ACM 95b/100b Winter 2004.	Sterl Phinney	3/5/2004

What is examinable on final exam:

It will be assumed that you can use all the midterm material (i.e. you
can solve second order ODEs by reduction of order, series, Laplace
transform, and so on, recognise singular points, find Wronskians, use
delta functions, etc.) as it arises in the course of solving these more
advanced problems.

Eigenvalue problems, eigenfunctions.
Linear Hilbert space of functions.
Inner product of functions in Hilbert Space -with and without 
 a weight function.
Orthogonality of functions, orthonormal basis functions on Hilbert Space.
Gram-Schmidt orthogonalisation procedure.
Completeness of basis functions.
Examples of complete basis sets: Fourier, Fourier-Bessel, Legendre
 polynomials, Chebyshev polynomials.
Generalised Fourier series.
Sturm Liouville problems: of regular, singular, periodic type.
Relation of S-L problem to self-adjoint operators (rule + boundary conditions).
  The bilinear Concomitant.
Theorems on Sturm-Liouville problems: orthogonality of eigenfunctions,
 real eigenvalues.
Theorems for regular S-L problem: uniqueness of eigenfunctions, infinite number
 of eigenvalues with no accumulation point. Eigenfunctions form complete basis.
What remains of theorems for irregular/periodic S-L problems.
Determining the expansion coefficients of eigenfunction expansions (aka
 generalised Fourier series).
Parseval's Theorem (for general complete eigenfunction expansions).
How S-L problems arise from PDEs of physics (heat, wave, Laplace).
Separation of variables: Finding the appropriate S-L problem and 
  eigenfunction expansions for PDEs
Fourier Series (period pi: sine or cosine, period 2 pi: sine & cosine, 
  complex exponential)
Fourier Transforms. Delta representation, Shifting, Derivatives, Convolution,
  multiplication. Application to PDE on infinite intervals.
Green's functions for S-L problems with homog BC: 
 When exist (when only solution to homog problem is y=0 -ie not eigenvalue).
 Expression in terms of y_1, y_2 (deduce, don't memorize!)
 When don't exist (when homog problem has nontrivial solution -ie 
 is eigenvalue), consequent requirements on f(x). Expression. 
 Eigenfunction expansion of Green's function.
 Green's functions and Inhomogeneous boundary conditions.


What isn't examinable (incomplete list!):
Details of proofs of theorems.
Details of examples given in class or on homeworks (e.g.
 Jacobians of spherical coordinates, Bessel solutions of 
 drum).