Solutions to Homework 12 are finally available here. I apologise for the long delay in typing these up!

A few notes about what I expect you to memorise for the final, and how much work I expect you to show:

-- I expect that you will memorise for the final everything that was required for the term test (basically the first page of the term test review sheet). You also need to memorise the relationship between P_lm and P_l, and all of the first few P_l and P_lm which are given on the term test review sheet (these are needed on the final!), and the normalisation integrals for the Bessel and spherical Bessel functions, as well as the relation d/dx (x^m J_m) = x^m J_(m - 1), which we use to expand functions in series of the J_m.

-- On that point, while you are welcome to memorise also the integral of things like rho^(m + 1) J_m(lambda_mi rho), if you need a result like this on the test I expect you to work it out, rather than just giving a memorised formula.

-- Similarly, if solving the Poisson, heat, or wave equations using orthogonal expansions, I expect you to show your work; for example, with orthogonal expansions, starting out by writing out the expansion and showing how the equations for the coefficients follow, rather than going directly to the equations for the coefficients. Also, if a problem says to solve the heat or wave equation `using Fourier transforms', this means you cannot just use the heat kernel or the representation formula for the wave equation.

-- As far as memorisation beyond the material on the term test goes, I expect you to memorise the new orthogonal sets we introduced, as well as all of the eigenfunctions and eigenvalues we introduced in class (as well as understanding the geometries and boundary conditions in which they can be used, of course). You also need to either memorise or be able to quickly rederive the eigenfunctions for the Laplacian on a cube with Neumann boundary conditions which we derived and used in Homeworks 10 and 11; so basically everything in the first two sections of p. 1 of the review sheet, as well as the information about spherical Bessel functions on p. 2.

-- -- I expect you to memorise the expression for the Green's function in R^3, as well as the formulas giving u in terms of its Laplacian and boundary values on p. 2 of the review sheet.

-- -- Fourier transforms: everything in this section on the review sheet should be memorised.

-- -- The expression for the heat kernel, as well as the representation formula for solutions to the wave equation on R^3, will be given on the test if needed.

-- Finally, a random note: if a sum is over odd indices, you may write is as sum_{i, j, k odd}, for example. There is no need to write out expressions like (2m + 1) for i, etc., as we have previously done.

Please let me know if you have any questions! See you at 4.00 --

Nathan