I'm reading "Modular Groups of Quantum Fields in Thermal States" by Borchers and Yngvason, and on page 19 I find:
Let's call $x=\beta p \in \mathbb{R}$, so I'm interested in the Fourier transform of $(1-e^{-x})^{-1}$.
I have tried a contour that avoids the origin with a small arc (in the upper-half plane) and closing with a large arc also in the upper half-plane, but the contribution from the small arc does not vanish, and this contour would also have infinite contributions from the residues at $2 \pi i n$, with $n$ a positive integer. I have also tried with a rectangular contour, of width $2L$ and height $2\pi$, with its lower horizontal segment an $\epsilon$ distance below the real axis (so to avoid the pole at the origin). The integrals along the horizontal segments can be related, and the integral along the left vertical segment goes to zero for $L\rightarrow \infty$, but the right vertical segment still contributes to the contour integral.
So, any suggestions on other possible contours?