Re: Bessel K expansion, large argument?

*To*: mathgroup at smc.vnet.net*Subject*: [mg70125] Re: [mg70088] Bessel K expansion, large argument?*From*: Carl Woll <carlw at wolfram.com>*Date*: Wed, 4 Oct 2006 05:59:51 -0400 (EDT)*References*: <200610031016.GAA06359@smc.vnet.net>

AES 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 > >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. > > In version 5.2 both: Series[z BesselK[1, z]/BesselK[0, z], {z, Infinity, 2}] and z BesselK[1, z]/BesselK[0, z] + O[z, Infinity]^3 produce 1/(1/z) + 1/2 - 1/(8*z) + (1/8)*(1/z)^2 + O[z,Infinity]^3 Carl Woll Wolfram Research

**References**:**Bessel K expansion, large argument?***From:*AES <siegman@stanford.edu>