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?

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

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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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...

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...

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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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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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?

how to convert from positive ones?

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

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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