Not to be confused with
Hermite's identity, a statement about fractional parts of integer multiples of real numbers.
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
![{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add4d3e70d16463f3a6941af196afc1e6c337ad1)
(in particular, A1,1, being an empty product, is 1). Then
![{\displaystyle \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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db60873e93178b3b00e9cdea04ca45727748be1)
The simplest non-trivial example is the case n = 2:
![{\displaystyle \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}).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfb44dd60f1ab92046cca46fe263014af8f9fb9)
Notes and references
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