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

Manuel Eberl
2022

 pdf  AFP entry

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},
}