In a mathematical-physical problem, the following double integral has to be determined: $$ \int_0^1 \int_0^\infty qk \, \exp \left( ikq \right) \, J_0 \left( k \sqrt{1-q^2} \right) \, \mathrm{d}k \, \mathrm{d}q \, , $$ which follows from an inverse Fourier transformation. A numerical integration of this double integral yields finite values. However, when proceeding analytically, it can be shown that the integral over $k$ is divergent. I was wondering whether an analytical evaluation of this double integral is amenable. Your help and hints are most welcome.
Thanks
HH