Cotangent

$\begin{figure}\begin{center}\BoxedEPSF{Cotangent.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{CotReIm.epsf scaled 700}\end{center}\end{figure}$

The function defined by $\cot x\equiv 1/\tan x$ , where $\tan x$ is the Tangent. The Maclaurin Series for cot is

$\displaystyle \cot x$	$\textstyle =$	$\displaystyle {1\over x}-{\textstyle{1\over 3}}x-{\textstyle{1\over 45}}x^3-{\textstyle{2\over 945}}x^5-{\textstyle{1\over 4725}}x^7-\ldots$
	$\textstyle \phantom{=}$	$\displaystyle -{(-1)^{n+1}2^{2n}B_{2n}\over(2n)!}-\ldots,$

where

is a Bernoulli Number.

$\begin{displaymath} \pi\cot (\pi x)={1\over x}+2x\sum_{n=1}^\infty {1\over x^2-n^2}. \end{displaymath}$

It is known that, for $n\geq 3$ , $\cot (\pi/n)$ is rational only for

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Circular Functions.'' §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.

Spanier, J. and Oldham, K. B. ``The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions.'' Ch. 34 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 319-330, 1987.