Hi, Mathematica 10.2.0.0 correctly gives for
(2 \[Pi])^(1/2) FourierTransform[Exp[-E^s + s x], s, y, Assumptions -> x > 0]
the output
Gamma[x + I y]
(the integral is a rewritten form of Euler's integral for the Gamma function), but for the inverse transform
(2 \[Pi])^(-1/2)
InverseFourierTransform[Gamma[x + I y], y, s, Assumptions -> x > 0]
Mathematica returns the input. Would it be an idea if this integral is added to Mathematica's database?