### 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.

### License

BSD License

### Topics

- Mathematics/Analysis

### Session Cotangent_PFD_Formula

- Cotangent_PFD_Formula

## Cite

×

Cotangent_PFD_Formula-AFP

    @article{Cotangent_PFD_Formula-AFP,
      author  = {Manuel Eberl},
      title   = {A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent},
      journal = {Archive of Formal Proofs},
      month   = {March},
      year    = {2022},
      note    = {\url{https://isa-afp.org/entries/Cotangent_PFD_Formula.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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