Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time)

Is there any way I can approximate power moments of $X$: i.e. $\mathbf{E}(X^{\alpha})$ where $\alpha$ is not necessarily an integer? (I realize that this will not always be convergent for higher moments, or necessarily very accurate, especially if $\alpha < 0$ and hence the function is not analytic. See notes in When do Taylor series approximations to expectations of (entire) functions converge?)

I tried to find some transformation of a taylor series, but couldn't figure it out. What I want to do is avoid solving an ODE in transformed $\log(X)$ variable if I can.

• I think If $X$ has some specific distributions, such as Gamma distribuiton then $E(X^{\alpha})$ can be analytically expressed but that only for $\alpha$ is a positive integer Aug 15 '15 at 9:20
• Thanks. Alas, no chance on an analytic distribution. Aug 15 '15 at 15:16