Stirling's series

Example

Let $\Gamma$ be the gamma function, let $\ln \Gamma$ be its principal branch on $\C \setminus (-\infty, 0]$, and let $B_{2k}$ be the Bernoulli numbers. Fix $\delta$ with $0 < \delta < \pi$. As $z \to \infty$ in the sector $\abs{\arg z} \le \pi - \delta$, $$\ln \Gamma(z) \sim \left(z - \tfrac{1}{2}\right) \ln z - z + \tfrac{1}{2} \ln(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k(2k-1) \, z^{2k-1}}.$$ This is an asymptotic expansion in the scale $z \ln z, z, \ln z, 1, z^{-1}, z^{-3}, z^{-5}, \ldots$. With $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$ and $B_6 = \frac{1}{42}$, the first correction terms are $$\frac{1}{12 z} - \frac{1}{360 z^3} + \frac{1}{1260 z^5}.$$