# AProof 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 Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.