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Jonas
science forum addict
Joined: 08 Sep 2005
Posts: 77
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 Posted: Thu Jun 22, 2006 3:02 pm Post subject:
Expansion of Factorial of a Fraction
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I looked on the internet and in my textbooks but I can't find what the
expansion of a factorial of a fraction looks like. I know that
5!=(5)(4)(3)(2)(1) but what is (1/2)!. Is the definiton of
n!=n(n-1)(n-2)(n-3)....(2)(1)?
I think that in order to get a complete understanding of how to deal
with (1/2)!, I'll have to review the gamma function but I would still
like to know how (1/2)! can be expanded.
Responses are appreciated. |
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Jonathan Cass
science forum beginner
Joined: 22 Jun 2006
Posts: 2
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| Posted: Thu Jun 22, 2006 3:06 pm Post subject:
Re: Expansion of Factorial of a Fraction
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As far as I know, (1/2)! doesn't have meaning in and of itself. It is only
when the correlation between factorials and the gamma function is realized
that one can begin to understand (1/2)!. Go straight to the gamma function
if you really want to see what fractional factorials look like because they
don't expand in any traditional sense.
-Jon
"Jonas" <sundet@yahoo.com> wrote in message
news:1150988556.551281.66430@y41g2000cwy.googlegroups.com...
| Quote: | I looked on the internet and in my textbooks but I can't find what the
expansion of a factorial of a fraction looks like. I know that
5!=(5)(4)(3)(2)(1) but what is (1/2)!. Is the definiton of
n!=n(n-1)(n-2)(n-3)....(2)(1)?
I think that in order to get a complete understanding of how to deal
with (1/2)!, I'll have to review the gamma function but I would still
like to know how (1/2)! can be expanded.
Responses are appreciated.
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Denis Feldmann
science forum addict
Joined: 23 Apr 2006
Posts: 87
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| Posted: Thu Jun 22, 2006 3:33 pm Post subject:
Re: Expansion of Factorial of a Fraction
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Jonas a écrit :
| Quote: | I looked on the internet and in my textbooks but I can't find what the
expansion of a factorial of a fraction looks like. I know that
5!=(5)(4)(3)(2)(1) but what is (1/2)!. Is the definiton of
n!=n(n-1)(n-2)(n-3)....(2)(1)?
I think that in order to get a complete understanding of how to deal
with (1/2)!, I'll have to review the gamma function
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This must be a new meaning of "review" I was not aware of.
but I would still
| Quote: | like to know how (1/2)! can be expanded.
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The fact that (1/2)!=sqrt(pi)/2 could give you a hint that your
defnition doesn't apply well in that case...
| Quote: |
Responses are appreciated.
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Ioannis
science forum Guru Wannabe
Joined: 24 Mar 2005
Posts: 246
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| Posted: Thu Jun 22, 2006 3:45 pm Post subject:
Re: Expansion of Factorial of a Fraction
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"Jonas" <sundet@yahoo.com> wrote in message
news:1150988556.551281.66430@y41g2000cwy.googlegroups.com...
| Quote: |
I looked on the internet and in my textbooks but I can't find what the
expansion of a factorial of a fraction looks like.
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That's only natural because the factorial function is not defined at such
values.
| Quote: | I know that
5!=(5)(4)(3)(2)(1) but what is (1/2)!. Is the definiton of
n!=n(n-1)(n-2)(n-3)....(2)(1)?
I think that in order to get a complete understanding of how to deal
with (1/2)!, I'll have to review the gamma function but I would still
like to know how (1/2)! can be expanded.
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The abused notation "a!", for a in R\[N U {0}] is nonsense, *unless* it is
understood that if it has to have any meaning, such a meaning must come from
the Gamma function which is the analytic continuation of the factorial
function.
In a similar spirit: The natural power function is defined for a in R and n
in N, as:
a^n = {a, iff n=1}
{a*a^{n-1}, iff n>1} [1]
Informally,
a^n = a*a*...*a (n-times) [2]
Similarly, you cannot "expand" a^{sqrt{2}}, into a form similar to [2],
because [1] is not defined for n in R\N.
It can make sense only through the real power function,
a^x = exp(log(a)*x),
by considering appropriate branches of the complex "log" function, when a <
0.
| Quote: | Responses are appreciated.
-- |
Ioannis |
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William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
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| Posted: Fri Jun 23, 2006 3:42 am Post subject:
Re: Expansion of Factorial of a Fraction
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On Thu, 22 Jun 2006, Jonas wrote:
| Quote: | I looked on the internet and in my textbooks but I can't find what the
expansion of a factorial of a fraction looks like. I know that
5!=(5)(4)(3)(2)(1) but what is (1/2)!. Is the definiton of
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Look up the Gamma function that gives x! for all x except
negative integers (which have +oo factorials).
Gamma(t) = integral(0,oo) x^(t-1) e^-x dx, t /= 0, -1, -2, ...
Gamma(n) = (n-1)! when n in N
(-1/2)! = Gamma(1/2) = sqr pi
(1/2)! = Gamma(3/2) = (sqr pi)/2 |
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