Re: How to calculate factorial of fractions?
- From: mike3 <mike4ty4@xxxxxxxxx>
- Date: Sat, 10 Nov 2007 02:13:36 -0800
On Nov 9, 7:18 pm, Sunil <sravip...@xxxxxxxxx> wrote:
I have seen several posts as to how to calculate factorial of
fractions like using gamma method. Can any one explain it using any
approximation such as sterling's?. I am able to calculate half
fractions using gamma but not the other fractions.
For example (3.142)!
You're right, you use the Gamma function:
3.142! = Gamma(4.142), exactly.
Stirling's series is an asymptotic series for the Gamma
function, which means that although it ultimately diverges,
it can be used to generate arbitrarily-good approximations
provided one makes the input to the function large enough.
Combined with the recurrence Gamma(z+1) = z*Gamma(z), one
can amplify the input to the size required for getting good
approximations.
See:
http://en.wikipedia.org/wiki/Asymptotic_expansion
Using the first few terms on that page even, we obtain
3.142! = Gamma(4.142) ~ 7.1918765295. The program I'm using
says the true value of Gamma(4.142) is closer to 7.1918772362,
so we'll just round the approximation from the asymptotic
to 7.191877. Not too bad, eh?
.
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