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 $\frac{1}{ib|\omega|^a - \omega}$

  • $\begingroup$ Just to be sure, what does $\mathbb{R}_0$ mean? $\endgroup$
    – Qmechanic
    Mar 4, 2023 at 14:15
  • $\begingroup$ The set of real numbers except $0$. $\endgroup$
    – Audrique
    Mar 4, 2023 at 14:30


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