nLab
cotangent function

Contents

Contents

Idea

In trigonometry the cotangent function is one of the basic trigonometric functions.

Definition

The contangent is the ratio of the cosine and the sine:

cot(x)=cos(x)sin(x) \cot(x) = \frac{\cos(x)}{\sin(x)}

Properties

Series expansion

The cotangent is the logarithmic derivative? of the sine function:

cotx=(log(sinx)).\cot x = (\log (\sin x))'.

Applying this observation to the Euler-Weierstrass product formula for the sine:

sin(πx)=πx n=1 (1x 2n 2)\sin (\pi x) = \pi x \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)

one obtains the following summation formula for the cotangent:

πcot(πx)=1x+ n=1 (1x+n+1xn)\pi\, \cot (\pi x) = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)

This expansion was used by Eisenstein as a starting point for developing the theory of trigonometric functions; Eisenstein’s account of elliptic functions, developed further by Weierstrass, Kronecker, and others, runs parallel to his trigonometric theory, as explained later by Weil. For some more details, see these notes by Varadarajan.

References

Last revised on May 11, 2019 at 04:02:48. See the history of this page for a list of all contributions to it.