Re: Bessel K expansion, large argument?
- To: mathgroup at smc.vnet.net
- Subject: [mg70221] Re: Bessel K expansion, large argument?
- From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
- Date: Sat, 7 Oct 2006 07:09:19 -0400 (EDT)
- References: <email@example.com>
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 > > 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. Several people have already mentioned using (1) w + 1/2 - 1/(8w) + 1/(8w^2). But you might well be interested to know that there is an approximation which, compared to (1), is both simpler and more accurate when |w| is large: (2) w + 1/2 - 1/(8(w+1)). [For large |w|, the errors in (1) and (2) are approximately 25/(128w^3) and 9/(128w^3), resp.] David W. Cantrell