Factorial on TI-83

Date: 09/24/97 at 09:37:51
From: Tom Carrigan
Subject: Definition of factorial on TI-83

What is the origin of the TI-83 definition of factorial (in increments
of 0.5) and where is it used? What resources would you suggest?

Date: 09/24/97 at 13:49:29
From: Doctor Bombelli
Subject: Re: Definition of factorial on TI-83

It depends on how advanced you want to get. There is a function called 
the Gamma Function defined in various ways:

For x>0, Gamma(x)=Integral{x=0 to infinity} t^x-1 e^-t  dt 

This function has a property called the Factorial property, namely 
that Gamma(x+1)=xGamma(x). By rewriting the factorial property as 
Gamma(x)=1/x Gamma(x+1) you can actually define the Gamma Function for 
x<0  [[if -1<x<0, Gamma(x+1) is defined, so Gamma(x)=1/x Gamma(x+1).  
Now if -2<x<-1, Gamma(x+1) is defined, so Gamma(x)=1/x Gamma(x+1), 
etc.]]

The function can also be defined as

Gamma(x)=e^(-bz) Product{n=1 to infinity} (1+z/n)^-1  e^(z/n)
(where b makes Gamma(1)=1)

or as

lim{n->infinity}  n!n^(z+1)/[z(z+1)...(z+n)]

Note that Gamma(n+1)=n! (let z=1 in the Factorial property), so the 
Gamma function generalizes the factorial function.

The fact that all the defintions are equivalent on their domains and 
that the factorial property holds are not easy!

With the definition(s), Gamma(1/2)=Sqrt[Pi], which may explain why the 
TI-83 uses the definition it does.  For those without a TI-83 manual 
the definition is

(n+1)!=n*n! recursively until n=0 or -1/2 and then 0!=1 and (-1/2)!=
Sqrt[Pi]

n!=n(n-1)(n-2)...2.1  if n>0 and an integer

n!=n(n-1)(n-2)...2.1 .Sqrt[Pi]  if n+1/2 >0 and an integer.

You can find information in most books on complex variables/analysis 
or in books about "advanced engineering mathematics".

-Doctor Bombelli,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 09/26/97 at 09:25:34
From: Tom Carrigan
Subject: thanks

Dear Dr. Math, 

Thanks very much for the very helpful answer to our question. 

Tom Carrigan, Library/Media 
Fred Floyd, Math 

Fox Lane High School
Bedford, New York