I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$):
$$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$
The $e^{-jk\sigma}$ term is making my usual approach to the problem ineffective. However, unless I am mistaken, this same term makes the above a forward Fourier transform where $\sigma$ would be the frequency term and $k$ the time/space term:
$$\rho^{-1} \int_{k_1}^{k_2} \rho k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k = \rho^{-1} \mathcal{F}_\sigma \left( \rho k J_n(\rho k) \right) $$
I've been looking for this transform pair (i.e., $\mathcal{F}_x(xJ_n(x))$ but have not been able to find anything. Is there a analytic solution to this?
I don't know if this helps but first try solving
$I(\sigma):=\int e^{-jk\sigma} J_n(\rho k)dk$
it is probably easier. Then use the fact that your integral above is simply $j\frac{dI}{d\sigma}$.