I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points.
The integral I need to solve is
$$\int_\infty^{-\infty} \frac{dk}{\sqrt{k^2+m^2}} \frac{e^{-\beta\sqrt{k^2+m^2}/2}}{1-e^{-\beta\sqrt{k^2+m^2}}} e^{i\sqrt{k^2+m^2}}e^{ikx}$$
(this is the thermal propagator from finite temperature field theory in 1+1 dimension).
$\beta$ and $m$ are positive numbers, $t$ and $x$ are not necessarily positive but we can take them to be positive. I have two branch point, $k=\pm i m$, and the branch cut is the line connecting these two points.
With $t$ and $x$ positive I can close the contour on the upper plane (I think), so that my integration contour will be formed by $\int_{-\infty}^0 + \int_0^{i m} + \int^0_{i m}+\int^{\infty}_0+\int_{C_R} = 2 \pi i \text{Res}_\infty$, where the two integrals on the imaginary axis don't cancel cause they are on the right and left part of the branch cut. The integral on $C_R$ goes to 0 when R goes to infinity.
Assuming this setup is correct, I have the problem that computing the integral on the imaginary axis is still something I cannot do. The square root has a sign of difference on the two sides, but this doesn't give me anything simple I can solve.
Any ideas?
