Hyperbolic Areakosekans and hyperbolic Areakosekans

Hyperbolic Areakosekans and hyperbolic Areakosekans belong to the area functions . They are the inverse of the hyperbolic secant and hyperbolic cosecant . They are written as functions or less often or and less often . ${\displaystyle \operatorname {arsech} }$ ${\displaystyle \operatorname {sech} ^{-1}}$ ${\displaystyle \operatorname {arcsch} (x)}$ ${\displaystyle \operatorname {csch} ^{-1}(x)}$

Definitions

The hyperbolic and hyperbolic areakoscans are usually defined as:

${\displaystyle \operatorname {arsech} (x)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)}$

${\displaystyle \operatorname {arcsch} (x)=\ln \left({\frac {1+{\sqrt {1+x^{2}}}}{x}}\right)}$

Here stands for the natural logarithm . ${\displaystyle \ln }$

properties

Graph of the hyperbolic Areasekans function

Graph of the hyperbolic areakosecan function

	Areasecans hyperbolicus	Hyperbolic areakosekans
Domain of definition	${\displaystyle 0<x\leq 1}$	${\displaystyle -\infty <x<+\infty \,;\,x\neq 0}$
Range of values	${\displaystyle 0\leq f(x)<+\infty }$	${\displaystyle -\infty <f(x)<+\infty \,;\,f(x)\neq 0}$
periodicity	no	no
monotony	strictly falling monotonously	${\displaystyle x\neq 0}$ strictly falling monotonously
Symmetries	no	Odd function ${\displaystyle f(x)=-f(-x)}$
asymptote	${\displaystyle f(x)\to 0}$ ; ${\displaystyle x\to +1}$	${\displaystyle f(x)\to 0}$ ; ${\displaystyle x\to \pm \infty }$
zeropoint	${\displaystyle x=1}$	no
Jump points	no	no
Poles	${\displaystyle x=0}$	${\displaystyle x=0}$
Extremes	no	no
Turning points	${\displaystyle x={\frac {1}{2}}{\sqrt {2}}}$	no

Special values

The following applies:

${\displaystyle \operatorname {arcsch} \,2=\ln \Phi }$

where denotes the golden ratio . ${\displaystyle \!\ \Phi }$

Series developments

${\displaystyle {\begin{alignedat}{2}\operatorname {arsech} (x)&=\ln \left({\frac {2}{x}}\right)-\sum _{k=1}^{\infty }{\frac {(2k-1)!!x^{2k}}{(2k)!!2k}}&\qquad \mathrm {f{\ddot {u}}r} \,0<x\leq 1\\\operatorname {arcsch} (x)&=\sum _{k=1}^{\infty }{\frac {P_{k-1}(0)}{k}}x^{k}\\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k}\cdot ({\tfrac {1}{2}})_{k-1}}{(2k-1)(k-1)!}}\,x^{1-2k}\end{alignedat}}}$

It is the th Legendre polynomial and represents the Pochhammer symbol . ${\displaystyle P_{k}}$ ${\displaystyle k}$ ${\displaystyle ({\tfrac {1}{2}})_{n}}$

Derivatives

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsech}}(x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}}$.

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsch} (x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}}$.

Integrals

Primitives of Areasekans hyperbolic and hyperbolic Areakosekans are:

${\displaystyle \int \operatorname {arsech} (x)\,\mathrm {d} x=x\cdot \operatorname {arsech} (x)-\arctan \left({\sqrt {{\frac {1}{x^{2}}}-1}}\right)+C}$

${\displaystyle \int \operatorname {arcsch} (x)\,\mathrm {d} x=x\cdot \operatorname {arcsch} (x)+\ln \left(x+x{\sqrt {1+{x}^{-2}}}\right)+C.}$

Conversion and relationships to other trigonometric functions

${\displaystyle \operatorname {arsech} \,(x)=\operatorname {arcosh} \left({\frac {1}{x}}\right)}$

${\displaystyle \operatorname {arcsch} \,(x)=\operatorname {arsinh} \left({\frac {1}{x}}\right)}$

Web links

• Eric W. Weisstein: Inverse Hyperbolic Secant and Inverse Hyperbolic Cosecant on MathWorld