Finding the Factorial of Non-Integers

Date: 01/06/2005 at 09:59:13
From: Raffaele
Subject: Fractions and factorials

Hi,

I know that non-negative integers can be computed as factorials.  For 
example, 5! = 5*4*3*2*1.  I have no problem with this but I notice on 
my calculator that if I type in say 3.5! it would return a defined 
number.

If you can find the factorial of non-negative REAL numbers then how do
you calculate them and is it possible to think about them geometrically?

Thanks in advance,

Raffaele

Date: 01/06/2005 at 11:07:22
From: Doctor Vogler
Subject: Re: Fractions and factorials

Hi Raffaele,

Thanks for writing to Dr Math.  That's a good question.  Technically,
the factorial only has meaning for nonnegative integers (0, 1, 2, 3,
etc.).  But there is a smooth function which is defined for all
positive real numbers (in fact, for all real and complex numbers
except the negative integers) and equals the factorial for all
positive integers.  The function is called the Gamma Function and
satisfies

  gamma(n + 1) = n!

for all nonnegative integers n.  So some computers and calculators
will, when asked to compute x!, prefer to give

  gamma(x + 1)

instead of saying "ERROR" (or "E"), since people don't like to get
errors.  For more information on the gamma function, search our
archives for "gamma function" or look at the MathWorld site at

  Gamma Function
    http://mathworld.wolfram.com/GammaFunction.html 

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/