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Re: NASTY INTEGRAL. need advice

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106090] Re: [mg105986] NASTY INTEGRAL. need advice
  • From: Mark McClure <mcmcclur at unca.edu>
  • Date: Fri, 1 Jan 2010 05:34:51 -0500 (EST)
  • References: <200912290619.BAA02684@smc.vnet.net>

On Tue, Dec 29, 2009 at 1:19 AM, pianoman2008sg
<pianoman2008sg at 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


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