TITLE: Fourier transform of complex Gamma function
QUESTION [8 upvotes]: I am wondering if it is known how to evaluate the Fourier transform of the complex Gamma function, i.e.
$$
\frac{1}{\pi}\int_0^\infty{\mathrm{e}^{\mathrm{i}\,t\,x}\;\Gamma(1+\mathrm{i}\,t)\;\mathrm{d}t},\quad x\in\mathbb{R}.
$$
I tried all sorts of substitutions and integral / sum representations for either factor, but it doesn't seem to work easily. Often I seem to encounter a double exponential $\mathrm{e}^{-\mathrm{e}^{-r}+r}$ in one form or another.
Many thanks.

REPLY [8 votes]: The real part of the integral is easily obtained thanks to the Fourier transform (see attachment).
The imaginary part leads to much more difficulties. We even not know if a closed form exists.
 
The formula below shows the imaginary part expressed as a Cauchy Principal Value. I doubt that a simpler closed form could be derived. 
The numerical tests are in very good agreement with this formula. Sorry, I don't presently publish the analytical calculus leading to the imaginary part because there is still a remaining theoretical difficulty.
 
[Typo corrected in the formula : 1/pi was missing ]