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

Theorem

$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^{d - 1} \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {z^{1 - 2 d} }$

where:

$\Gamma$ is the Gamma function

$\Ln$ is the principal branch of the complex logarithm

$B_{2 n}$ is the $2n$th Bernoulli number

$\OO$ is Big-O notation.

The validity of the material on this page is questionable. In particular: For what domain is this valid? This cannot be true on $\C$. The left hand side is singular at each $z \in \Z_{<0}$ but the right hand side is bounded. I think, the constant of $\OO$ must depend on the choice of domain, not true for $\C \setminus \Z_{<0}$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

This theorem requires a proof. In particular: Resembles Binet's Formula for Logarithm of Gamma Function You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Approximation to Stirling's Formula for Gamma Function