I am trying to calculate the Inverse Laplace Transform of $$F(\lambda)=\exp\left[\mu(e^{-\lambda}-1)\right],$$ where $\mu$ is just a constant, to obtain $f(Q)$. My attempt so far brought me to a fairly recognizable integral (since $F(s)$ does not have any singularities) $$\int_{-\infty}^\infty e^{Q\lambda}\exp\left[\mu(e^{-\lambda}-1)\right]d\lambda$$
By setting $e^{-\lambda}=s$ I obtained after some manipulations
$$\int_0^\infty s^{Q+1}e^{\mu s}ds.$$
From here on is the part where I am stuck. I know I could do another change of variables to arrive at an expression containing the Gamma-function or I could use Feynman's Integration trick to derive under the integral but I don't see how either of those options would give me a decent anwers. My supervisor hinted me that I should find that $f(Q)=e^{-\mu}\mu^Q/Q!$, a Poisson distribution.