How to Invert a Rapidly Decaying Function Without Poles?

[Charles49]

I know that there is the Bromwich integral which inverts the Laplace Transform but it requires the function to have poles.

I am wondering if there is a formula for inverting a function which decays rapidly but has no poles?

 

[hunt_mat]

Not that I know of, I am looking at inverse Laplace transforms too.

Does the function have any real zeros? You might try doing a keyhole contour around the zeros.

Are you certain that this in Laplace invertable?