  In 1991, Vladimir Maz'ya, Serguei Nazarov and Boris Plamenevskij developed
the theory of compound asymptotics for elliptic boundary value problems in
singularly perturbed domains. They considered a harmonic function whose domain
contains a small inclusion. We applied this technique in the analysis of a
Nematic liquid crystal with a small colloidal inclusion. However, we realised
that the Maz'ya, Nazarov and Plamenevskij did not consider the asymptotic
analysis of Robin boundary conditions, which corresponded to weak anchoring in
the context of liquid crystals.
  In this piece we shall derive an asymptotic approximation to a harmonic
function, in a domain with a small circular inclusion of radius $\epsilon>0$,
with inhomogenous Robin boundary conditions and a corresponding parameter
$\kappa>0$. We shall then prove that the difference between the exact solution
and the approximation is uniformly bounded and derive the rate as a function of
$\epsilon$ and $\kappa$.
