I define the following function:
Nat2[s_] := InverseFourierTransform[FourierTransform[1/t, t, w] (-I w)^(s - 1), w, x]/Cos[Pi (s - 1)] /. x -> 1
I expected it to always give the same as
Gamma[x] but it gives some values at negative integers as well. How should I interpret this? Does it mean that embeeded Gamma is poorly realized and should be artificially extended to cover broader range of arguments?