In the study of Conformal Field Theory in physics, one encounters the following function
$$ \left( \frac{1}{\sinh(x+i\epsilon)} \right)^{2\Delta}. $$ It appears as the correlation function of certain special kinds of variables. Here $\Delta$ is a positive real number and $\epsilon$ should be thought of as a regulating parameter that is small but positive, and is taken to zero at the end of the calculation.
The question is about the Fourier transform of this function. It can be found in Equation 5 of http://arxiv.org/pdf/1407.3415v1.pdf. I am reproducing it below.
$$ \int_{-\infty}^{\infty} dx\, e^{-i\omega x} \left(\frac{1}{\sinh (x + i \epsilon)} \right)^{2\Delta} = (-1)^{-\Delta} \frac{2^{2\Delta-1}}{\Gamma(2\Delta)}\exp \left[ - \frac{\pi \omega}{2} \right] \Gamma\left( \Delta + \frac{i \omega}{2\pi} \right) \Gamma\left( \Delta - \frac{i \omega}{2\pi} \right) $$
How can one derive this last formula?