I'm trying to invert the following Laplace Transform using the Residue Theorem:
$$ F(s) = \frac{K_1(\sqrt s)}{\sqrt s K_0(\sqrt s)} $$
where $K_0()$ and $K_1()$ are the Modified Bessel Functions of the Second Kind of Orders $0$ and $1$, respectively.
There appears to be a pole at $s=0$, but the singularity remains since the ratio of the Bessel functions still tends to $\infty$ as $s \rightarrow 0$, so L'Hôpital's Rule won't help here. I'm looking into how to address the residue at $\infty$ following the approach in this link , but unless I can resolve the residue at $s=0$, I don't see a viable path forward.
Any suggestions on how to resolve this, or an alternative analytical approach would be most welcome. Thanks