# How to perform an inverse Laplace transform

I'm trying to work out the inverse Laplace transform $$f(z)=\mathcal{L}^{-1}\left\{s^2\log\left(1-\frac{z}{s}\right)\right\}.$$

To make sure this was first possible I turned to Mathematica and entered:

InverseLaplaceTransform[s^2Log[1-z/s],s,z]


I get an expression with terms like

DiracDelta[0], HeavisideTheta[0], HeavisideTheta[-z], DiracDelt'[0],


and so on. I've looked these generalised functions up online and understand the basics of them. However, I was wondering is this just some over complicated expression which Mathematica has cooked up, or is there a more natural way to write the inverse Laplace transform in terms of elementary functions? If this is the only way to write the function, then how can I interpret these generalised functions since from what I have read they are discontinuous at $0$. Maybe this will depend on context?

Transform $s$ or transform $z$ ? –  doraemonpaul Mar 18 at 14:04