Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Conditional convergence of J1(kr)k

  1. Sep 3, 2009 #1

    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.

  2. jcsd
  3. Sep 3, 2009 #2


    User Avatar
    Science Advisor
    Gold Member


    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!

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook