y = \operatorname{arsinh} x

Inverse Hyperbolic Sine $y = \operatorname{arsinh} x$

$\operatorname{arsinh} x$ , the inverse hyperbolic sine or area hyperbolic sine, is the inverse of $\sinh x = \dfrac{e^x - e^{-x}}{2}$ ¹. Since $\sinh$ increases monotonically over the whole real line, the inverse is uniquely determined without any restriction.

Definition and closed form

$y = \operatorname{arsinh} x$ is the value $y$ with $x = \sinh y$ . Solving the resulting quadratic in $e^y$ produces a closed form written with a logarithm.

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

The quantity $x^2 + 1$ under the root is always positive, so this expression makes sense for every real number.

Domain and range

The domain is all real numbers
The range is all real numbers
It increases monotonically
It is an odd function

Since $\sinh$ maps the whole real line one-to-one onto the whole real line, its inverse has the same reach. Of all the inverse hyperbolic functions, this is the only one whose domain carries no restriction at all.

Symmetry and monotonicity

Since $\operatorname{arsinh}(-x) = -\operatorname{arsinh} x$ , the function is odd and its graph is symmetric about the origin. The derivative is as follows.

\frac{d}{dx}\operatorname{arsinh} x = \frac{1}{\sqrt{x^2 + 1}}

It is always positive, so the function increases monotonically over the whole line. The denominator never vanishes, so there is no point where the tangent turns vertical, as it does for $\operatorname{arcosh}$ .

How fast it grows

Range	Approximation	Behavior
near the origin	$\operatorname{arsinh} x \approx x$	tangent to the line $y = x$
$x$ large	$\operatorname{arsinh} x \approx \ln(2x)$	grows slowly, like a logarithm

For large $x$ we have $\sqrt{x^2+1} \approx x$ , so the growth settles into that of a logarithm. There is no asymptote, vertical or horizontal.

Notable values

$x$	$\operatorname{arsinh} x$
$0$	$0$
$1$	$\ln(1 + \sqrt{2}) \approx 0.8814$
$2$	$\ln(2 + \sqrt{5}) \approx 1.4436$
$10$	$\approx 2.9982$

Multiplying $x$ tenfold from $1$ to $10$ raises the value only a little over threefold, which is the gentleness one expects of a logarithm.

Applications

Its most basic use is as the result of an integral.

\int \frac{dx}{\sqrt{x^2 + 1}} = \operatorname{arsinh} x + C

Computing the arc length of a catenary
Calculations in special relativity and of geodesics
The inverse hyperbolic sine transformation for data of wide range, a substitute for the logarithm that also accepts negative values

Inverse hyperbolic functions, Wikipedia

Inverse Hyperbolic Sine y=arsinh⁡xy = \operatorname{arsinh} xy=arsinhx