Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter $\epsilon$ and we denote by $u_\epsilon$ the corresponding solution. The behavior of $u_\epsilon$ for $\epsilon$ small and positive can be described in terms of real analytic functions of two variables evaluated at $(\epsilon,1/\log\epsilon)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by $u_\epsilon$ for $\epsilon$ small and describe $u_\epsilon$ by real analytic functions of $\epsilon$. Then it is natural to ask what happens when $\epsilon$ is negative. The case of boundary data depending on $\epsilon$ is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.