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Bessel integral divergence

  1. Feb 11, 2014 #1

    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 gonna 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!
  2. jcsd
  3. Feb 12, 2014 #2
    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
  4. Feb 12, 2014 #3
    Ah, oops i'm sorry. I had meant a 0th order bessel, not 2nd order! It's J[0,2*pi*phi*r]. =(
  5. Feb 12, 2014 #4
    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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