= Periodic electrostatic Green function with one compact dimension
{title2=$\phi(r,z)=\frac{q}{2\pi r}\frac{\sinh(r/R)}{\cosh(r/R)-\cos(z/R)}$}
For the <Poisson equation> $\nabla_4^2\phi=-4\pi Rq\,\delta^3(\mathbf x)\delta_{2\pi R}(z)$ on $\mathbb R^3\times S^1$, expand the periodic delta as $(2\pi R)^{-1}\sum_ne^{inz/R}$. The $n$th <Kaluza-Klein mode> has radial potential $q e^{-|n|r/R}/(2\pi r)$, since $(\nabla_3^2-m^2)e^{-mr}/r=-4\pi\delta^3(\mathbf x)$. Summing the geometric series gives the displayed <Green function>. At equal compact coordinate it reduces to $q\coth(r/(2R))/(2\pi r)$. Thus the interaction energy changes from $q_1q_2/(2\pi r)$ at large separation to $q_1q_2R/(\pi r^2)$ at small separation, with normalization fixed by the stated Poisson equation.
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