Hey! Do you guys know if there is a physical formulae for calculating the factorial of something? I know that for instance 3! = 1 x 2 x 3 = 6 but is there a formulae for that? I tried to come up with one but no success (YET! I''m a little optimistic Any way if anyone knows... Please post! Thanks for your input! ----------=Last Attacker=---------- E-mail: laextr@icqmail.com

The formula for n! is

         | n > 0 -> n x f(n-1)f(n) = | n = 0 -> 1       | n < 0 -> undefined

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[edited by - darookie on July 18, 2003 7:48:59 AM]

there is an aproximation formula (stirlington):

n! = (n/e)^n * sqrt(2*pi*n)


T2k

[edited by - T2k on July 18, 2003 7:50:52 AM]

Stirling's Formula : n! ~ sqrt(2πn) (n/e)ⁿ

Note that it is an equivalence, not an equality.

See also mathworld.

edit: reformulation & formatting.

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[edited by - Fruny on July 18, 2003 8:18:23 AM]

n! = 1 x 2 x 3 x ... x n is a formula (in fact it´s even the definition of the factorial; it derives from statistics and gives you the number of permutations of a given set of n elements) so what are you asking for ?

For very big numbers you can use the approximation ln(n!) = n(ln(n) -1). This should be an approximation of the so called Stirling Formula (try googling for it, the version I found in my book looks wrong). Problem with this is that I can´t really tell you if it´s useable with computers since it´s commonly used in analytical calculation using numbers that your computer can´t display (say n = 10^23 where 60! = 10^82, so guess (10^23)! ...).

If for some unknown reason you want to compute something like (2.3)! (the factorial of real numbers) you can use the gamma function. To qoute from my math book:

G(x>0) = Integrate(0 to infinity){ t^(x-1)*exp(-t) dt },

G(n+1) = n!
xG(x) = G(x+1)
( maximum respect if your trying came even close to this ;] )

One last word: Maybe this explains why I didn´t really get your question, but: What the hell is a physical formula ??? As far as I got it physics is a way to translate nature into a mathematical desription and applying boudary conditions to it that seem appropriate (setting inappropriate boundary conditions strange things might occur like you end up becoming famous ).

EDIT: Looks like some other people responded to your post while I was writing. I´d like to add something to their replies:
As far as I got it the Stirling Formular is not an approximation. It´s correct but it contains an infinite sum. The equations given above are approximations of it by taking the first summand and ignoring the rest of the sum. If I´m wrong about this feel free to correct me; I´m always interested in learning.

[edited by - Atheist on July 18, 2003 8:18:38 AM]

Wow!

I never thought that there even was going to be a formulae, but thanks very much you guys!

To me a formulae is not a short cut but a one-line execuation of an equasion to get to an answer, like 2*3 is a shortcut to get to 6 but is infact a non-loop solution to 2+2+2 to get to 6.

When I asked for a formulae for n! I hoped that it would be something like the 2*3 thing (above), to get an answer without going through a loop.

Thanks for your input!

----------=Last Attacker=----------

E-mail: laextr@icqmail.com

[edited by - Last Attacker on July 18, 2003 8:25:26 AM]

We are, the knights who say... n!

quote:

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Original post by Dredge-Master
We are, the knights who say... n!

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Good one !

Maybe I just skipped over it, but no one has mentioned the gamma function yet.
<a href="http://mathworld.wolfram.com/GammaFunction.html">here</a>

I think products are easier than integrals.

Brendan

It was mentioned, but as a solution to taking factorials of real numbers. But hey, if it works on real numbers it''ll work on integers too. I would love to see how the gamma function works. *clicks link quickly* =)