y = \operatorname{arcsch} x

The Inverse Hyperbolic Cosecant $y = \operatorname{arcsch} x$

$\operatorname{arcsch} x$ , the inverse hyperbolic cosecant, is the inverse of $\operatorname{csch} x = \dfrac{1}{\sinh x}$ ¹. Since $\operatorname{csch}$ maps each branch of $x \neq 0$ monotonically onto $y \neq 0$ , the inverse has domain $x \neq 0$ and range $y \neq 0$ as well.

Definition and closed form

The equation $\operatorname{csch} y = x$ is the same as $\sinh y = \dfrac{1}{x}$ , so $\operatorname{arcsch} x = \operatorname{arsinh}\dfrac{1}{x}$ . Written with a logarithm it takes the following form.

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

Since $\operatorname{arsinh}$ is defined for every real number, a value exists as soon as $\dfrac{1}{x}$ does. That is the reason the domain is $x \neq 0$ .

Domain and range

The domain is $x \neq 0$
The range is $y \neq 0$
It decreases monotonically on each branch
It is an odd function

Symmetry and asymptotes

Since $\operatorname{csch}$ is odd, so is its inverse, and the graph has point symmetry about the origin.

Approach	$\operatorname{arcsch} x$
$x \to 0^{+}$	$\to +\infty$
$x \to 0^{-}$	$\to -\infty$
$x \to \pm\infty$	$\to 0$

The $y$ -axis is a vertical asymptote and the $x$ -axis a horizontal one.

Monotonicity and concavity

The derivative is $-\dfrac{1}{|x|\sqrt{1+x^{2}}}$ , negative throughout the domain, so the function decreases on both branches. The second derivative carries the sign of $x$ , so the curve is concave up for $x > 0$ and concave down for $x < 0$ , with no inflection point.

How gently it diverges

Where the original $\operatorname{csch} x$ blew up near the origin like $\dfrac{1}{x}$ , the inverse diverges no faster than $\ln\dfrac{2}{x}$ .

$x$	$\operatorname{arcsch} x$
$0.001$	$\approx 7.6009$
$0.5$	$\approx 1.4436$
$1$	$\ln(1+\sqrt{2}) \approx 0.8814$
$2$	$\approx 0.4812$

The value $\operatorname{arcsch}(0.001) \approx 7.6009$ agrees with $\ln 2000 \approx 7.6009$ . Shrinking $x$ by a factor of a thousand raises the value only to about $8$ , which is the slowness of a logarithm laid bare. It is a concrete case of the general fact that taking an inverse replaces a violent divergence with a logarithmic one.

Behavior far out

For large $|x|$ the quantity $\dfrac{1}{x}$ is small, so $\operatorname{arsinh} u \approx u$ applies and $\operatorname{arcsch} x \approx \dfrac{1}{x}$ .

Range	$\operatorname{csch} x$	$\operatorname{arcsch} x$
near the origin	$\approx \dfrac{1}{x}$ , an inverse proportion	$\approx \ln\dfrac{2}{x}$ , a logarithm
far out	$\approx 2e^{-x}$ , exponential decay	$\approx \dfrac{1}{x}$ , an inverse proportion

Between the function and its inverse, the roles played near the origin and far out are exactly exchanged.

The full family of inverse hyperbolic functions

With this the six inverse hyperbolic functions are complete. The range of each hyperbolic function has passed straight over to become the domain of its inverse.

Function	Domain	Range
$\operatorname{arsinh} x$	all real numbers	all real numbers
$\operatorname{arcosh} x$	$x \geq 1$	$y \geq 0$
$\operatorname{artanh} x$	$-1 < x < 1$	all real numbers
$\operatorname{arcoth} x$	$\|x\| > 1$	$y \neq 0$
$\operatorname{arsech} x$	$0 < x \leq 1$	$y \geq 0$
$\operatorname{arcsch} x$	$x \neq 0$	$y \neq 0$

The prefix ar is short for area, from the area of a hyperbolic sector. It is the counterpart of the arc in the names of the inverse trigonometric functions, which refers to the length of a circular arc.

Inverse hyperbolic functions, Wikipedia

The Inverse Hyperbolic Cosecant y=arcsch⁡xy = \operatorname{arcsch} xy=arcschx