Cotangent and the Herglotz trick

Abstract

What is the most interesting formula involving elementary functions? In his beautiful article [2], whose exposition we closely follow, Jürgen Elstrodt nominates as a first candidate the partial fraction expansion of the cotangent function:

$$\pi \cot \pi x = \frac{1} {x} + \sum\limits_{n = 1}^\infty {\left( {\frac{1} {{x + n}} + \frac{1} {{x - n}}} \right)\left( {x \in \mathbb{R}\backslash \mathbb{Z}} \right).} $$