Formulate of Stirling

See also: Stirling

The formula of Stirling , of the name of the Mathematician James Stirling, gives a equivalent Factorielle to the Voisinage of the real Infini (when N tends towards the infinite one):

$\backslash \; lim\_\; \{N\; \backslash \; to\; +\; \backslash \; infty\}\; \{N\; \backslash !\; \backslash \; over\; \backslash \; sqrt\; \{2\; \backslash \; pi\; N\}\; \backslash ;\; \backslash \; left\; (\backslash \; frac\; \{N\}\; \{E\}\; \backslash \; right)\; ^\; \{N\}\}\; =\; 1$

that one often finds written as follows:

$n\; \backslash !\; \backslash \; sim\; \backslash \; sqrt\; \{2\; \backslash \; pi\; N\}\; \backslash ,\; \backslash \; left\; (\{N\; \backslash \; over\; E\}\; \backslash \; right)\; ^n$

Continuous version

The preceding formula is a particular case, for a whole argument, asymptotic formula of Stirling for the function Γ of Euler.

History

The formula was initially discovered by Abraham de Moivre in the form

$n\; \backslash !\; \backslash \; sim\; C\; \backslash ;\; n^\; \{n+1/2\}\; \backslash ,\; \backslash \; mathrm\; \{E\}\; ^\; \{-\; N\}$,

where $C$ is a real constant (nonnull).

The contribution of Stirling was to give a development of $\backslash \; ln\; (N!)$ with any order and to allot the value $C\; =\; \backslash \; sqrt\; \{2\; \backslash \; pi\}$ to the constant. The traditional demonstration of this is given in the article Intégrales of Wallis.

Foot-note

One can improve quality of the approximation of Stirling by using the development of function Γ; one finds:

$n\; \backslash !\; =\; \backslash \; sqrt\; \{2\; \backslash \; pi\; N\}\; \backslash ,\; \backslash \; left\; (\{N\; \backslash \; over\; E\}\; \backslash \; right)\; ^n\; \backslash \; left\; +\; \backslash \; frac\; \{1\}\; \{12\; \backslash \; N\}\; +\; \backslash \; frac\; \{1\}\; \{288\; \backslash \; n^2\}\; -\; \backslash \; frac\; \{139\}\; \{51\; \backslash \; 840\; \backslash \; n^3\}\; -\; \backslash \; frac\; \{571\}\; \{2\; \backslash \; 488\; \backslash \; 320\; \backslash \; n^4\}\; +\; \backslash \; frac\; \{163\; \backslash \; 879\}\; \{209\; \backslash \; 018\; \backslash \; 880\; \backslash \; n^5\}\; +\; \backslash \; mathcal\; \{O\}\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{n^5\}\; \backslash \; right)\; \backslash \; right$

The formula of Euler-MacLaurin makes it possible to lead to the result the order which one wants.

(Sloane' S.A. 001163 and A001164).

Numerical calculations

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