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

Let $n\ge 3$. Let $\Omega^i$ and $\Omega^o$ be open bounded connected subsets of $\mathbb{R}^n$ containing the origin. Let $\epsilon_0>0$ be such that $\Omega^o$ contains the closure of $\epsilon\Omega^i$ for all $\epsilon\in]-\epsilon_0,\epsilon_0[$. Then, for a fixed $\epsilon\in]-\epsilon_0,\epsilon_0[\setminus\{0\}$ we consider a Dirichlet problem for the Laplace operator in the perforated domain $\Omega^o\setminus\epsilon\Omega^i$. We denote by $u_\epsilon$ the corresponding solution. If $p\in\Omega^o$ and $p\neq 0$, then we know that under suitable regularity assumptions there exist $\epsilon_p>0$ and a real analytic operator $U_p$ from $]-\epsilon_p,\epsilon_p[$ to $\mathbb{R}$ such that $u_\epsilon(p)=U_p[\epsilon]$ for all $\epsilon\in]0,\epsilon_p[$. Thus it is natural to ask what happens to the equality $u_\epsilon(p)=U_p[\epsilon]$ for $\epsilon$ negative. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality $u_\epsilon(p)=U_p[\epsilon]$ for $\epsilon$ negative depends on the parity of the dimension $n$.