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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


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    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:

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