= Static Green function across a circle compactification
{title2=$G_{\rm stat}(r,0)=[4\pi Rr]^{-1}\coth(\pi r/R)$}
At zero frequency, sum the <Kaluza-Klein tower> of three-spatial-dimensional Yukawa kernels: $[4\pi Rr]^{-1}(1+2\sum_{n\geq1}e^{-2\pi nr/R})$. The geometric series gives the displayed function. It behaves as $1/(4\pi^2r^2)$ for $r\ll R$, the four-spatial-dimensional Coulomb kernel, and as $1/(4\pi Rr)$ for $r\gg R$, the three-spatial-dimensional kernel with a zero-mode normalization factor.
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