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Re: Bessel K expansion, large argument?


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.
>  
>
In version 5.2 both:

Series[z BesselK[1, z]/BesselK[0, z], {z, Infinity, 2}]

and

z BesselK[1, z]/BesselK[0, z] + O[z, Infinity]^3

produce

1/(1/z) + 1/2 - 1/(8*z) + (1/8)*(1/z)^2 + O[z,Infinity]^3

Carl Woll
Wolfram Research


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