Harmonic potential between a grounded line and a circular boundary ($V=V_0\log |(z+ib)/(z-ib)|/\mathrm{arcosh}(y_0/R)$)

In the upper half-plane outside a circle of radius $R$ centered at $i{y}_0$, with $y_0>R$, set $b=\sqrt{y_0^2-R^2}$. The displayed harmonic function is zero on the real axis and equals $V_0$ on the circle. Indeed, the Apollonius circle with ratio $P>1$ has center $ib(P^2+1)/(P^2-1)$ and radius $2bP/(P^2-1)$; hence $\log P=\mathrm{arcosh}(y_0/R)$. The logarithmic singularity at $ib$ is inside the excluded circle, and the one at $-ib$ is below the real axis. The real part of a local holomorphic logarithm is harmonic even if its imaginary part is multivalued. The potential tends to zero at infinity. Finite boundary values alone do not impose a growth condition at infinity; uniqueness in the unbounded domain needs such an additional condition, for example boundedness with the prescribed limit.