Re: BesselK problems at large argument?
- To: mathgroup at smc.vnet.net
- Subject: [mg78911] Re: [mg78860] BesselK problems at large argument?
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Thu, 12 Jul 2007 05:26:20 -0400 (EDT)
- Reply-to: hanlonr at cox.net
If I understand your integrals, they can be done symbolically.
$Version
5.2 for Mac OS X (June 20, 2005)
Clear[expr];
expr[n_,a_, w_]=Integrate[r^n * BesselK[0, w * r]^2,
{r, a, Infinity}, Assumptions->{w>0, a>0}]
(1/4)*Sqrt[Pi]*w^(-n - 1)*MeijerG[{{}, {1, (n + 2)/2}},
{{0, (n + 1)/2, (n + 1)/2, (n + 1)/2}, {}}, a^2*w^2]
Table[expr[1,a,w],{a,.5,1.5,.25},
{w,.5,1.5,.25}]
Table[expr[3, a, w], {a, .5, 1.5, .25},
{w, .5, 1.5, .25}]
Bob Hanlon
---- AES <siegman at stanford.edu> wrote:
> Optical fiber mode calculations, at least if done programmed
> straightforwardly, require evaluating integrals of r BesselK[0, w r]^2
> and r^3 BesselK[0, w r]^2 from a finite value r = a out to r = infinity,
> for smallish (order unity) values of w and a , using NIntegrate and with
> all quantities real.
>
> Trying to do this for different parameter values, however, I keep
> running into messages saying things like "Integral failed to converge to
> desired precision," or at other times
>
> BesselK[0, r] is not numerical at {r} = {5.`*^8}
>
> I don't think there's a programming error involved, because for some
> values of the parameters everything comes out just fine.
>
> I'm aware that BesselK[ ] has a simple asymptotic form at large r , and
> I suppose I could mess around trying to switch to it at large r , or
> figure out some way to truncate the integral at some large outer limit.
>
> But, hey, BesselK is a nice smooth, non-oscillatory (and long known)
> function, which decreases very rapidly at large r. Shouldn't these
> integrals work OK -- or more important, is there an option I can set to
> make them go OK?
>
> [This is still 5.2 on Mac OS 10.3.9.]
>