I'm interested in computing the following integral:
$$\displaystyle \int_{|k| \geqslant 1} \frac{\sin(\rho|k-\alpha|)}{|k-\alpha|^{3/2}} \ \mathrm{d}k,$$
where $\rho$ is constant, and $\alpha \in \mathbb{R}^2$ is such that $|\alpha| \geqslant 1$.
I would like to make the substitution $\tau = |k - \alpha|$. It looks like we may assume WLOG that $(\alpha_1, \alpha_2) = (r,0)$ for $r \geqslant 1$. Then the integral transforms to
$$\displaystyle \int_{0}^{2\pi} \int_{?}^{\infty} \frac{\sin(\rho \tau)}{\sqrt{\tau}} \ \mathrm{d}\tau \ \mathrm{d}\theta,$$
where the $?$ indicates that I don't know what the lower limit should be. Can anyone suggest what it is? I suspect it should be either $|1 + \alpha|$ or $|1 - \alpha|$.