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Conditional convergence of J1(kr)k

  1. Sep 3, 2009 #1
    Hey!

    I need to perform the following integration:

    [tex]\int\limits_0^{\infty} J_1(k r)k dk[/tex]

    where [tex]J_1(x)[/tex] is the cylindrical Bessel function of the first kind. This is an oscillatory function with amplitude decreasing as [tex]1/\sqrt{x}[/tex]. Due to the multiplication of [tex]k[/tex] the integrand however, becomes an oscillatory function which increases as a function of [tex]x[/tex]. How can I perform this integration? I was thinking about multiplying the integrand by [tex]e^{-k d}[/tex], performing the integration and then taking the limit as [tex]d[/tex] approaches zero, but I can't figure out how to evaluate the integral.

    I hope someone has a suggestion.

    René
     
  2. jcsd
  3. Sep 3, 2009 #2

    jasonRF

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    Repetit

    Are you sure this integral converges? I am skeptical. If you think about the integral as a sequence of integrals between each zero crossing, you need to sum an alternating series. Let [tex] a_n [/tex] denote the [tex] n^{th}[/tex] term in the series. Since [tex]lim_{n\rightarrow \infty} a_n \neq 0 [/tex] the series doesn't converge. This isn't really a proof - I leave that to you. But I think you need to prove that this integral converges before you spend much time trying to evaluate it!

    Jason
     
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