Hermite's cotangent identity - Alchetron, the free social encyclopedia

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

A n , k = ∏ 1 ≤ j ≤ n j ≠ k cot ⁡ ( a k − a j )

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

cot ⁡ ( z − a 1 ) ⋯ cot ⁡ ( z − a n ) = cos ⁡ n π 2 + ∑ 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 ) .

References

Hermite's cotangent identity Wikipedia

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