Message boards : Science (non-SETI) : Non-integer factorials?
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Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0 |
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? |
William Rothamel Send message Joined: 25 Oct 06 Posts: 3253 Credit: 1,297,049 RAC: 341 |
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 |
Clyde C. Phillips, III Send message Joined: 2 Aug 00 Posts: 1851 Credit: 5,955,047 RAC: 0 |
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. |
Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0 |
My intuition tells me that rational, non-integer factorials would be useful in determining series convergences--albeit with some fiddling with scale factors. Thank you. I'll try to find the formula for practical use on that field... |
Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0 |
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... |
HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 866,353 RAC: 268 |
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)!. 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. Manned mission to Mars in 2019 Petition <-- Sign this, please. |
HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 866,353 RAC: 268 |
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. Manned mission to Mars in 2019 Petition <-- Sign this, please. |
Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0 |
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? |
Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0 |
how to convert from positive ones? Oh, I did found that -z! = pi/sin(-z*pi)*(-z)/z! |
HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 866,353 RAC: 268 |
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. Manned mission to Mars in 2019 Petition <-- Sign this, please. |
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Non-integer factorials?
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