Namespaces
Variants
Actions

Secant

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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

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

References

[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. §4.3
How to Cite This Entry:
Secant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Secant&oldid=44685
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article