Does anybody know if there is an analytical solution to the following integral of Bessel functions:
$$\int J_m^*(kx) \, J_m(kx) \, x \,dx,$$
where $m$ is integer and the problem is that $k$ may be complex and so the values of $J_m(kx)$?
I know the solution of similar integral $\int J_m(kx)^2 \, x \,dx = J_m(kx)^2-J_{m-1}(kx)\,J_{m+1}(kx) + C$, however I really need it for the situation where one Bessel function is conjugate (so the integral is always real).
(I have browsed all the textbooks available to me as well as the internet resources, and the question here is the last resort)