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
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