I have now finished marking your homework assignments; please let me know if you have any questions about my comments. (On one of the papers I wrote that assuming u = u(z) for problem 1 would not work, which was incorrect -- my apologies to whomever got that particular comment! On the other hand, while that assumption winds up being correct, it is not clear from the statement of the problem that this should be so: were we working with Dirichlet boundary conditions, for example, it would not be true; and anyway that was not how I intended the problem to be approached.)

Some general comments:

-- In the definition of the Fourier transform in multidimensional spaces (such as R^3), the quantity in the exponent is -2 pi i k dot x, where k dot x indicates the dot product of the vector k (which is the parameter of the Fourier transform) with the vector x (which is the integration variable). It is NOT just |k| |x| as it includes also information about the angle between the two vectors. In general, in spherical coordinates this is equal to the rather unruly expression (assuming k = (k_1, k_2, k_3)) k_1 r sin theta cos phi + k_2 r sin theta sin phi + k_3 r cos theta. Only when there is spherical symmetry (as in problem 2) can we reduce it to |k| |x| cos theta.

-- Please note that the definition of the Fourier transform we are using in class (and which, therefore, I will insist on your using on the final and would vastly prefer to see you use on the homework) has a 2pi in the exponent and no factor of 2pi in front of the integral (for either the forwards or inverse transform). This is a different convention from that in the textbook, which means that results in the textbook cannot be used directly in either homework or on the final. The lecture notes should contain corresponding results for our definition of the Fourier transform for all of the properties we need; if you find something missing please let me know.

-- The main point of problem 1 was to compute a set of eigenfunctions for the Laplacian which satisfy homogeneous Neumann boundary conditions (as opposed to the homogeneous Dirichlet conditions we have used so far). Unfortunately it seems that almost no-one realised this. I highly recommend reworking this problem (even if you have already read the solutions) by first finding an appropriate set of eigenfunctions. It is quite likely that something like this will show up on the final as it is a good way for me to test whether you understand the theory behind the calculations we do (using the Dirichlet eigenfunctions quite strongly indicates that there are, at the least, some gaps in one's understanding).

-- Please remember that e^(ix) = cos x + i sin x. This gives rise to the formulas sin x = 1/(2i) (e^(ix) - e^(-ix)) and cos x = 1/2 (e^(ix) + e^(-ix)). Going backwards is a simplification which I would expect you to do on the final.

Please let me know if you have any questions! I will post the next homework later on this evening (though, as I mentioned, it will only be for credit if the vote in class tomorrow on adding two additional homeworks to the marking scheme goes through).

See you tomorrow!

Nathan