Secant

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

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

The function defined by $\sec x\equiv {1/\cos x}$ , where $\cos x$ is the Cosine. The Maclaurin Series of the secant is

$\begin{eqnarray*} \sec x &=& {(-1)^nE_{2n}\over (2n)!}x^{2n}\\ &=&1+{\textsty... ...extstyle{61\over 720}}x^6+{\textstyle{277\over 8064}}x^8+\ldots, \end{eqnarray*}$

where $E_{2n}$ is an Euler Number.

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 Secant $\sec(x)$ and Cosecant $\csc(x)$ Functions.'' Ch. 33 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 311-318, 1987.