# Stirlings Formula

| Stirlings formula is an indispensable tool for all combinatorial and statistical problems because it allows to deal with factorials, i.e. expressions based on the definition 1 · 2 · 3 · 4 · 5 · .... · N := N!
| It exists in several modifications; all of which are approximations with different degrees of precision. It is relatively easy to deduce its more simple version. We have
ln x! | = | ln 1 + ln 2 + ln 3 + .... + ln x | = | x S 1 | ln y
|  | With y = positive integer running from 1 to x
|  | For large y we may replace the sum by an integration in a good approximation and obtain
x S 1 | ln y | » | x ó õ 1 | (ln y) · dy
|  | With (ln y) · dy = y · ln y – y, we obtain
ln x! | » | x · ln x – x + 1
--- | --- | ---
| This is the simple version of Stirlings formula. it can be even more simplified for large x because then x + 1 << x · ln x; and the most simple version, perfectly sufficient for many cases, results:
ln x! | » | x · ln x
--- | --- | ---
| However!! We not only produced a simple approximation for x!, but turned a discrete function having values for integers only, into a continuous function, giving numbers for something like 3,141! - which may or may not make sense.
|  | This may have dire consequences. Using the Strirling formula you may, e.g., move from absolute probabilities (always a number between 0 and 1) to probability densities (any positive number) without being aware of it.
| Finally, an even better approximation exists (the prove of which would take some 20 pages) and which is already rather good for small values of x, say x > 10:
x! | » | (2p)1/2 · x(x + ½) · e– x
--- | --- | ---

With frame

2.1.1 Simple Vacancies and Interstitials

Solution to Basic Exercise 2.1-4

© H. Föll (Defects - Script)