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{{distinguish2|[[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]].<ref>Warren P. Johnson, "Trigonometric Identities à la Hermite", ''[[American Mathematical Monthly]]'', volume 117, number 4, April 2010, pages 311–327</ref> Suppose ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> are [[complex number]]s, no two of which differ by an integer multiple of ''π''. Let | |||
: <math> A_{n,k} = \prod_{\begin{smallmatrix} 1 \le j \le n \\ j \neq k \end{smallmatrix}} \cot(a_k - a_j) </math> | |||
(in particular, ''A''<sub>1,1</sub>, being an [[empty product]], is 1). Then | |||
: <math> \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).</math> | |||
The simplest non-trivial example is the case ''n'' = 2: | |||
: <math> \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). \, </math> | |||
== Notes and references == | |||
{{reflist}} | |||
[[Category:Trigonometry]] |
Revision as of 22:44, 28 January 2014
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
(in particular, A1,1, being an empty product, is 1). Then
The simplest non-trivial example is the case n = 2:
Notes and references
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- ↑ Warren P. Johnson, "Trigonometric Identities à la Hermite", American Mathematical Monthly, volume 117, number 4, April 2010, pages 311–327