Logarithmic Dirichlet potential between a circle and a line ($V=V_0\log |(z-a)/(z+a)|/\mathrm{arcosh}(d/R),a=\sqrt{d^2-R^2}$)

In $\mathrm{Re}z>0$ outside the circle $|z-d|=R$ with $d>R>0$, the displayed bounded harmonic function is zero on the imaginary axis and $-V_0$ on the circle. Put ${\tau }_0=\mathrm{arcosh}(d/R)$. The circle is the Apollonius circle $|(z-a)/(z+a)|=e^{-{\tau }_0}$; its centre and radius are $a\coth {\tau }_0=d$ and $a/\sinh {\tau }_0=R$. The point $a$ lies inside the excluded disk and $-a$ outside the half-plane, so the logarithmic modulus is harmonic throughout the physical domain. Local branches of the complex logarithm suffice; its real part is globally single-valued even though a global complex logarithm need not exist. The potential tends to zero at infinity. A boundedness or far-field condition is needed to infer uniqueness from the finite boundary data. This is also the geometry of cylinder-plane image charges.