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