Stirling formula

2010 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL]

$$\newcommand{\abs}[1]{\left|#1\right|} \newcommand{\Re}{\mathrm{Re}}$$ An asymptotic representation which provides approximate values of the factorials $n! = 1 \ldots n$ and of the gamma-function for large values of $n$. This representation has the form $$n! = \sqrt{2\pi n}\; n^n e^{-n} e^{\theta(n)}, \tag{^*}$$ where $\abs{\theta(n)} < 1/12n$. The asymptotic equalities $$n! \approx \sqrt{2\pi n}\; n^n e^{-n}, \quad n \rightarrow \infty,$$ $$\Gamma(z+1) \approx \sqrt{2\pi z}\; z^z e^{-z}, \quad z \rightarrow \infty,\; |{\arg z}|<\pi,$$ hold, and mean that when $n\rightarrow\infty$ or $z \rightarrow \infty$, $|{\arg z}|<\pi$, the ratio of the left- and right-hand sides tends to one.

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

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

References

 [Br] N.G. de Bruijn, "Asymptotic methods in analysis", Dover, reprint (1981) [MaMa] G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of $n!$" Amer. Math. Monthly, 97 (1990) pp. 826–829 [Na] 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=44695
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article