I want to calculate this integral
$$I\stackrel{\text{def}}{=}\int_{-\infty}^{\infty}\frac{\mathrm{exp(-\mathrm{i}\mathbf{q}\mathbf{r}})}{-\mathrm{i}\omega+\mathbf{q}^{3-\eta}(-\mathrm{i}\omega)^{\eta/3}} \, \mathrm{d}^3\mathbf{q}$$ where $q$ and $\omega$ are complex variables, while $\eta$ is a real such that $1<\eta <2$. After a change of variable it is possible to show that (if I made any mistake) $I$ is equal to $$I=\frac{4\pi}{r}\int_{0}^{\infty}\frac{q \sin(qr)}{-\mathrm{i}\omega+q^{3-\eta}(-\mathrm{i}\omega)^{\eta/3}}\,\mathrm{d}q$$ At this stage I don't know how to calculate this integral. I would want to apply Residue theorem but I don't know how to do because of the non-integer power in the denominator ($\eta$). I know that there is a branch cut but that's all.
Does any body have an idea? Even if it is not exactly this integral I am interested in any suggestion about this kind of integral with non integral powers at the denominator.