Mathematica can compute a certain Fourier transform but misses the inverse

Ideas
1

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?

1 Like
2

I would write out the corresponding Integrals and see what they do.

Integrate sometimes won’t do integrals you would expect it to because it handles input generically. This is often the case with certain orthogonal functions which are nonzero at isolated points.