I'm looking for a survey of methods to solve Laplace equation in two dimensions.
Is there a book describing them with hints regarding their applicability for various cases?
I mean analytical methods, not numerics.
Since Community bumped the question, I might as well put something relevant here. My reaction to seeing "two dimensions" and "Laplace equation" in the same sentence is "ooh, I can use conformal maps to do that". The idea is that since the Laplace equation is invariant under conformal maps, we can transform our boundary value problem into a BVP on a nicer domain, a disk or a halfplane. This works very well for the Dirichlet problem; for the Neumann problem the boundary conditions must be multiplied by the derivative of the conformal map, which makes life a bit harder.
Here, I am giving you a link. See the file. You will get some hint. http://newton.ex.ac.uk/teaching/CDHW/EM/CW970317-2.pdf