FW: Numerical Hypergeometric Functions

["Daniel Lozier" <lozier@nist.gov>]

Sender: Nico.Temme@cwi.nl
Subject: Numerical Hypergeometric Functions



 >  Sender: David Strozzi <dstrozzi@MIT.EDU>
 >  Subject: Numerical Hypergeometric Functions
 >
 >
 >  I am looking for a way to numerically compute the hypergeometric
 >  function (the famous 2F1).  I need it for complex parameters (namely
the
 >  a and b in F(a,b;c;z) Abramowitz and Stegun notation) and
analytically
 >  continued outside the unit disk (along the negative real axis).
 >
 >  Any ideas?  Is there code for this on the web?  Is there a good
article
 >  or book that can be followed in a somewhat cookbook manner?
 >
 >  The function I need happens to be of the form F(a , b;
a+b; -real^2),
 >  and this can be "simplified" (A+S 15.3.10) to an infinite sum of Psi
 >  functions.
 >
 >  Thanks a lot.
 >
 >  --
 >  David Strozzi                             dstrozzi@mit.edu
 >  home: 472 Putnam Ave., Apt. 2B     office: Bldg. NW 16-223
 >  617-876-4929                                  617-452-2343
 >  Cambridge, MA 02139           http://www.mit.edu/~dstrozzi

David,

Let z_0=(1-sqrt(5)/2=-0.618...

You may use (15.1.1) when the argument z = -real^2 satsfies,
z_0 \le z \le 0 and (15.3.8) if z< z_0 (unless a=b \pm n).
Then the argument of the series is always less than |z_0|
and convergence is good, unless a and/or b are large.

Your choice 15.3.11 is good if z is close to unity.

The best paper on evaluating F(a,b;c;z) is
Robert C. Forrey
Computing the hypergeometric function
Journal of Computational Physics, Vol. 137, 79 - 100 (1997).

Regards,

	Nico Temme

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