I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by
\begin{equation}
 M\_X(t) = \text{E} \exp(tX)
\end{equation}
Since I have never done anything like this before, I am searching for some good references for this, especially references with worked examples.



Inversion of Moment-generating functions (aka Laplace transform of prob dist)
The inverse Laplace transform takes place in the complex domain and includes the residue theorem. You can probably do that analytically (as the above forwarded you to Wiki). But usually you can perhaps see how your usual transforms look like and try to invert by converting the result to a commonly used form which is easily transformed back. 


For example, if your result is a polynomial, you can separate it to partial fractions on the laplace domain and then the conversion back doesn't need to go through the complex domain and the definition but rather via known formulas.


