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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>.
| | James Stirling (1692-1770, Scotland) |
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| :<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> |
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| using [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] one has
| | Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] |
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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> | | :<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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| where ''B''<sub>1</sub> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −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> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −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'',
| | :<math>~\approx \int_1^N \ln x dx</math> |
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| :<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
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| after some further manipulation one arrives at (apparently Stirling's contribution was the prefactor of <math>\sqrt{2 \pi}</math>)
| | :<math>~= \left[ x \ln x - x \right]_1^N</math> |
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| :<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
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| where
| | :<math>~= N \ln N -N +1</math> |
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| :<math>\frac{1}{12N+1} < \lambda_N < \frac{1}{12N}.</math>
| | Thus, for large ''N'' |
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| For example:
| | :<math>\ln N! \approx N \ln N -N</math> |
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| {| border="1"
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| | N || N! (exact) || N! (Stirling) || Error (%)
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| |1 || 1 || 0.92213700 || 8.44
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| |2 || 2 || 1.91900435 || 4.22
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| |3 || 6 || 5.83620959 || 2.81
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| |4 || 24 || 23.5061751 || 2.10
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| |5 || 120 || 118.019168 || 1.67
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| |6 || 720 || 710.078185 || 1.40
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| |7 || 5040 || 4980.39583 || 1.20
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| |8 || 40320 || 39902.3955 || 1.05
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| |9 || 362880|| 359536.873 || 0.93
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| |10 || 3628800 || 3598695.62 || 0.84
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| |}
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| When one is dealing with numbers of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact.
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| In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100; for N=100 the
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| percentage error is approximately 0.083%.
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| ==Gosper’s formula==
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| 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>:
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| :<math>n! \approx \sqrt{2 \pi \left( n + \frac{1}{6} \right)} \; \left( \frac{n}{e} \right)^n</math>
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| Which results in:
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| {| border="1"
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| | N || N! (exact) || N! (Gosper)
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| |1 || 1 || 0.99602180
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| |2 || 2 || 1.99736305
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| |3 || 6 || 5.99613535
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| |4 || 24 || 23.9908895
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| |5 || 120 || 119.970030
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| |6 || 720 || 719.872829
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| |7 || 5040 || 5039.33747
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| |8 || 40320 || 40315.9028
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| |9 || 362880 || 362850.646
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| |10 || 3628800 || 3628560.82
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| |}
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| ==Applications in statistical mechanics==
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| *[[Ideal gas Helmholtz energy function]]
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| ==References==
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| <references/> | |
| [[Category: Mathematics]] | | [[Category: Mathematics]] |