View source for Secant

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One of the [[Trigonometric functions|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

$$x=\frac\pi2(2n+1),\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}\tag{*}$$

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 \ref{*}.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1965)  pp. §4.3</TD></TR>
</table>

[[Category:Special functions]]