= Yukawa kernel convolution
{c}
{title2=$\frac1{4\pi}\int\frac{e^{-m|x-y|}}{|x-y|}\frac{e^{-\beta|y|}}{|y|}d^3y=\frac{e^{-m|x|}-e^{-\beta|x|}}{|x|(\beta^2-m^2)}$}
For $m,\beta>0$, set $K_m(r)=e^{-mr}/r$. Distributionally, $(\Delta-m^2)K_m=-4\pi\delta$, so the <modified Helmholtz equation> with unit delta source has <Green function> $-K_m/(4\pi)$. The difference $[K_\beta-K_m]/(\beta^2-m^2)$ solves the equation with source $K_\beta$: the two delta terms cancel. It has no added point source at zero and decays at infinity. The <Green function> representation proves the displayed convolution with its positive sign.
At coincident rates, the continuous limit is $e^{-mr}/(2m)$; at $r=0$, it is $1/(m+\beta)$. The latter also follows by directly integrating $e^{-(m+\beta)r'}dr'$ after the spherical factors cancel. These limits prevent the formula's apparent singular denominators from being mistaken for physical divergences.
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