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\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 $x$ 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 $B_n$ is a Bernoulli Number.

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

It is known that, for $n\geq 3$, $\cot (\pi/n)$ is rational only for $n = 4$.

See also Hyperbolic Cotangent, Inverse Cotangent, Lehmer's Constant, Tangent


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.

© 1996-9 Eric W. Weisstein