Stirling's Approximation, etc.

Stirling's approximation for N!:

N! ~ sqrt(2 π N).(N/e)N + ...

hence

loge(N!) ~ N.loge(N) - N + 0.5 loge(N) + 0.5 loge(2 pi) +...

 

function Stirling(N) // JavaScript
 { return
    (N+0.5)*Math.log(N) - N + Math.log(2*Math.PI)/2;
 }

Factorial is generalized by the Γ function to real, and even complex, values. For a positive integer, n, Γn=(n-1)!.

Notes

D. E. Knuth in The Art of Computer Programming, Fundamental Algorithms, Vol.1, p.46, (1969), gives the reference as:

James Stirling. Methodus Differentialis, p.137, (1730).

On page 111 of his book, Knuth derives a more accurate approximation:

N! = sqrt(2 π N) (N/e)^N {1 + 1/12N + 1/288N² - 139/51840N³ - 571/2488320N⁴ + O(1/N⁵)}