Integrals of the Bessel functions of the first kind

  • #1
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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]

and

[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.
 

Answers and Replies

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

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

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

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:

bessel.JPG
 

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