Re: factorial
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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]
> Reply To: 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-
>