# Integration of Bessel's functions

• A
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.

Last edited:

fresh_42
Mentor
• 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.

• 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.

• vanhees71 and Haborix
Good to hear! Special functions, when they are useful, always seemed like magic (rigorous magic) to me.

• tworitdash