Hyperbolic secant and hyperbolic cosecant

Hyperbolic secant (blue) and hyperbolic cosecant (red)

The hyperbolic cosecant (csch) and hyperbolic secant (six) functions are hyperbolic functions . They result from the reciprocal of the hyperbolic sine or hyperbolic cosine .

Definitions

${\displaystyle {\begin{aligned}\operatorname {sech} \ x&={\frac {2}{e^{x}+e^{-x}}}={\frac {1}{\cosh x}}\\\operatorname {csch} \ x&={\frac {2}{e^{x}-e^{-x}}}={\frac {1}{\sinh x}}\end{aligned}}}$

properties

	Hyperbolic secant	Hyperbolic cosecan
Domain of definition	${\displaystyle -\infty <x<+\infty }$	${\displaystyle -\infty <x<+\infty \,;\,x\neq 0}$
Range of values	${\displaystyle 0<f(x)\leq 1}$	${\displaystyle -\infty <f(x)<+\infty \,;\,f(x)\neq 0}$
periodicity	no	no
monotony	${\displaystyle x<0}$strictly monotonically increasing strictly monotonically decreasing ${\displaystyle x>0}$	${\displaystyle x>0}$strictly monotonically falling strictly monotonically falling ${\displaystyle x<0}$
Symmetries	Mirror symmetry to the y-axis	Point symmetry to the coordinate origin Axial symmetry to y = ${\displaystyle -x}$
asymptote	${\displaystyle f(x)\to 0}$ For ${\displaystyle x\to \pm \infty }$	${\displaystyle f(x)\to 0}$ For ${\displaystyle x\to \pm \infty }$
zeropoint	no	no
Jump points	no	no
Poles	no	${\displaystyle x=0}$
Extremes	Maximum at x = 0	no
Turning points	${\displaystyle x=\pm \ln {(1+{\sqrt {2}})}}$	no

Inverse functions

The inverse functions are the corresponding area functions :

${\displaystyle {\begin{aligned}x&=\operatorname {arsech} \ y\\x&=\operatorname {arcsch} \ y\end{aligned}}}$

Derivatives

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {sech} \ x&=-\operatorname {sech} \ x\cdot \operatorname {tanh} \ x=-{\frac {\sinh x}{\cosh ^{2}x}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {csch} \ x&=-\operatorname {csch} \ x\cdot \operatorname {coth} \ x=-{\frac {\cosh x}{\sinh ^{2}x}}=-\operatorname {csch} \ x\cdot {\sqrt {1+\operatorname {csch} ^{2}x}}\\\end{aligned}}}$

Integrals

${\displaystyle {\begin{aligned}\int \operatorname {sech} x\ \mathrm {d} x&=\arctan \left(\sinh x\right)+C\\\int \operatorname {csch} x\ \mathrm {d} x&=\ln \left|\operatorname {tanh} \,{\frac {x}{2}}\right|+C\end{aligned}}}$

Series developments

${\displaystyle {\begin{aligned}\operatorname {sech} \ x&=\sum _{k=0}^{\infty }(-1)^{k}{\frac {(8k+4)\pi }{(2k+1)^{2}\pi ^{2}+4x^{2}}}\\\operatorname {csch} \ x&=1/x+\sum _{k=1}^{\infty }(-1)^{k}{\frac {2x}{k^{2}\pi ^{2}+x^{2}}}\end{aligned}}}$

Complex argument

${\displaystyle {\begin{aligned}&\operatorname {sech} (x+\mathrm {i} y)={\frac {2\cosh(x)\cos(y)}{\cosh(2x)+\cos(2y)}}+\mathrm {i} {\frac {-2\sinh(x)\sin(y)}{\cosh(2x)+\cos(2y)}}\\&\operatorname {sech} (\mathrm {i} y)=\sec(y)\\&\operatorname {csch} (x+\mathrm {i} y)={\frac {2\sinh(x)\cos(y)}{\cosh(2x)-\cos(2y)}}+\mathrm {i} \;{\frac {-2\cosh(x)\sin(y)}{\cosh(2x)-\cos(2y)}}\\&\operatorname {csch} (\mathrm {i} y)=-\mathrm {i} \csc(y)\end{aligned}}}$

See also

• Trigonometric functions
• Circle and hyperbolic functions .

Web links

• Eric W. Weisstein : Hyperbolic Cosecant . In: MathWorld (English).