# Stirling formula

From Encyclopedia of Mathematics

An asymptotic representation which provides approximate values of the factorials and of the gamma-function for large values of . This representation has the form

(*) |

where . The asymptotic equalities

hold, and mean that when or , the ratio of the left- and right-hand sides tends to one.

The representation (*) was established by J. Stirling (1730).

#### Comments

See Gamma-function for the corresponding asymptotic series (Stirling series) and additional references.

#### References

[a1] | N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981) |

[a2] | G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of " Amer. Math. Monthly , 97 (1990) pp. 826–829 |

[a3] | V. Namias, "A simple derivation of Stirling's asymptotic series" Amer. Math. Monthly , 93 (1986) pp. 25–29 |

**How to Cite This Entry:**

Stirling formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Stirling_formula&oldid=13618

This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article