# What is the representation of the Harmonic Number being used by Mma in this result?

The Fourier Transform of the function

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


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

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


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*m))] - PolyGamma[0, (I*p)/(2*Sqrt*m)] + PolyGamma[0, (1/4)*(2 - (I*Sqrt*p)/m)] + PolyGamma[0, (1/4)*(2 + (I*Sqrt*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 $$n\rightarrow \infty$$

$$H_z=\Psi(z+1)+\gamma$$

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

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