Stirling's formula

This work contains a proof of Stirling's formula both for the factorial $n!\sim \sqrt{2\pi n}(n/e{)}^n$ on natural numbers and the real Gamma function $\mathrm{\Gamma }(x)\sim \sqrt{2\pi /x}(x/e{)}^x$. The proof is based on work by Graham Jameson.

This is then extended to the full asymptotic expansion $$\begin{gathered} \log \mathrm{\Gamma }(z)=(z-\frac{1}{2})\log z-z+\frac{1}{2}\log (2\pi )+\sum _{k=1}^{n-1}\frac{B_{k+1}}{k(k+1)}{z}^{-k} \\ -\frac{1}{n}{\int }_0^{\mathrm{\infty }}{B}_n([t])(t+z{)}^{-n}\text{d}t \end{gathered}$$ uniformly for all complex $z\ne 0$ in the cone $\text{arg}(z)\le \alpha$ for any $\alpha \in (0,\pi )$, with which the above asymptotic relation for Γ is also extended to complex arguments.