Periodic electrostatic Green function with one compact dimension ($\varphi (r,z)=\frac{q}{2\pi r}\frac{\sinh (r/R)}{\cosh (r/R)-\cos (z/R)}$)

For the Poisson equation ${\nabla }_4^2\varphi =-4\pi Rq{\delta }^3(\mathbf{x}){\delta }_{2\pi R}(z)$ on ${\mathbb{R}}^3\times S^1$, expand the periodic delta as $(2\pi R{)}^{-1}{\sum }_n{e}^{inz/R}$. 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 $q_1{q}_2/(2\pi r)$ at large separation to $q_1{q}_{2}R/(\pi r^2)$ at small separation, with normalization fixed by the stated Poisson equation.