Secant

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:

$$\sec x=$$

$$=\frac{\pi}{(\pi/2)^2-x^2}-\frac{3\pi}{(3\pi/2)^2-x^2}+\frac{5\pi}{(5\pi/2)^2-x^2}-\mathinner{\ldotp\ldotp\ldotp\ldotp}$$

#### Comments[edit]

The series expansion is valid in the domain of definition of $\sec$, i.e. not for the points \eqref{*}.

#### References[edit]

[a1] |  K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)

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[a2] |  M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) pp. §4.3