Diffusion in a disk with inclusion: Evaluating Green's functions

We give exact Green's functions in two space dimensions. We work in a scaled domain that is a circle of unit radius with a smaller circular "inclusion", of radius a, removed, without restriction on the size or position of the inclusion. We consider the two cases where one of the two boundaries is absorbing and the other is reflecting. Given a particle with diffusivity D, in a circle with radius R, the mean time to reach the absorbing boundary is a function of the initial condition, given by the integral of Green's function over the domain. We scale to a circle of unit radius, then transform to bipolar coordinates. We show the equivalence of two different series expansions, and obtain closed expressions that are not series expansions.