From: "Charles H. Giffen" <chg4k@virginia.edu>
Subject: Re: Bessel function algoritm
Date: Tue, 06 Jul 1999 14:19:35 -0400
Newsgroups: sci.math
To: normanfowler@my-deja.com

normanfowler@my-deja.com wrote:
> 
> Does anyone know of a good algorithm for
> numerically calculating standard Bessel
> functions?  In particular, in BASIC or Visual
> Basic?
> 
> Any help would be appreciated.
> 
> Sincerely,
> 
> Norman Fowler
> 
> Sent via Deja.com http://www.deja.com/
> Share what you know. Learn what you don't.

One has the asymptotic expansion

J[p](x) = (2/(\pi x))^{1/2}\cos( x - \pi/4 - p\pi/2 ) 

            + r(x)/x^{3/2},

where  r(x)  is bounded as  x --> infinity.

This suggests using the series expansion of  J[p](x)  for
|x| less than some constant and using the above asymptotic
expansion for large  |x|  (how large the constant is will
be determined by the accuracy desired in the computation of
J[p](x) ).

Also, for non-negative integers  p, one has

	J[1](x) = -J[0]'(x)
and
	J[p+1](x) = 2pJ{p](x)/x - J[p-1](x),

so by computing first just  J[0](x) and J[1](x), one can
compute  J[p](x)  recursively.

--Chuck Giffen