Integration of Bessel's functions

  • #1
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Summary:
For implementing a mode-matching technique in EM simulation, I want to get a closed-form equation of the integral of [tex]\int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho [/tex].
I can only find a solution to [tex] \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho [/tex] with the Lommel's integral . The closed form solution to [tex] \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho [/tex] I am not able to find anywhere. Is there any way in which I can approach this problem from scratch? Here, [tex] J_m [/tex] is the Bessel function of the first kind of order m.
 
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Answers and Replies

  • #3
183
134
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.
 
  • #4
68
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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.
 
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Likes vanhees71 and Haborix
  • #5
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Good to hear! Special functions, when they are useful, always seemed like magic (rigorous magic) to me.
 

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