A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent

 

Title: A Proof from THE BOOK: The Partial Fraction Expansion of the Cotangent
Author: Manuel Eberl
Submission date: 2022-03-15
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.

BibTeX:
@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   = mar,
  year    = 2022,
  note    = {\url{https://isa-afp.org/entries/Cotangent_PFD_Formula.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License