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Difference between revisions of "Secant"

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One of the [[Trigonometric functions|trigonometric functions]]:
 
One of the [[Trigonometric functions|trigonometric functions]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836401.png" /></td> </tr></table>
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$$y=\sec x=\frac{1}{\cos x};$$
  
another notation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836402.png" />. Its domain of definition is the whole real line apart from the points
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another notation is $\operatorname{sc}x$. Its domain of definition is the whole real line apart from the points
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$x=\frac\pi2(2n+1),\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}\tag{*}$$
  
The secant is an unbounded even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836404.png" />-periodic function. The derivative of the secant is
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The secant is an unbounded even $2\pi$-periodic function. The derivative of the secant is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836405.png" /></td> </tr></table>
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$$(\sec x)'=\frac{\sin x}{\cos^2x}=(\tan x)(\sec x).$$
  
 
The indefinite integral of the secant is
 
The indefinite integral of the secant is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836406.png" /></td> </tr></table>
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$$\int\sec xdx=\ln\left|\tan\left(\frac\pi4+\frac x2\right)\right|+C.$$
  
 
The secant can be expanded in a series:
 
The secant can be expanded in a series:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836407.png" /></td> </tr></table>
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$$\sec x=$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836408.png" /></td> </tr></table>
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$$=\frac{\pi}{(\pi/2)^2-x^2}-\frac{3\pi}{(3\pi/2)^2-x^2}+\frac{5\pi}{(5\pi/2)^2-x^2}-\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
  
  
  
 
====Comments====
 
====Comments====
The series expansion is valid in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083640/s0836409.png" />, i.e. not for the points (*).
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The series expansion is valid in the domain of definition of $\sec$, i.e. not for the points \ref{*}.
  
 
====References====
 
====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>
 
<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>

Revision as of 14:02, 13 November 2014

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

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

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

$$\int\sec xdx=\ln\left|\tan\left(\frac\pi4+\frac x2\right)\right|+C.$$

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

[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=13004
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