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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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