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