Hermite's cotangent identity

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{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})

(in particular, A_1,1, 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

gollark: I'm not sure about reproducible (the build config for a project probably won't specify precise versions for all compilation stuff) but declarative, yes.

gollark: Basically, you specify all your configuration in a declarative purely-functional language, which is then translated into actual system config.

gollark: One really cool thing I use on a server is NixOS.

gollark: Yes, because they compile the binaries on their end.

gollark: I use Arch Linux, which is rolling release (there are no fixed versions of the distro, you just update your packages whenever they get updated), very flexible (you install it manually, choosing partition layout, DE, extra software, etc), and has the AUR, a giant collection of user-packaged software.

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