OK, math geniuses/professors!?

Question

Something I've been wondering about: what do you find is a good function to use for the approximation of f(x) = x factorial?
I've tried Stirling's approximation, and the Gamma function, which are both quite good; yet Stirling's is not as accurate, and the Gamma Function, while very cool, cannot be integrated nor differentiated due to the multiplicity of its equation and variables.
So...any suggestions for further exploration on the calculus of
f(x) = x!
?
Note: please do not simply link me to factorial explanation sites; I've already been to most of them.

Answer 1

Stirling's formula is actually provable, like they show here, for example

http://www.sosmath.com/calculus/sequence/stirling/stirling.html

so i doubt very much you'll find a significant improvement on that. x^x * e^ (-x) is pretty much a given, and will be a part of any formula. sqrt(2*PI*n) seems like an insignificant constant that can be played with a bit

Answer 2

I'm presuming you already know what a factorial is since you've been to those sites. That being the case, are you wanting to us it in a computer program, or just find a short cut? More details would be helpful.

Answer 3

Try reading the relevant sections of Chapter 1, Volume 1
of Knuth's "Art of Computer Programming".
There you will find an excellent discussion about
x!. Good luck!

Answer 4

Have you, by chance, read this page:

http://www.luschny.de/math/factorial/approx/SimpleCases.html

Especially the last set of equations at the bottom of the page? Perhaps, these will be helpful to you.

HTH

Charles