John Britton
2004-05-05 17:34:14 |
factorial!
I was sitting in AP calc today and we stumbled upon someting very interesting, on a TI-83 Plus you can evaluate .5! but you cannot evaluate any other decimal. The calculator says it is 0.886226925 but I don't understand why this would even be somthing it could do. I tried it on a few other calculators and I got the same answer and on google.com too. I am really confused please help! |
Tristan
2004-05-05 18:16:26 |
Re: factorial!
You know that's funny, because I was wondering the exact same thing!
I didn't really understand the calculus reasons behind it, but apparently the calculator needs to be able to calculate the factorial of multiples of 1/2 for some kind of function. |
John Britton
2004-05-05 21:55:56 |
Re: factorial!
It's very odd. I searched google and google can evaluate factorials of any decimal... one more reason i love google!... the best resource I could find was http://mathworld.wolfram.com/Factorial.html but I couldn't find anything on how they are evaluated for non-integers, there must be a method because the same values are given on different calculators. |
Geoff
2004-05-06 05:32:59 |
Re: factorial!
i believe its an infinite series equation
really? you cant find out how to do it via google? |
Brian Smith
2004-05-06 10:00:32 |
Re: factorial!
The generalization of the factorial is the gamma function. See:
http://mathworld.wolfram.com/GammaFunction.html |
John Britton
2004-05-06 10:43:11 |
Re: factorial!
Thank you. |
Jer
2004-10-08 12:38:45 |
Re: factorial!
That gamma function is hard to work with in general.
.5! = sqrt(pi)/2
1.5! = 3*sqrt(pi)/4
2.5! = 5*sqrt(pi)/8
etc.
but I can figure out the exact values of any other non-integers.
Using google it is apparent that a! = a*(a-1)!
So what I would like is a way of finding the exact values of other rational numbers less than 1.
|
Charlie
2004-10-08 13:48:23 |
Re: factorial!
Some formulae are given at the site Brian Smith mentions above. I don't know if there are any closed form exact values for other rational numbers. Numerical approximations can be obtained, if x is large enough, from the following formula that approximates the natural logarithm of x factorial (i.e., the natural log of the Gamma function of x + 1):
ln(x) * (x + .5) + (-x + 1 / (12 * x) - 1 / (360 * x^2) + 1 / (1260 * x^5)) + ln(2*pi) / 2
If x is in the hundreds this should be quite good. So if you want ln((.7)!) for example, make x=100.7, apply the above formula, and then subtract ln(100.7), then subtract ln(99.7), ... subtract ln(1.7). Then take the natural antilog to get (.7)!. (should be something like .9086387328530) Obviously this is more easily done via computer or programmable calculator.
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Charlie
2004-10-08 13:50:08 |
Whoops
Make that formula
ln(x) * (x + .5) + (-x + 1 / (12 * x) - 1 / (360 * x^3) + 1 / (1260 * x^5)) + ln(2*pi) / 2
(note the x^3 rather than x^2)
|
Jer
2004-10-12 16:17:04 |
Re: factorial!
Thanks Charlie |
