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# Difference between revisions of "Stirling's approximation"

Carl McBride (talk | contribs) m (Added applications section.) |
Carl McBride (talk | contribs) (Added a table) |
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:<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 | ||

+ | |||

+ | :<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 (%) | ||

+ | |- | ||

+ | |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 ]](<math>10^{23}</math>) this formula is essentially exact. | ||

==Applications in statistical mechanics== | ==Applications in statistical mechanics== | ||

*[[Ideal gas Helmholtz energy function]] | *[[Ideal gas Helmholtz energy function]] | ||

[[Category: Mathematics]] | [[Category: Mathematics]] |

## Revision as of 20:14, 4 November 2008

James Stirling (1692-1770, Scotland)

Because of Euler-MacLaurin formula

where *B*_{1} = −1/2, *B*_{2} = 1/6, *B*_{3} = 0, *B*_{4} = −1/30, *B*_{5} = 0, *B*_{6} = 1/42, *B*_{7} = 0, *B*_{8} = −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*,

after some further manipulation one arrives at

where

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 () this formula is essentially exact.