View source for Stirling formula

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$$
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\Re}{\mathrm{Re}}
$$
An asymptotic representation which provides approximate values of the
factorials $n! = 1 \ldots n$ and of the
[[Gamma-function|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
$$
n! \approx  \sqrt{2\pi n}\; n^n e^{-n},  \quad n \rightarrow \infty,
$$
$$
\Gamma(z+1) \approx  \sqrt{2\pi z}\; z^z e^{-z},  \quad 
\Re z \rightarrow +\infty,
$$
hold, and mean that when $n\rightarrow\infty$ or $\Re z \rightarrow \infty$, the ratio of the left- and right-hand sides tends to one.

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



====Comments====
See
[[Gamma-function|Gamma-function]] for the corresponding asymptotic
series (Stirling series) and additional references.

====References====
{|
|-
|valign="top"|{{Ref|Br}}||valign="top"| N.G. de Bruijn, "Asymptotic methods in analysis", Dover, reprint (1981)
|-
|valign="top"|{{Ref|MaMa}}||valign="top"| G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of $n!$" ''Amer. Math. Monthly'', '''97''' (1990) pp. 826–829
|-
|valign="top"|{{Ref|Na}}||valign="top"| V. Namias, "A simple derivation of Stirling's asymptotic series" ''Amer. Math. Monthly'', '''93''' (1986) pp. 25–29
|-
|}