An Introduction to Laplace Transforms and Fourier Series : Review
Laplace Transformation is a useful tool in problem solving. From the probabilistic theory perspective, Moment generating function is nothing but a linear combination of two Laplace transforms. In fact moment generating function has a more direct relationship with Laplace-Stieltjes transform. Most of the calculations involving MGF and convolutions become easy with the use of Laplace Transform. I think one of the most powerful ways to use Laplace transforms is solving Partial Differential equations.