Integration of Bessel's functions

[tworitdash]

TL;DR Summary

For implementing a mode-matching technique in EM simulation, I want to get a closed-form equation of the integral of $$\int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho$$.

I can only find a solution to $$\int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho$$ with the Lommel's integral . The closed form solution to $$\int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho$$ I am not able to find anywhere. Is there any way in which I can approach this problem from scratch? Here, $$J_m$$ is the Bessel function of the first kind of order m.

 

[fresh_42]

I would try the integral representations:
https://en.wikipedia.org/wiki/Bessel_function#Bessel's_integrals

If you find the proof for Lommert, it might be that it can be adapted to $x^{-1}$ instead of $x$.

 

[Haborix]

It looks to me like you could use a recurrence relation (see this link) once or twice and arrive at a sum of integrals in the form you know.

 

[tworitdash]

It looks to me like you could use a recurrence relation (see this link) once or twice and arrive at a sum of integrals in the form you know.

Perfect. Thanks! I decomposed my problem and I got the form of Lommel's integrals for all my problems. I verified with numerical solutions for my EM problems and the Lommel's integrals work like magic.

 

[Haborix]

Good to hear! Special functions, when they are useful, always seemed like magic (rigorous magic) to me.