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)^-1log |epsilon|), with $r_0 in mathbbR$. 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.

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 e small we write u(epsilon)(p) as a convergent power series of epsilon and of 1/(r(0) + (2 pi)(-1) log vertical bar epsilon vertical bar), with r(0) is an element of 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.