I'm working through this paper: https://www.nature.com/articles/srep35222
In it, the authors have the following result from an integral obtained via the residue theorem (Eq. 24 in the attached paper: there is a missing factor of $\exp(i\lambda s)$ in their paper which I have restored here):
$$ \begin{align} I &=-\frac{1}{16\pi^2 k^2}\int_{-\infty}^\infty \frac{e^{i \lambda s}}{\sinh^2(s/2k-i\epsilon) - \frac{r^2}{k^2}\sin^2\Delta\theta/2}\mathrm{d}s \\ &=\frac{2\pi i }{16\pi^2 r\sqrt{1 + \frac{r^2\sin^2\frac{\Delta\theta}{2}}{k^2}} \sin \frac{\Delta\theta}{2}} \sum_{n=0}^\infty e^{-2n\pi k \lambda} \bigg[ e^{-2ik\lambda\sinh^{-1}\frac{r\sin\frac{\Delta\theta}{2}}{k}} - \text{c.c} \bigg] \end{align} $$
I am unsure how they have proceeded from the first line to the second line. Particularly, I don't know how the infinite series arises - there seem to be two poles at $s = \pm 2k \sinh^{-1} ( \frac{r}{k} \sin\frac{\Delta\theta}{2} )$, not an infinite number of them.
Any guidance on how this result is achieved would be very much appreciated!