Stirling formula

2020 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).

Comments

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