Proofs from THE BOOK

pp 149-154

Cotangent and the Herglotz trick

  • Martin AignerAffiliated withFU Berlin
  • , Günter M. ZieglerAffiliated withFU Berlin

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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:
$$ \pi \cot \pi x = \frac{1}{x} + \sum^{\infty}_{n=1} (\frac{1}{x+n} + \frac{1}{x-n}) \quad (x \in \mathbb{R}\textbackslash \mathbb{Z}). $$