Re: Bessel K expansion, large argument?

• To: mathgroup at smc.vnet.net
• Subject: [mg70128] Re: Bessel K expansion, large argument?
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Wed, 4 Oct 2006 06:00:14 -0400 (EDT)
• Organization: Universitaet Hannover
• References: <eftdhr\$6b9\$1@smc.vnet.net>

```AES schrieb:
> 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.
>

Series expansion at infinity is a bit noisy

Normal@Series[1/z  BesselK[1, 1/z]/BesselK[0, 1/z], {z, Infinity, 4}]

[...]

so do it by inversion z->1/z at 0

Normal@Series[1/z  BesselK[1, 1/z]/BesselK[0, 1/z], {z, 0, 4}]

1/2 + 1/z - z/8 + z^2/8 - (25*z^3)/128 + (13*z^4)/32/.z->1/w

1/2 + w - 1/(8*w) + 1/(8*w^2) - 25/(128*w^3) + 13/(32*w^4)

--

Roland Franzius

```

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