## Story

I want to prove Euler's reflection formula by showing that

\begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*}

is constant, where $s = \sigma + it$. It's easy to see that $f$ is entire and $f(s + 1) = f(s)$, so for fixed $t$ we have $f \in C(\mathbb{R})^\infty$ and $f(\sigma + it)$ is 1-periodic. Therefore $f$ has a rapidly decreasing Fourier series for fixed $s$

\begin{equation*} f(s) = \sum _{n \in \mathbb{Z}} c_n(t) e^{i 2 \pi n \sigma}. \end{equation*}

Let's have a look at the $m$-th term. By definition we have \begin{equation*} f_m(s) = \left(\int _0 ^1 f(s + x) e^{- i 2 \pi m x} dx\right) e^{i 2 \pi m \sigma} = c_m(t) e^{i 2 \pi m \sigma}. \end{equation*}

## Problem

Now to my problem and question: **Is $f_m$ analytic and why**? I can't find an argument and I need one to justify the use of the Cauchy-Riemann equations later on.