I am struggling to evaluate the following integral
\begin{align} I = \int_{-\infty}^{\infty} \frac{e^{i\kappa_0\sqrt{r^2+t^2}}}{(r^2+t^{2})^{3/4}}e^{-ik t} d t. \end{align}
I have tried standard method of residues but that doesnt seem to work as the result comes out as zero due to the fact that the $t\pm ir$ factor doesn't cancel completely with that of the denominator. I have a feeling this integral can be expressed in in terms of generalized functions.