LaplaceTransform and InverseLaplaceTransform?

Questions
,
1

I wonder if the knowledge basis of the LaPlaceTransform anf the InverseLaplaceTransform is complete. In Particular, I Have tried to find the InverseLaplaceTransform of s^{-\alpha} without any result if 0< \alpha <1. I need to use Caputo fractional derivative, so if I am true, how can I complement the knowledge basis of LaplaceTransform and InverseLaplaceTransform. Has a specific package developped for this subject ?

Thanks

2

You can add definitions to InverseLaplaceTransform by something like:

Unprotect[InverseLaplaceTransform]
InverseLaplaceTransform[s_^-a_, a_, t_] := outcome

But be careful what you redefine, if you do it wrong it can lead to disastrous results…