Is there any way to solve this integral?
$\int_{-\infty}^{\infty}\frac{e^{qiy -K(\sqrt{\lambda -a-iy}-\sqrt{\lambda})}}{a+iy} dy$ where $K,\lambda, a$ and $q$ are real numbers and $K>0$, $a>0$, $\lambda > 0$ and $q<0$
I have tried the standard contour approaches, but the branch cut makes it complicated on the lower half plane, and the integrand grows unbounded on the upper half plane.