I'm trying to evaluate a particularly ugly inverse Laplace transform ($L^{-1}[\frac{s^c}{ln(s)}]$), and since all indications point to its being analytically unsolvable exactly, I'm trying to find some way of approximating it.
Post's inversion formula seemed like an interesting alternative to the standard inverse Laplace transform, but it requires evaluating a derivative of infinite order. Specifically, the formula is:
$f(t) = L^{-1}[F(s)] = \lim_{k\to\infty} \frac{(-1)^k}{k!} \left(\frac{k}{t}\right)^{k+1} F^{(k)}\left(\frac{k}{t}\right)$
For any function such that:
$\sup_{t>0}\frac{f(t)}{e^{at}} < 0$
It seems obvious to use the Grunwald-Letnikov derivative to evaluate the continuous k-th order differential in the inversion, but from what I've read this calculation is at the heart of why Post's inversion is rarely practical. My question is, could this be used as an approximation which gets better with increasing finite k, or is there something inherent in the infinite limit that makes this unapproximable?