NASTY INTEGRAL. need advice

[pianoman2008sg]

hello guys,
i wonder if somebody can help me. i have come across a "nasty" looking
integral that contains two modified bessel functions.
see below:

Integrate:
[( k^2 + c^2)^(0.5 p) * B1 * B2, {k, -\[Infinity], + \[Infinity]} ],
where

B1 =Bessel K[ b + 1/2, m*Sqrt (k^2 + c^2)]

B2 = Bessel K[ a + 1/2, n*Sqrt (k^2+c^2)]

a,b,c,m& n are constants.

any advise on how to do this on mathematica will be most
appreciated!
cheers
pm

[Mark McClure]

On Tue, Dec 29, 2009 at 1:19 AM, pianoman2008sg
<pianoma...@gmail.com> wrote:
> hello guys,
> i wonder if somebody can help me. i have come across a "nasty" looking
> integral that contains two modified bessel functions.
> see below:

I see no reason to believe that this integral can be done
analytically. Is there any reason
a numerical approximation won't do? As a function of the parameters,
you could set up
such an approximation like so:

f[a_?NumericQ, b_?NumericQ, c_?NumericQ, m_?NumericQ,
n_?NumericQ, p_?NumericQ] := NIntegrate[(k^2 + c^2)^(0.5 p)*
BesselK[b + 1/2, m*Sqrt[k^2 + c^2]]*
BesselK[a + 1/2, n*Sqrt[k^2 + c^2]],
{k, -Infinity, Infinity}]

While you don't have a formula for f, you can treat it like an
ordinary function in many ways.
For example, here's a plot of f as a function of the variable p:

Plot[f[1, 1, 1, 1, 1, p], {p, -2, 2}]

Here's a plot of the derivative of f with respect to p:

Plot[Derivative[0, 0, 0, 0, 0, 1][f][1, 1, 1, 1, 1, p], {p, -2, 2}]

Have fun,
Mark