product of different order Bessel function integral

Question

$\displaystyle w = \int_0^\infty r\; J_\mu(ar)\;J_\theta(br)\; \text{d}r $

I'd like to solve this integral ,where a and b are real and positive constant. any information regarding this integral help me alot.

Answer 1

WolframAlpha gives the integral (care to not confuse $a$ and $\alpha$ in the formula).
http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/21/02/02/0001/MainEq1.L.gif

In case of $r$ tending to infinity the function to be integrated doesn't tends to $0$ (equivalent shown below). So, the integral in not convergent : it contains a sinusoidal component.

If we deduct this non convergent component from the integral, the calculus at limit will be possible and the formula given for $\alpha<2$ should probably be extended to $\alpha=2$ (to be checked, I didn't it).