**2022: SklogWiki celebrates 15 years on-line**

# Stirling's approximation

**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)