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'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770)<ref>J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)</ref>.
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James Stirling (1692-1770, Scotland)
  
:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>
+
:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k</math>
  
using [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] one has
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Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula]
  
:<math>\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 ,</math>
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:<math>\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</math>
  
 
where ''B''<sub>1</sub> = &minus;1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = &minus;1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = &minus;1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.
 
where ''B''<sub>1</sub> = &minus;1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = &minus;1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = &minus;1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''.
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Then, for large ''N'',
 
Then, for large ''N'',
  
:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
+
:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N</math>
 
 
after some further manipulation one arrives at (apparently  Stirling's contribution was the prefactor of <math>\sqrt{2 \pi}</math>)
 
 
 
:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
 
 
 
where
 
 
 
:<math>\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.</math>
 
 
 
For example:
 
 
 
{| border="1"
 
|-
 
| N || N! (exact) || N! (Stirling)  || Error (%)
 
|-
 
|1 ||  1  || 0.92213700 ||  8.44
 
|-
 
|2 ||  2  || 1.91900435 ||  4.22
 
|-
 
|3 ||  6  || 5.83620959 ||  2.81
 
|-
 
|4 ||  24  || 23.5061751 ||  2.10
 
|-
 
|5 ||  120  || 118.019168 || 1.67
 
|-
 
|6  || 720  ||  710.078185 || 1.40
 
|-
 
|7  || 5040  || 4980.39583  || 1.20
 
|-
 
|8  ||  40320 ||  39902.3955 || 1.05
 
|-
 
|9  ||  362880||  359536.873  || 0.93
 
|-
 
|10  || 3628800  ||  3598695.62  || 0.84
 
|}
 
 
 
When one is dealing with  numbers of the order of the [[Avogadro constant ]](<math>10^{23}</math>)  this formula is essentially exact.
 
In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100;  for N=100 the
 
percentage error is approximately 0.083%.
 
==Gosper’s formula==
 
Gosper’s formula <ref>[http://www.pnas.org/content/75/1/40 R. William Gosper, Jr. "Decision procedure for indefinite hypergeometric summation", PNAS '''75''' pp. 40-42 (1978)]</ref><ref>[http://dx.doi.org/10.1016/j.amc.2009.12.013  Cristinel Mortici "Best estimates of the generalized Stirling formula", Applied Mathematics and Computation '''215''' pp. 4044-4048 (2010)]</ref>:
 
 
 
:<math>n! \approx \sqrt{2 \pi \left( n + \frac{1}{6} \right)} \;  \left( \frac{n}{e} \right)^n</math>
 
 
 
Which results in:
 
 
 
{| border="1"
 
|-
 
| N || N! (exact) || N! (Gosper)
 
|-
 
|1 ||  1        || 0.99602180
 
|-
 
|2 ||  2        || 1.99736305
 
|-
 
|3 ||  6        || 5.99613535
 
|-
 
|4 ||  24        || 23.9908895
 
|-
 
|5 ||  120      || 119.970030
 
|-
 
|6  || 720      || 719.872829
 
|-
 
|7  || 5040      || 5039.33747
 
|-
 
|8  ||  40320    || 40315.9028
 
|-
 
|9  ||  362880  || 362850.646
 
|-
 
|10  || 3628800  || 3628560.82
 
|}
 
==Applications in statistical mechanics==
 
*[[Ideal gas Helmholtz energy function]]
 
==References==
 
<references/>
 
 
[[Category: Mathematics]]
 
[[Category: Mathematics]]

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