# Re: factorial

```Is there a Ti that can do the gamma function?  Hp's have had that for a
decade!  Is there a Ti-92 owner who can/can't do it?

Thanks,
Mark P. Wilson
E-mail:  im1077@exmail.usma.edu

> ----------
> From:         David Starr[SMTP:burke@THERAMP.NET]
> Sent:         Saturday, January 24, 1998 3:48 PM
> To:   CALC-TI@LISTS.PPP.TI.COM
> Subject:      Re: factorial
>
> Michael Xu Wang wrote:
> >
> > does anyone know how the calculators calculate the factorial of a
> > floating number, like 2.1!=2.19762...?
>
> The factorial only has the non-negative integers as a domain, but the
> gamma function is often used to calculate "factorials" of numbers
> which
> can't be determined using the typical n!=n*(n-1)*(n-2)*...*3*2*1.  The
> gamma function is defined at all points on the real number line except
> the negative integers (it may also be used to compute the factorial of
> complex arguments).  The gamma function is defined as
> Gamma(x)=Int[0...infinity,t^(x-1)*e^(-t)dt].  Using some integration
> by
> parts you can easily see that Gamma(x+1)=x*Gamma(x), which for
> positive
> integers is the recursive definition of the factorial function.  Using
> this, the relationship between factorial and gamma may be expressed as
> Gamma(x+1)=x!.  Since
> Int[0...infinity,e^(-t)*t^(-1/2)dt]=2*Int[0...infinity,e^(-x^2)dx],
> Gamma(1/2)=sqrt(pi).  Using this with the recursive formula, any
> numbers
> which are multiples of 1/2 (e.g., -5/2, 7/2) may have their gamma
> values
> computed in terms of sqrt(pi).  As an example
> (1/2)!=Gamma(3/2)=1/2*Gamma(1/2)=Sqrt(pi)/2.  The gamma function can
> be
> expressed in many other ways.  Two of the most common are
> Gamma(x)=Limit[n->infinity,n!*n^x/(x*(x+1)*(x+2)*...*(x+n-1)*(x+n))]
> and
> 1/Gamma(x)=x*e^(_gamma_*x)*ProductSummation[n=1...infinity,(1+x/n)*e^(
> -x/n)],
> where _gamma_ is Euler's constant (defined as -d/dx(Gamma(1))), which
> may be expressed as Limit[n->infinity,Sum[k=1...n,1/k]-ln(n)].  It is
> approximately 0.5772156649....  (It may be rational or irrational-no
> one
> yet knows.)
>
> -David-
>

```