Non-integer factorials?


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Profile Sergej O. S.
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Message 824881 - Posted: 30 Oct 2008, 11:25:01 UTC

Hi. I'm writing a program (as one of my hobbies) that calculates factorials. But people are telling me that non-integer factorials have no use - the Gamma function instead - G(z)=(z-1)!.

Is it true that non-integer factorials have no use?
And if they have, then how, where?

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Message 824914 - Posted: 30 Oct 2008, 13:57:18 UTC - in response to Message 824881.

My intuition tells me that rational, non-integer factorials would be useful in determining series convergences--albeit with some fiddling with scale factors.

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BILL

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Message 824961 - Posted: 30 Oct 2008, 17:59:12 UTC

So, what is 2.5!? 1 x 2 x (3)^1/2? Probably not, because that last factor is probably calculated by a different rule. Maybe non-integer factorials never have been defined.
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Message 824977 - Posted: 30 Oct 2008, 18:48:08 UTC - in response to Message 824914.

My intuition tells me that rational, non-integer factorials would be useful in determining series convergences--albeit with some fiddling with scale factors.

Regards,

BILL

Thank you. I'll try to find the formula for practical use on that field...

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Message 824979 - Posted: 30 Oct 2008, 18:53:43 UTC - in response to Message 824961.

So, what is 2.5!? 1 x 2 x (3)^1/2? Probably not, because that last factor is probably calculated by a different rule. Maybe non-integer factorials never have been defined.

Is 96% true. Exactly, 2.5! = 0.5 * 1.5 * 2.5 * pi^0.5

Unfortunately, values other than integer + 0.5 doesn't work that way, or I don't see how...

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Message 825010 - Posted: 30 Oct 2008, 21:31:15 UTC - in response to Message 824881.
Last modified: 30 Oct 2008, 21:33:31 UTC

Hi. I'm writing a program (as one of my hobbies) that calculates factorials. But people are telling me that non-integer factorials have no use - the Gamma function instead - G(z)=(z-1)!.

Is it true that non-integer factorials have no use?
And if they have, then how, where?


The gamma function is the generalization of the factorial function.

Some gadgets and formulas:
[tex]
n! = \Gamma(n+1)
\Gamma(p) = \int_{0}^{\infty}e^{-x}x^{p-1}\,\text{d}x
\Gamma(p+1) = p*\Gamma(p)
\Gamma(1/2) = \sqrt{\pi}
[/tex]

2.5!? Hmm. Let's see:
[tex]
2.5! = \Gamma(2.5+1) = \Gamma(3.5) = \Gamma(7/2) = (5/2)\Gamma(5/2) = (5/2)(3/2)\Gamma(3/2) = (5/2)(3/2)(1/2)\Gamma(1/2) = (5/2)(3/2)(1/2)\sqrt{\pi} = (15/8)\sqrt{\pi} \approx 3.3233509704478425511840640312646.
[/tex]

Henri.
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Message 825017 - Posted: 30 Oct 2008, 21:40:11 UTC - in response to Message 824979.

Unfortunately, values other than integer + 0.5 doesn't work that way, or I don't see how...


[tex]
0.6\Gamma(0.6) = \Gamma(0.6+1) = \Gamma(1.6) = from the table book = 0.89352.
[/tex]

[tex]
\Gamma(0.6) = \Gamma(0.6+1)/0.6 = (1/0.6)\Gamma(1.6) = (1/0.6)*0.89352 = 1.4892.
[/tex]

[tex]
\Gamma(-1/2) = \Gamma(-(1/2) + 1)/(-1/2) = -2\Gamma(1/2) = -2\sqrt{\pi}.
[/tex]

Henri.
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Message 825042 - Posted: 30 Oct 2008, 22:37:11 UTC - in response to Message 825017.
Last modified: 30 Oct 2008, 22:40:08 UTC

Yes, Henri.
My program is fairly precise by now: 17-18 digits. Not as much as Windows Calculator (32), but does it instantly for big numbers.

By the way, you don't know how to calculate nagative factorials? Or how to convert from positive ones?

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Message 825661 - Posted: 1 Nov 2008, 10:38:46 UTC - in response to Message 825042.
Last modified: 1 Nov 2008, 10:39:36 UTC

how to convert from positive ones?

Oh, I did found that
-z! = pi/sin(-z*pi)*(-z)/z!

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Message 826126 - Posted: 2 Nov 2008, 16:54:41 UTC - in response to Message 825042.

Hi!

By the way, you don't know how to calculate nagative factorials? Or how to convert from positive ones?


[tex]
\Gamma(-1/2) = \Gamma(-(1/2) + 1)/(-1/2) = -2\Gamma(1/2) = -2\sqrt{\pi}.
[/tex]

[tex]
\Gamma(-1.4) = \Gamma(-1.4+1)/-1.4 = -\Gamma(-0.4)/1.4 = -\Gamma(-0.4+1)/1.4(-0.4) = (1/(1.4*1.4))\Gamma(0.6) = 2.65929.
[/tex]

Hope This Help,
Henri Tapani Heinonen.
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