section 2.3 "The Fourier Integral" (The Fourier Integral Theorem), "Fourier Transform as a Unitary Transformation".
  6. Friday 4/10/09: In Section 2.1.2, pages 56-58 "Poisson's summation formula". In Section 2.3,  "Fourier Transform via Parceval's Relation".
  7. Monday 4/13/09: Section 2.4 "Orthonormal Wave Packet Representation", Section 2.4.1 "Construction", Section 2.4.2 "Definition and Properties".
  8. Wednesday 4/15/09: Section 2.4.3 "Phase Space Representation".
  9. Friday 4/17/09: Handout on "Wavelets and Multiresolution analysis", Section 2.5 "Orthonormal Wavelet Representation". Reminder: Know the  Cauchy-Goursat Theorem.
  10. Monday 4/20/09: Section 2.6.7 "The Pyramid Algorithm", Section 2.6.8 "The Requirement of Commensurability" and "the scaling equation".
  11. Wednesday 4/22/09: Section 4.1 "The Adjoint of an Operator", Section 4.1.1 "Adjoint Boundary Conditions", Section 4.1.2 "Second Order Operator and the Bilinear Concomitant"
  12. Friday 4/24/09: Section 4.2 "Green's Function and its Adjoint", Section 4.2.1 "Translation Invariant Systems".
  13. Reminder: Know (i) how to solve Euler's differential equation , (ii) the Cauchy Goursat theorem, (iii) Cauchy's integral theorem, (iii) how to find the Residue of a simple pole. Monday 4/27/09: Section 4.3 "Pictorial Definition of a Green's Function": Section 4.3.1 "The Simple String and Poisson's Equation", Sectiion 4.3.2 "Point Force Applied to the System", Section 3.4 "Properties and Utility of Green's function", the "Fundamental Theorem ofor Green's Functions"
  14. Wednesday 4/29/09: Section 4.5 "Construction of the Green's function"
  15. Friday 5/1/09: Section 4.6 "Unit Impulse Response: General Homogeneous Boundary conditions", Section 4.7 "Totally Inhomogeneous Boundary Value Problem", Theorem 4.4.2 "Uniqueness of a Green's Function" (review), Section 4.8.2 "Spectral Resolution of the Green's Function". Reminder about prerequisites: From Math 602 you should already know Section 3.3.3, " Basic Properties of a Sturm-Liouville Eigenvalue Problem".
  16. Monday 5/4/09: Section 4.8 "Spectral Representation", Section 4.8.3 "Green's Function as the Fountainhead of the Eigenvalues and Eigenvectors of a System", Section 4.8.4 "String with Free Ends"
  17. Wednesday 5/6/09: Section 4.9 "Boundary Value Problem via Green's Function: Integral Equations."
  18. Friday 5/8/09: Section 4.10 "Singular Boundary value Problem: Infinite Domain", Section 4.10.1 "Review of Branches, Branch  Cuts, and Riemann Sheets, Section 4.10.2 "Square integrability", Section 4.10.3 "Infinite String", Section 4.10.4 "Infinite String as the Limit of a Finite String"
  19. Monday 5/11/09: Section 4.11" Spectral Representation", Section 4.11.2 "Contour Integration around a Branch Cut", Section 4.11.3 "Fourier Sine Theorem"
  20. Wednesday 5/13/09: Start reading Chapter 5 "Special Function  Theory".  The purpose of this chapter is to develop a most powerful method for putting the nature of waves and their propagation into a mathematically tractable form.  Pages 279-280 are reminders from linear algebra (Math 601 and 602). The powerful method is illustrated by means of the Helmholtz equation. Thus you should assimilate Sections 5.1.1 "Cartesian vs. Polar coordinates", Section 5.1.2 "Degenerate Eigenvalues", Section 5.1.3 "Complete Set of Commuting Operators", Section 5.1.4 "Translations and Rotations in the Euclidean Plane"
  21. Friday 5/15/09: Section 5.1.5 "Symmetries of the Solutions to the Helmholtz Equation", Section 5.1.6 "Wanted: Rotation Invariant Solutions to the Helmholtz Equation"
  22. Monday 5/18/09: Section 5.2  "Properties of Hankel and Bessel Functions"
  23. Wednesday 5/20/09: Section 5.2 "Property 4 - Property 13"
  24. Friday 5/22/09: Section 5.2 "Property 14 - Property 18"                                                                                                                                       
    25. Monday (No Classes: Memorial Day)
  25. Wednesday 5/27/09: Section 5.4 "Properties 19 and 20: Translation of Cylinder Harmonics, a.k.a. the cylindrical addition theorem",
  26. Friday 5/29/09:  "Properties 21, 22: Completeness and the Bessel transform"
  27. Monday: Section 5.5 "Method of Steepest Descent and Stationary Phase (a.k.a. short wave length, high frequency approximation)"
  28. Wednesday: Section 5.7, p 355-357: "Spherical Systems"; Section 5.7.1 "Spherically Symmetric Solutions", Section 5.8 "Static Solutions"
  29. Friday: Section 5.8.1 "Static Multipole Fields", Section 5.8.2 "Translation of Spherical Harmonics", a.k.a. the "Spherical Addition Theorem"