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\begin{document}

\title{Stirling's formula}
\author{Manuel Eberl}
\maketitle

\begin{abstract}
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 $\Gamma(x)\sim \sqrt{2\pi/x} (x/e)^x$. The proof is based on work by Graham Jameson~\cite{jameson}.

This is then extended to the full asymptotic expansion
\begin{multline*}
\log\Gamma(z) = \big(z - \tfrac{1}{2}\big)\log z - z + \tfrac{1}{2}\log(2\pi) + \sum_{k=1}^{n-1} \frac{B_{k+1}}{k(k+1)} z^{-k}\\
{} - \frac{1}{n} \int_0^\infty B_n([t])(t + z)^{-n}\,\text{d}t
\end{multline*}
uniformly for all complex $z\neq 0$ in the cone $\text{arg}(z)\leq \alpha$ for any $\alpha\in(0,\pi)$, with which the above asymptotic relation for $\Gamma$ is also extended to complex arguments.
\end{abstract}

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