From: mathar@mpia-hd.mpg.de (Richard Mathar)
Subject: Re: Modified Bessel Function: a name
Date: 8 Jan 2000 20:22:31 GMT
Newsgroups: sci.math.num-analysis

 "Pawel F. Gora" <gora@if.uj.edu.pl> wrote:
|> During some calculations I came across the function K_n(z), a 
|> solution to the modified Bessel equation. Mathematica evaluates
|> it as BesselK[n,z] and calls it a modified Bessel function of
|> the second kind; my Mathematica manual (written by some Polish
|> authors, so probably not the one most of you are using) additionally
|> calls it Macdonald function. Since I am interested in asymptotic
|> properties of this function (see below), I checked other sources
|> as well. Now, Bateman (H. Bateman, Higher Transcendental Functions, 
|> Vol. II, McGraw-Hill 1953) calls _the_same_ function a modified Bessel
|> function of the third kind, or Basset's function. Gradshteyn
|> and Ryzhik (Tables of Integrals etc) call it modified Hankel's
|> function.
|> 
|> I am a bit confused: What is then the name of this function?
|> Or perhaps, What name of this function is now most commonly used?
|> 
|> What I am really interested in is not the name, though, but the
|> asymptotics. How does K_n(z) behave for n real, z real, positive
|> and small (z --> 0+)? The function appears to have a pole at zero,
|> but does it diverge as z^{-s}? Or logarithmically? Or in a more
|> complicated manner? Bateman gives formulae for large |z|, and
|> so do Gradshteyn and Ryzhik; I can't find any sources for the
|> case I am interested in. Any references (on-line or published
|> work) will be greatly appreciated.

K_0(z)=-{ln(z/2)+\gamma}I_0(z)+(1/4)z^2+O(z^4)
where \gamma is Euler's constant=0.577..
This formula is 9.6.13 in the book by Abramowitz and STegun
who just call it a Modified Bessel Function. This is equivalent
to eq. 8.447.3 in the book by Gradsteyn and Ryzhik who refer to it as
"Cylinder function of an imaginary argument".
Physicists sometimes call it McDonald function (but they tend to use
some people's names who've indulged on their properties early, whereas
mathematicians tend to use the more qualified and coherent names...).
A good numerical implementation that avoids some hassle one may have with
extinction of digits caused by recurrence formulas is
the Cephes code in the netlib.
I_0(z) is the usual modified Bessel function of order 0.