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 ε and we denote by u_ε the corresponding solution. The behavior of u_ε for ε small and positive can be described in terms of real analytic functions of two variables evaluated at (ε, 1/log ε). 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_ε for ε small and describe u_ε by real analytic functions of ε. Then it is natural to ask what happens when ε is negative. The case of boundary data depending on ε 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.