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# Difference between revisions of "Stirling's approximation"

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James Stirling (1692-1770, Scotland)

$\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .$

Because of Euler-MacLaurin formula

$\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,$

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

$\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .$

after some further manipulation one arrives at

$N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}$

where

$\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.$

For example:

 N N! (exact) N! (Stirling) Error (%) 5 120 118.019168 1.016 6 720 710.078185 1.014 7 5040 4980.39583 1.012 8 40320 39902.3955 1.010 9 362880 359536.873 1.009 10 3628800 3598695.62 1.008

As one usually deals with number of the order of the Avogadro constant ($10^{23}$) this formula is essentially exact.
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