Integral of 2 Bessel functions of different orders

  • #1
68
14
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} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho [/tex]
with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when [tex] m = n[/tex] (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel functions.). However, all that I get with the problem having Bessel's functions with different orders is some hyper-geometric functions. Is there any other way to solve it when [tex] m != n[/tex]


Here, [tex] J_m [/tex] is the Bessel function of the first kind of order m.
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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What's wrong with hypergeometric functions?
 
  • #3
68
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What's wrong with hypergeometric functions?
Well, I just realized I got a hypergeometric function when the orders are of the form m - 1 and m + 1. For any other arbitrary case, I haven't seen any solution with hypergeometric functions.
 
  • #4
phyzguy
Science Advisor
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1,828
I was able to find a solution when a=b, otherwise no.
 

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