(Comp.sys.hp48)
Item: 625 by edp@alien.enet.dec.com
Author: [Eric Postpischil]
  Subj: The factorial (gamma) function
  Date: Fri Feb 14 1992

In article <1992Feb11.201938.11196@athena.cs.uga.edu>,
rollie@marie.stat.uga.edu (Rollie Smith) writes...

> The factorial function on the hp48 does something for non integers.

Mathematicians have something called the gamma function, which is
written with the capital Greek letter gamma.  The definition is:
gamma(x) = integral from 0 to infinity of e^(-t)*t^(x-1) dt.  It so
happens that gamma(x+1) = x! for integers x.  Also, gamma(x+1) =
x*gamma(x) even for non-integers.  Anyway, when you use x! for
non-integers, you get gamma(x+1).

The gamma function turns up in the volume of a sphere in a general
number of dimensions.  The volume of a "sphere" with radius r in n
dimensions is r^n*pi^(n/2) / gamma(n/2+1).  If you evaluate that for n
= 2 and 3, you will get the formulae for the area and volume of a
circle and sphere, respectively.

If you ever have a product such as 1*3*5*7*... or 2*5*8*11*... that
you want to evaluate, the gamma function might be useful.  Since x! =
x*(x-1)!, (7/2)! = (7/2)*(5/2)*(3/2)*(1/2)*(-1/2)!, so 1*3*5*7 = 2^4 *
(7/2)! / (-1/2)!. Similarly, 2*5*8*11*...*(3*i-1) = 3^i * ((3*i-1)/3)!
/ (-1/3)!.


                                       -- edp (Eric Postpischil)
                                       "Always mount a scratch monkey."
                                       edp@alien.enet.dec.com