Re: Bessel K expansion, large argument?
- To: mathgroup at smc.vnet.net
- Subject: [mg70119] Re: Bessel K expansion, large argument?
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 4 Oct 2006 05:59:28 -0400 (EDT)
- Organization: The University of Western Australia
- References: <eftdhr$6b9$1@smc.vnet.net>
In article <eftdhr$6b9$1 at smc.vnet.net>, AES <siegman at stanford.edu> wrote: > 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 You have switched from z to w or do you mean something else here? > 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. Since z has magnitude larger than unity, expand about Infinity: Series[ z BesselK[ 1, z ] / BesselK[ 0, z ], {z, Infinity, 3} ] Normal[ % ] /. z -> w 1 25 1 1 w + - - ------ + ---- - --- 2 3 2 8 w 128 w 8 w Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul