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’: π cot(πz) = 1/z + ∑ (1/(z+n) + 1/(z-n)) where the sum ranges over all integers n > 0.
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   = {March},
  year    = {2022},
  note    = {\url{https://isa-afp.org/entries/Cotangent_PFD_Formula.html},
            Formal proof development},
  ISSN    = {2150-914x},
}