Causal diffusion from a finite-duration planar point source ($b=\frac{r^2}{4D}$)

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 $p_0[\mathrm{Ei}(-b/(t-t_0))-\mathrm{Ei}(-b/t)]/(4\pi D)$, where $b=r^2/(4D)$. The exponential integral expansion gives $p_0\ln [t/(t-t_0)]/(4\pi D)$ when $b/(t-t_0)\ll 1$.