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?

  • $\begingroup$ It's all fine! In[49]:= $Version Out[49]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" In[48]:= InverseFourierTransform[ FourierTransform[1/t, t, w] (-I w)^(s - 1), w, x]/ Cos[Pi (s - 1)] /. x -> 1 Out[48]= -Cos[[Pi] s] Gamma[s] Sec[[Pi] (-1 + s)] In[53]:= Simplify[%] Out[53]= Gamma[s] $\endgroup$ – Dr. Wolfgang Hintze Nov 1 '14 at 11:12
  • $\begingroup$ @Dr. Wolfgang Hintze have u tried to feed negative integers to the Nat2 function? $\endgroup$ – Anixx Nov 1 '14 at 11:21
  • $\begingroup$ @ 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. $\endgroup$ – Dr. Wolfgang Hintze Nov 1 '14 at 11:38
  • $\begingroup$ @Dr. Wolfgang Hintze and what? $\endgroup$ – Anixx Nov 1 '14 at 12:09

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