Re: Bessel K expansion, large argument?

• To: mathgroup at smc.vnet.net
• Subject: [mg70124] Re: Bessel K expansion, large argument?
• From: bghiggins at ucdavis.edu
• Date: Wed, 4 Oct 2006 05:59:45 -0400 (EDT)
• References: <eftdhr\$6b9\$1@smc.vnet.net>

Try the following:

f[z_] := z BesselK[1, z]/BesselK[0, z]

f2 = f[r Exp[\[ImaginaryI] Î¸]] /. r -> 1/p

Now let us expand this function about p=0

asympf[z_] = Normal[Series[f2, {p, 0, 5}]] /.
p^(x_.) -> 1/(z^x/E^(x*I*Î¸))

1/2 - 1073/(1024*z^5) + 13/(32*z^4) - 25/(128*z^3) +
1/(8*z^2) - 1/(8*z) + z

This expression can be written as

z + 1/2 - 1/(8*z) + (1/(8*z))^2 - 25/(128*z^3) + 13/(32*z^4) -
1073/(1024*z^5)

Let see how good it is

In[12]:=
f[10.*E^(I*(Pi/4))]

Out[12]=
7.562333392224384 + 7.078789829741519*I

In[25]:=
asympf[10.*E^(I*(Pi/4))]

Out[25]//InputForm=

7.562333868323298 + 7.078787343994053*I

More terms in the expansion gives improved accuracy. Hope this helps

Brian

PS: Note that the function has an esential singularity at p->0, so the
expansion is not strictly valid, but it works!

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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