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MathGroup Archive 2006

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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

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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