## Tricks for inverting a Laplace Transform, part II: Products and Convolutions

*EDIT: In the meanwhile, I have continued the series of posts on Laplace Transform inversion. You can find the subsequent articles here: part I (guesses based on series expansions), part III, part IV (substitutions), part V (pole decomposition). Enjoy!*

Following the previous post on inverting Laplace transforms, here is another trick up the same alley. This one actually considers a generalization of the previous case

**Find such that .**

As usual, the built-in InverseLaplaceTransform function from* Mathematica 8* fails to give a result. To obtain a closed formula manually, note that each of the factors can be easily inverted:

has the solution

.

Hence, using the fact that Laplace transforms of convolutions give products, the solution for can be written as a convolution:

.

Computing the integral gives the following expression for in terms of the hypergeometric function :

Enjoy!!

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