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 ε and we denote by u_ε the corresponding solution. If p is a point of the domain, then for ε small we write u_ε(p) as a convergent power series of ε and of 1/(r₀ + (2π)^-1 log |ε|), with r₀ ∈ 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_ε(p) in the case of a ring domain.