Let X be a poisson distributed variable with parameter $\lambda$. I would like to calculate the expectation of $E[e^{-ax}]$, where $a$ is a constant. I use the Taylor series approximation as follows, although I get a really ugly result. Is there a better way to solve it so that the answer is more convenient?
$$ E[e^{-ax}] = \sum_{i=0}^\infty e^{-ai}\frac{e^{-\lambda}\lambda^i}{i!} = e^{-\lambda}\sum_{i=0}^\infty \frac{ (\lambda/e^a)^i }{i!} $$
which is the Taylor series for $e^{\lambda/e^a}$. Thus,
$$ E[e^{-ax}] = e^{-\lambda} e^{\lambda/e^a} $$
Is this correct? or is there a better / more correct way to do it?