Non-integer factorials?

Message boards : Science (non-SETI) : Non-integer factorials?

Author	Message
Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0	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?
	ID: 824881 ·

William Rothamel Send message Joined: 25 Oct 06 Posts: 2126 Credit: 1,088,132 RAC: 145	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. Regards, BILL
	ID: 824914 ·

Clyde C. Phillips, III Send message Joined: 2 Aug 00 Posts: 1851 Credit: 5,955,047 RAC: 0	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. ____________
	ID: 824961 ·

Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0	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...
	ID: 824977 ·

Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0	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...
	ID: 824979 ·

HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 774,636 RAC: 270	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. ____________ Manned mission to Mars in 2019 Petition* <-- Sign this, please.
	ID: 825010 ·

HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 774,636 RAC: 270	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. ____________ Manned mission to Mars in 2019 Petition* <-- Sign this, please.
	ID: 825017 ·

Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0	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?
	ID: 825042 ·

Sergej O. S. Volunteer tester Send message Joined: 29 Oct 08 Posts: 123 Credit: 44,886 RAC: 0	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(-zpi)(-z)/z!
	ID: 825661 ·

HTH Volunteer tester Send message Joined: 8 Jul 00 Posts: 690 Credit: 774,636 RAC: 270	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/(1.41.4))\Gamma(0.6) = 2.65929. [/tex] Hope This Help, Henri Tapani Heinonen. ____________ Manned mission to Mars in 2019 Petition* <-- Sign this, please.
	ID: 826126 ·

Message boards : Science (non-SETI) : Non-integer factorials?