Stirling formula
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
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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
| [1] | N.G. de Bruijn, "Asymptotic methods in analysis", Dover, reprint (1981) |
| [2] | G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of $n!$" Amer. Math. Monthly, 97 (1990) pp. 826–829 |
| [3] | V. Namias, "A simple derivation of Stirling's asymptotic series" Amer. Math. Monthly, 93 (1986) pp. 25–29 |
 |  |
