Yukawa kernel convolution ($\frac{1}{4\pi }\int \frac{e^{-m|x-y|}}{|x-y|}\frac{e^{-\beta |y|}}{|y|}{d}^{3}y=\frac{e^{-m|x|}-e^{-\beta |x|}}{|x|({\beta }^2-m^2)}$)

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 $e^{-(m+\beta )r^{\prime }}d{r}^{\prime }$ after the spherical factors cancel. These limits prevent the formula's apparent singular denominators from being mistaken for physical divergences.