= Harmonic potential between a grounded line and a circular boundary
{title2=$V=V_0\log|(z+ib)/(z-ib)|/\operatorname{arcosh}(y_0/R)$}
In the upper half-plane outside a circle of radius $R$ centered at $iy_0$, 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 $\log P=\operatorname{arcosh}(y_0/R)$. 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.
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