For the function (all variables real, and $m>0$)
$f(x) = \frac{m}{\sqrt{\lambda }} \tanh \left(\frac{m \sqrt{x^2}}{\sqrt{2}}\right)$,
the Fourier Transform is given by (result from Mathematica, for all variables being real) as
$\displaystyle{\frac{\sqrt{2 \pi } m \delta (p)}{\sqrt{\lambda }}+\frac{H_{\frac{1}{4} \left(\frac{i \sqrt{2} p}{m}-2\right)}}{2 \sqrt{\pi } \sqrt{\lambda }}+\frac{H_{\frac{1}{4} \left(-\frac{i p \sqrt{2}}{m}-2\right)}}{2 \sqrt{\pi } \sqrt{\lambda }}-\frac{H_{\frac{i p}{2 \sqrt{2} m}}}{2 \sqrt{\pi } \sqrt{\lambda }}-\frac{H_{-\frac{i p}{2 \sqrt{2} m}}}{2 \sqrt{\pi } \sqrt{\lambda }}}$
where $H$ is Harmonic Number.
Question 1: Does the inverse Fourier Transform (edited: of the resulting FT) exist, given the restrictions on the arguments of $H$?
Question 2: Does this imply that inverse Fourier Transform may not exist, even if the forward transform does exist?
Just in case, Mathematica will not evaluate the inverse FT of the result shown above.