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