The homework for this topic is slightly smaller in volume than the previous ones so that you can focus on completing the project.

There will be two lectures about distributions which are optional however, very important and interesting.

You can separate variables for Laplace's equation in simple domains. During the lecture I have shown you how to do it for a circle in polar coordinates. Prob. 3 concerns a square and that is simple. Just remember to correctly apply boundary conditions. That is, on the boundary for which a non-zero condition is given you do not impose any condition for the ODE problem. You do this at the end by using Fourier series.

A hint for Prob. 4: first find eigenfunctions of the Laplacian, i.e. $u_{xx}+u_{yy} = \lambda u$ and then expand $f$ in the resulting orthogonal series.

The problems on the circle are very similar. The thing that you should remember is to choose an appropriate special solution of Euler's equation - the one that is bounded in the given domain.

The reflection principle is very useful. It works by placing a negative charge, i.e. adding a opposite sign of the unit charge potential, at some point to cancel the potential on a specified boundary. This is very simple for a half-space.

To obtain a two-dimensional Green's function for the Laplacian you can follow the calculations from the lecture but in 2D. These are completely analogous. The answer contains a logarithm.

Remember that you can always contact me or refer to the literature.