= Causal diffusion from a finite-duration planar point source
{title2=$b=\frac{r^2}{4D}$}
For the planar <heat equation> with point injection of strength $p(s)$, zero initial data gives $\Psi(t,\mathbf r)=\int_0^t p(s)e^{-r^2/[4D(t-s)]}/[4\pi D(t-s)]\,ds$. If injection is $p_0$ until $t_0$ and zero later, then for $t>t_0$ this equals $p_0[\operatorname{Ei}(-b/(t-t_0))-\operatorname{Ei}(-b/t)]/(4\pi D)$, where $b=r^2/(4D)$. The <exponential integral> expansion gives $p_0\ln[t/(t-t_0)]/(4\pi D)$ when $b/(t-t_0)\ll1$.
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