Message boards :
Science (non-SETI) :
Non-integer factorials?
Message board moderation
Author | Message |
---|---|
Sergej O. S. 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: 3756 Credit: 1,999,735 RAC: 4 |
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. 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. 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 Send message Joined: 8 Jul 00 Posts: 691 Credit: 909,237 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)!. 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 Send message Joined: 8 Jul 00 Posts: 691 Credit: 909,237 RAC: 0 |
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. 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. 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 Send message Joined: 8 Jul 00 Posts: 691 Credit: 909,237 RAC: 0 |
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. |
©2024 University of California
SETI@home and Astropulse are funded by grants from the National Science Foundation, NASA, and donations from SETI@home volunteers. AstroPulse is funded in part by the NSF through grant AST-0307956.