I want to reproduce Eq. (11) of this paper. It is the result of solving the integral $$ \int_{0}^{\Lambda} \text{d}q \frac{q (e^{i q r} - e^{-iqr})}{q^2 - x} $$ (where $\Lambda = \pi$ and $x = \omega/\sqrt{\Delta k}$, with $\omega$ real and $\Delta k$ complex). The authors state that they use the Residue theorem and the method of steepest descent, and they assume $r \gg 1$. I see that I can write this as a closed curve around the pole $\sqrt{x}$, but I have no idea how to apply the method of steepest descent to this integral.
The integral should give, according to the authors, $$ e^{i\omega r\Re(\sqrt{\Delta k})/|\Delta k|}e^{-r\omega |\Im(\sqrt{\Delta k}|)/|\Delta k|}. $$