This question results from some afterthoughts from an earlier problem I posted in the mathematica-section (https://mathematica.stackexchange.com/q/130387/34999). So, I want to do the fourier-transformation of
$e^\frac{i c t}{|k|}$ where k is the variable. Now, the problem with this one is that the absolute value is a non-meromorphic function, so there is in principle no analytical continuation to complex values of $k$. Using $e^{i c t/\sqrt{k^2}}$ instead allows to cut the complex plane in two, divided by a vertical branch cut through the origin, so that the function is analytical in each half of the plane.
Is this way of dealing with the problem the legit one indeed?
The result of the calculation in mathematica is the sum of a $KelvinKei$, a $KelvinKer$ function and a Dirac $\delta-$distribution and is even als should be. At first sight, this results seem to be consistent with a numerical fourier transform as well. Now, if I do an inverse fouriertransform of this result, I don't retrieve the original function; I get some general $MeijerG$ symbols instead that don't give any output at values of $k$,$c$,$t$ where I know the original function definitely exists.
So, does a fourier transform fix a fixed bijective correspondence between two functions/distributions which should be valid throughout whole parameter space?