1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integrals of the Bessel functions of the first kind

  1. Jun 26, 2015 #1
    Hi Physics Forums.

    I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck.

    [itex]f(x,a) = \int_0^\infty\frac{t\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]


    [itex]g(x,a) = \int_0^\infty\frac{t^2\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]

    where [itex]J_0(x)[/itex] is the Bessel function of the first kind.
  2. jcsd
  3. Jun 26, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    This one is 6.532.4 in Gradshteyn and Ryzhik, equal to ##K_0(ax)## for ##a>0, \text{Re}~x>0##.

    I think this one falls into a class of integrals solved in Watson's A Treatise on the Theory of Bessel Functions, expressible as hypergeometric functions. I attached the relevant page:

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Integrals Bessel functions Date
A Lebesgue measure and integral Jan 14, 2018
I Integration above or below axis Dec 17, 2017
Integral representation of modified Bessel function of the second kind Nov 9, 2010
Integrals with bessel functions Mar 4, 2009