# Should this be equal to Gamma function or not?

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?

• It's all fine! In:= \$Version Out= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" In:= InverseFourierTransform[ FourierTransform[1/t, t, w] (-I w)^(s - 1), w, x]/ Cos[Pi (s - 1)] /. x -> 1 Out= -Cos[[Pi] s] Gamma[s] Sec[[Pi] (-1 + s)] In:= Simplify[%] Out= Gamma[s] – Dr. Wolfgang Hintze Nov 1 '14 at 11:12
• @Dr. Wolfgang Hintze have u tried to feed negative integers to the Nat2 function? – Anixx Nov 1 '14 at 11:21
• @ Anixx: The function Gamma[s] has simple poles at the integers <=0. I suggest you read about the properties of the Gamma function in the Help section which also includes a Plot for illustration. – Dr. Wolfgang Hintze Nov 1 '14 at 11:38
• @Dr. Wolfgang Hintze and what? – Anixx Nov 1 '14 at 12:09