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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 |
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