Planck units: Difference between revisions

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, ..., a_n are complex numbers, no two of which differ by an integer multiple of π. Let

A_{n, k} = \prod_{\begin{matrix} 1 \leq j \leq n \\ j \neq k \end{matrix}} \cot (a_{k} - a_{j})

(in particular, A_1,1, being an empty product, is 1). Then

\cot (z - a_{1}) \dots \cot (z - a_{n}) = \cos \frac{n π}{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

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

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

[1]

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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 &agrave; la Hermite", ''[[American Mathematical Monthly]]'', volume 117, number 4, April 2010, pages 311&ndash;327</ref>  Suppose ''a''<sub>1</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub> are [[complex number]]s, no two of which differ by an integer multiple of&nbsp;''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&nbsp;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&nbsp;''n''&nbsp;=&nbsp;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]]

Planck units: Difference between revisions

Revision as of 22:44, 28 January 2014

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