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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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