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MathGroup Archive 2006

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Re: Bessel K expansion, large argument?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70120] Re: Bessel K expansion, large argument?
  • From: Joel Storch <jstorch at earthlink.net>
  • Date: Wed, 4 Oct 2006 05:59:35 -0400 (EDT)
  • References: <eftdhr$6b9$1@smc.vnet.net>

The fuction in question has an esential singularity at the origin.
To obtain the desired expansion, use the Mathematica command
     Series[z BesselK[1,z]/BesselK[0,z],{z,Infinity,2}]
and you will obtain: z+1/2-1/(8z)+1/(8z^2)

AES 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.
> 


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