Can the Divergence of a Bessel Integral Be Prevented?

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Discussion Overview

The discussion revolves around the convergence of a Bessel integral, specifically the integral of the form Integrate[ (1/r) * J[0,2*pi*phi*r] ] from 0 to ∞ with respect to r. Participants explore whether it is possible to prevent the integral from diverging, considering different orders of Bessel functions and the implications of the limits of integration.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses concern about the divergence of the integral and seeks insights on preventing it.
  • Another participant provides a calculation suggesting that the integral is convergent for a second order Bessel function, but later corrects themselves to indicate they meant a zeroth order Bessel function.
  • A different participant asserts that the integral is divergent when the lower boundary is set to 0, stating that nothing can be done to change this fact.
  • There is a suggestion that to obtain a finite value, one must choose a lower boundary greater than 0.

Areas of Agreement / Disagreement

Participants do not reach consensus on whether the integral can be made to converge, with some arguing it is divergent while others present calculations suggesting convergence under different conditions.

Contextual Notes

The discussion highlights the dependence on the order of the Bessel function and the limits of integration, which are critical to the convergence of the integral.

tx213
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Hi,

I would like to confirm my intuition about a bessel integral from you guys.

The integral is: Integrate[ (1/r) * J[2,2*pi*phi*r] ] from 0 → ∞ with respect to r.

J[2,2*pi*phi*r] is a second order bessel. Integrals with 1/x from 0 to Inf are divergent. Sure enough, this one is going to diverge so mathematica says. But is there anything I might be able to do to stop this integral from diverging?

Thanks in advance for any insight!
 
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Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2
 
JJacquelin said:
Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2

Ah, oops I'm sorry. I had meant a 0th order bessel, not 2nd order! It's J[0,2*pi*phi*r]. =(
 
In this case, the integral is divergent when the lower boundary for r is =0. That is a fact and you can do nothing against a fact.
If you want to obtain a finite value, you have to chose a boundary higher than 0.
 

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