Stirling's Formula/Also defined as

Stirling's Formula: Also defined as

Stirling's formula can also be seen reported variously as:

$\ds n!$

$\sim$

$\ds \sqrt {2 \pi} \, n^n n^{1/2} e^{-n}$

$\ds n!$

$\sim$

$\ds n^n e^{-n} \sqrt {2 \pi n}$

$\ds n!$

$\sim$

$\ds \sqrt {2 \pi n} \, n^n e^{-n}$

Other variants are sometimes encountered, for example:

$\ds \lim_{n \mathop \to \infty} \dfrac {n!} {e^{-n} n^n \sqrt {2 \pi n} } = 1$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {I}$. The Gamma function: $4$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 16$: Asymptotic Expansions for the Gamma Function: $16.16$
1970: N.G. de Bruijn: Asymptotic Methods in Analysis (3rd ed.) ... (previous) ... (next): $1.1$ What is asymptotics? $(1.1.1)$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.2$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 25$: The Gamma Function: Asymptotic Expansions for the Gamma Function: $25.16.$