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# Difference between revisions of "Stirling's approximation"

Carl McBride (talk | contribs) m (Slight tidy.) |
Carl McBride (talk | contribs) (Added Gosper’s formula) |
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− | '''Stirling's approximation''' | + | '''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>. |

:<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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:<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 | + | 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> | :<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math> | ||

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|- | |- | ||

| N || N! (exact) || N! (Stirling) || Error (%) | | N || N! (exact) || N! (Stirling) || Error (%) | ||

+ | |- | ||

+ | |1 || 1 || 0.92213700 || 8.44 | ||

|- | |- | ||

|2 || 2 || 1.91900435 || 4.22 | |2 || 2 || 1.91900435 || 4.22 | ||

Line 49: | Line 51: | ||

In [[Computer simulation techniques | computer simulations]] the number of atoms or molecules (N) is invariably greater than 100; for N=100 the | 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%. | 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== | ==Applications in statistical mechanics== | ||

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

+ | ==References== | ||

+ | <references/> | ||

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

## Latest revision as of 12:33, 31 January 2011

**Stirling's approximation** is named after the Scottish mathematician James Stirling (1692-1770)^{[1]}.

using Euler-MacLaurin formula one has

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 (apparently Stirling's contribution was the prefactor of )

where

For example:

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 () this formula is essentially exact. In 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[edit]

Gosper’s formula ^{[2]}^{[3]}:

Which results in:

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[edit]

## References[edit]

- ↑ 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)
- ↑ R. William Gosper, Jr. "Decision procedure for indefinite hypergeometric summation", PNAS
**75**pp. 40-42 (1978) - ↑ Cristinel Mortici "Best estimates of the generalized Stirling formula", Applied Mathematics and Computation
**215**pp. 4044-4048 (2010)