Proposition 4

Tag 08JF, permanent. Cite it as [Bourbaki, Tag 08JF].

Along the filter $\mathfrak{F}$ one has (for every integer $p \geqslant 1$ ) the asymptotic expansion

\log \Gamma(z) \equiv z \log z - z - \frac{1}{2} \log z + \frac{1}{2} \log 2\pi + \sum_{k=1}^{p} \frac{b_{2k}}{2k(2k-1)} \frac{1}{z^{2k-1}} + O \left( \frac{1}{(\omega(z))^{2p}} \right) \quad (\text{mod. } 2\pi i)

(Stirling’s expansion).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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