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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: <eftdhr$6b9$1@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
>
> 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