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MathGroup Archive 2006

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Re: Bessel K expansion, large argument?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70127] Re: Bessel K expansion, large argument?
  • From: "sashap" <pavlyk at gmail.com>
  • Date: Wed, 4 Oct 2006 06:00:11 -0400 (EDT)
  • References: <eftdhr$6b9$1@smc.vnet.net>

In v 5.2:

In[1]:=
Series[z*(BesselK[1, z]/BesselK[0, z]),
  {z, Infinity, 3}]

<messages suppressed>

Out[1]=
SeriesData[z, Infinity, {1, 0, 1/2, 0, -1/8, 0, 1/8,
   0, -25/128}, -2, 7, 2]

thus the answer

 w + 1/2 - 1/8 1/w +1/64/w^2 + ...

Oleksandr Pavlyk
Special Functions Developer
Wolfram Research


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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