In Mathematica:
LaplaceTransform[Exp[-Exp[-t]], t, s]
Out:= Gamma[s]-Gamma[s,1]
But performing InverseLaplaceTransform
on the result does not give me any output. Even InverseLaplaceTransform[Gamma[s], s, t]
does not give any output, but InverseLaplaceTransform[-Gamma[s,1], s, t]
returns Exp[-Exp[-t]]
which is identical to my input. Does that mean InverseLaplaceTransform[Gamma[s], s, t]
=0 according to Mathematica?
Integration by summing residues shows that inverse Laplace transform of Gamma function is: $$\mathbf{L^{-1}}[\Gamma(s)]=\int_{\gamma-i\infty}^{\gamma+i\infty}ds~e^{st}\Gamma(s)\\ =\sum_{n=0}^\infty Res_{s=-n}[e^{st}\Gamma(s)]\\ =\sum_{n=0}^\infty e^{-nt}\frac{(-1)^n}{n!}=\sum_{n=0}^\infty \frac{(e^{-t})^n}{n!}=e^{-e^{-t}}$$
in which Gamma[s,1]
has no role! I have verified by numerically inverting the Laplace transform (using Talbot's method) that presence of Gamma[s,1]
indeed doesn't matter at all (i.e. whether I invert Gamma[s]
or Gamma[s]-Gamma[s,1]
I get the same number and this matches with Exp[-Exp[-t]]
). I know this is a broad question, but what is going on with Mathematica here?