Stirling's Formula for Gamma Function

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Theorem

Let $\Gamma$ denote the Gamma function.


Then:

$\Gamma \left({x + 1}\right) = \sqrt {2 \pi x} \, x^x e^{-x} \left({1 + \dfrac 1 {12 x} + \dfrac 1 {288 x^2} - \dfrac {139} {51 \, 480 x^3} + \cdots}\right)$


Proof


Also known as

This formula is also known as Stirling's asymptotic series.


Source of Name

This entry was named for James Stirling.


Sources