How to integrate the following function?

Question

Let's have the integral $$ I(\mathbf r, \omega) = \int \limits_{-\infty}^{\infty} e^{i(\mathbf k \cdot \mathbf r )}\frac{\sin(\omega \sqrt{\kappa^2 + k^2})}{\sqrt{\kappa^{2} + k^{2}}}\frac{d^{3}\mathbf k}{(2 \pi)^{3}}, \quad k = |\mathbf k|. $$ How to evaluate it? The result must contain the sum of Dirac delta of $\omega^2 - \mathbf r^{2}$ and Hankel function: $$ I(\mathbf r, \omega) = \frac{1}{2\pi}\frac{\omega }{|\omega|}\delta (\mathbf r^2 - \omega^{2}) - \frac{\kappa}{4 \pi}\frac{\omega }{|\omega|}Re\left[ \frac{H_{1}^{(1)}(i\kappa \sqrt{\mathbf r^2 - \omega^{2}})}{i \sqrt{\mathbf r^{2} - \omega^{2}}}\right]. $$