= Logarithmic Dirichlet potential between a circle and a line
{title2=$V=V_0\log|(z-a)/(z+a)|/\operatorname{arcosh}(d/R),\quad a=\sqrt{d^2-R^2}$}
In $\operatorname{Re}z>0$ 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 $\tau_0=\operatorname{arcosh}(d/R)$. 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>.
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