Integral of 2 Bessel functions of different orders

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Discussion Overview

The discussion revolves around the integral of two Bessel functions of different orders, specifically the expression \(\int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho\). Participants explore potential solutions, particularly focusing on cases where the orders \(m\) and \(n\) differ.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant mentions finding a solution using Lommel's integral for the case when \(m = n\) but struggles with the case when \(m \neq n\), noting that it leads to hypergeometric functions.
  • Another participant questions the validity or usefulness of hypergeometric functions in this context.
  • A later reply indicates that a hypergeometric function was found specifically for the case where the orders are \(m - 1\) and \(m + 1\), but no solutions have been identified for other arbitrary cases.
  • One participant claims to have found a solution when \(a = b\), but states that no solution exists otherwise.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of hypergeometric functions and the conditions under which solutions can be found. There is no consensus on a general solution for the case when \(m \neq n\).

Contextual Notes

The discussion highlights limitations in finding solutions for arbitrary orders of Bessel functions and the dependence on specific relationships between the orders.

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TL;DR
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 \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho
with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (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 m != nHere, J_m is the Bessel function of the first kind of order m.
 
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What's wrong with hypergeometric functions?
 
phyzguy said:
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.
 
I was able to find a solution when a=b, otherwise no.
 

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