The Fourier Transform of the function

F[x_] = (m/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*m)/Sqrt[2]]

where all variables are real, and $m>0$ is given by (Mma 11.0)

   (2*Sqrt[Pi]*Sqrt[\[Lambda]]) - 
   (2*Sqrt[Pi]*Sqrt[\[Lambda]]) + 
  HarmonicNumber[(1/4)*(-2 - (I*Sqrt[2]*p)/m)]/
   (2*Sqrt[Pi]*Sqrt[\[Lambda]]) + 
  HarmonicNumber[(1/4)*(-2 + (I*Sqrt[2]*p)/m)]/

For all variables set to real (i.e. m also real), the result is in terms of polygamma functions.

(1/(2*Sqrt[Pi]*Sqrt[\[Lambda]]))* (-PolyGamma[0, -((I*p)/(2*Sqrt[2]*m))] - PolyGamma[0, (I*p)/(2*Sqrt[2]*m)] + PolyGamma[0, (1/4)*(2 - (I*Sqrt[2]*p)/m)] + PolyGamma[0, (1/4)*(2 + (I*Sqrt[2]*p)/m)])

Mma will not evaluate the InverseFourierTransform however, for either of the two results above. Is that because (for the first result, for e.g.) that the integral representation of Harmonic Number, $\gamma=(H_{\text{n}} - ln(\text{n}))$ in the limit $n\rightarrow \infty$


$\displaystyle{\Psi_{z+1}=-\gamma + \int_0^1 \left( \frac{1-t^z}{1-t}\right)}{\text d}t$ $~~~Re(z)>0$

which may imply that the (inverse) Fourier Transform does not exist, since the integration is from $-\infty$ to $+\infty$?


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