Re: Bessel K expansion, large argument?
- To: mathgroup at smc.vnet.net
- Subject: [mg70128] Re: Bessel K expansion, large argument?
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Wed, 4 Oct 2006 06:00:14 -0400 (EDT)
- Organization: Universitaet Hannover
- References: <eftdhr$6b9$1@smc.vnet.net>
AES schrieb:
> The function
>
> z BesselK[ 1, z ] / BesselK[ 0, z ]
>
> with z complex, magnitude several times unity or larger, and argument
> between -90 and 90 degrees, appears in optical fiber mode calculations.
>
> Experience shows that a quite good approximation to this is just
>
> w + 1/2
>
> Can anyone suggest a next term or two in the expansion, e.g.
>
> w + 1/2 + a/w + b/w^2 ???
>
> Been trying to get Mathematica to tell me this, but not figuring out how
> to get the Series command to do what I want.
>
Series expansion at infinity is a bit noisy
Normal@Series[1/z BesselK[1, 1/z]/BesselK[0, 1/z], {z, Infinity, 4}]
[...]
so do it by inversion z->1/z at 0
Normal@Series[1/z BesselK[1, 1/z]/BesselK[0, 1/z], {z, 0, 4}]
1/2 + 1/z - z/8 + z^2/8 - (25*z^3)/128 + (13*z^4)/32/.z->1/w
1/2 + w - 1/(8*w) + 1/(8*w^2) - 25/(128*w^3) + 13/(32*w^4)
--
Roland Franzius