Let $a \in \mathbb{R}_0, b>0$, then consider the integral
$$I(t) = \int_{-\infty}^\infty d\omega e^{i\omega t}\frac{1}{ib|\omega|^a - \omega},$$
with $t>0$. I would like to understand the behavior for small times and big times however I don't really know what to do. Someone suggested using the steepest descent method but I don't really understand what to do in my case since the integral involves a branch cut.
Potentially useful answer: Inverse Fourier transform of $\frac{1}{ib|\omega|^a - \omega}$
