Factorial on TI-83Date: 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 |
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