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Inversion of Moment-generating functions (aka Laplace transform of prob dist)

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.