I was studying theory of electromagnetic from the book Time-harmonic Electromagnetic Fields by Roger Harrington and what caught up my attention was following identity: $$ \frac{\mathrm{e}^{-\mathrm{i}k_{0}r}}{r} = \frac{1}{2\mathrm{i}}\int_{-\infty}^{\infty}\mathrm{H}_{0}^{\left(2\right)} \left(\rho\,\sqrt{\,k_{0}^{2} - k_{z}^{2}\,}\,\right)\mathrm{e}^{\mathrm{i}k_{z}z}\,\mathrm{d}k_{z} $$ where $k_{0}$ is wave number in vacuum, $\mathrm{H}_{0}^{\left(2\right)}\left(x\right)$ is Hankel function of second kind and $$ r^{2} = \rho^{2} +z^{2} $$ Proving mentioned identity is quite trivial given the fact that we can calculate electromagnetic fields of a point source using other methods. However in order to solve other problems I found myself in need to know following integral: $$ \int_{-\infty}^{\infty}\mathrm{H}_{n}^{\left(2\right)} \left(\rho\,\sqrt{\,k_{0}^{2} - k_{z}^{2}\,}\,\right)\mathrm{e}^{\mathrm{i}k_{z}z}\,\mathrm{d}k_{z} $$ Evaluating this integral maybe possible in a similar fashion but I was unable to reach it.
So I would be more than grateful if anyone could help.
Regards.
