One of the trigonometric functions:
$$y=\sec x=\frac{1}{\cos x};$$
another notation is $\operatorname{sc}x$. Its domain of definition is the whole real line apart from the points
$$x=\frac\pi2(2n+1),\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}\label{*}\tag{*}$$
The secant is an unbounded even $2\pi$-periodic function. The derivative of the secant is
$$(\sec x)'=\frac{\sin x}{\cos^2x}=(\tan x)(\sec x).$$
The indefinite integral of the secant is
$$\int\sec x\,dx=\ln\left|\tan\left(\frac\pi4+\frac x2\right)\right|+C.$$
The secant can be expanded in a series:
$$\sec x=$$
$$=\frac{\pi}{(\pi/2)^2-x^2}-\frac{3\pi}{(3\pi/2)^2-x^2}+\frac{5\pi}{(5\pi/2)^2-x^2}-\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
The series expansion is valid in the domain of definition of $\sec$, i.e. not for the points \eqref{*}.
[a1] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
[a2] | M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) pp. ยง4.3 |