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In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite.[1] Suppose a1, ..., an are complex numbers, no two of which differ by an integer multiple of π. Let

\( A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j}) \)

(in particular, A1,1, being an empty product, is 1). Then

\( \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}). \)

The simplest non-trivial example is the case n = 2:

\( \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).\, \)

Notes and references

Warren P. Johnson, "Trigonometric Identities à la Hermite", American Mathematical Monthly, volume 117, number 4, April 2010, pages 311–327

List of things named after Charles Hermite

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

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