Proposition 5

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

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

\frac{\Gamma'(z)}{\Gamma(z)} = \log z - \frac{1}{2z} - \sum_{k=1}^{p} \frac{b_{2k}}{2k} \frac{1}{z^{2k}} + O \left( \frac{1}{(\omega(z))^{2p+1}} \right).

Instead of prop. 2 of VII, p. 318, one uses the formula

\int_{x}^{x+1} \frac{\Gamma'(t)}{\Gamma(t)} \, dt = \log \Gamma(x+1) - \log \Gamma(x) = \log x

in order to determine the constant.

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

Cited by

Nothing.