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

$\operatorname{arsinh} x$ (the inverse hyperbolic sine, or area hyperbolic sine) is the inverse of the hyperbolic sine $\sinh x = \dfrac{e^x - e^{-x}}{2}$. Because $\sinh$ is strictly increasing over all real numbers, its inverse is uniquely defined without any restriction.

Definition and closed form

$y = \operatorname{arsinh} x$ is defined as the value $y$ satisfying $x = \sinh y$. Solving with exponentials gives the closed logarithmic form

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

Domain and range

The domain is all real numbers $(-\infty, \infty)$, and the range is also all real numbers $(-\infty, \infty)$, since $\sinh$ maps the whole real line one-to-one onto itself.

Symmetry and monotonicity

$\operatorname{arsinh}$ is an odd function: $\operatorname{arsinh}(-x) = -\operatorname{arsinh} x$, so its graph is symmetric about the origin. Its derivative

is always positive, so the function is strictly increasing everywhere.

Limits and growth rate

Near the origin $\operatorname{arsinh} x \approx x$, so the curve is tangent to the line $y = x$. For large $x$, since $\sqrt{x^2 + 1} \approx x$, we have $\operatorname{arsinh} x \approx \ln(2x)$, meaning it grows logarithmically and very slowly. It has no vertical or horizontal asymptote.

Notable points

The graph passes through the origin $(0, 0)$. For example, $\operatorname{arsinh} 1 = \ln(1 + \sqrt{2}) \approx 0.8814$.

Relations and applications

It belongs to the family of inverse hyperbolic functions alongside $\operatorname{arcosh}$ and $\operatorname{artanh}$. It arises as the value of the integral $\displaystyle\int \frac{dx}{\sqrt{x^2 + 1}}$, appearing in arc-length calculations for the catenary and in computations in special relativity and geodesics.