# AProof From the BOOK: The Partial Fraction Expansion of the Cotangent

March 15, 2022

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### Abstract

In this article, I formalise a proof from THE BOOK; namely a formula that was called ‘one of the most beautiful formulas involving elementary functions’:

$\pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^\infty\left(\frac{1}{z+n} + \frac{1}{z-n}\right)$

The proof uses Herglotz's trick to show the real case and analytic continuation for the complex case.