Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

We consider the Dirichlet problem for the Laplace equation in a planar domain with a small hole. The diameter of the hole is proportional to a real parameter $epsilon$ and we denote by $u_epsilon$ the corresponding solution. If $p$ is a point of the domain, then for $epsilon$ small we write $u_epsilon(p)$ as a convergent power series of $epsilon$ and of $1/(r_0+(2pi)^{-1}log |epsilon|), with $r_0 in mathbb{R}$. The coefficients of such series are given in terms of solutions of recursive systems of integral equations. We obtain a simplified expression for the series expansion of $u_epsilon(p)$ in the case of a ring domain.