= Point-source potential with one compact spatial dimension
{title2=$G(r,0)=\frac{\coth(\pi r/L)}{4\pi Lr}$}
For three noncompact spatial coordinates of radius $r$ and a circle of circumference $L$, the positive Laplace Green function on the source slice is $G(r,0)=\sum_{n\in\mathbb Z}[4\pi^2(r^2+n^2L^2)]^{-1}=(4\pi Lr)^{-1}\coth(\pi r/L)$. At $r\ll L$ this is $1/(4\pi^2r^2)$; at $r\gg L$ it is $1/(4\pi Lr)$ with exponentially small corrections. The long-distance coefficient also follows by replacing the dense image sum with $L^{-1}\int_{-\infty}^{\infty}dy/(r^2+y^2)=\pi/(Lr)$. Electric and gravitational potentials inherit this crossover after multiplying by their signed source couplings.
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