A Proof From the BOOK: The Partial Fraction Expansion of the Cotangent

Manuel Eberl 🌐

March 15, 2022

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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.

BSD License


Theories of Cotangent_PFD_Formula