Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function

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Theorem

$\displaystyle \Ln \paren {\Gamma \paren z} = \paren {z - \dfrac 1 2} \Ln \paren z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^{d - 1} \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \mathcal O \paren {z^{1 - 2 d} }$

where:

$\Gamma$ is the Gamma function
$\Ln$ is the principal branch of the complex logarithm
$B_{2 n}$ is the $2n$th Bernoulli number
$\mathcal O$ is Big-O notation.


Proof


Also see