Inverse Hyperbolic Cosine $y = \operatorname{arcosh} x$

$\operatorname{arcosh} x$ (the inverse hyperbolic cosine, or area hyperbolic cosine) is the inverse of the hyperbolic cosine $\cosh x = \dfrac{e^x + e^{-x}}{2}$. Since $\cosh$ is an even function taking the same value on both sides of $0$, it cannot be inverted directly. We therefore restrict it to the branch $x \ge 0$ (the principal branch) and call that inverse $\operatorname{arcosh}$.

Definition and closed form

$y = \operatorname{arcosh} x$ is the value $y$ satisfying $x = \cosh y$ with $y \ge 0$. Using logarithms it takes the closed form

For the square root to be real we need $x \ge 1$; for $x < 1$ there is no real value.

Domain and range

The domain is $x \ge 1$, that is $[1, \infty)$. The range is $y \ge 0$, that is $[0, \infty)$.

Monotonicity and symmetry

The derivative

is always positive, so the function is strictly increasing over its whole domain. Unlike $\operatorname{arsinh}$ it is neither even nor odd and has no symmetry.

Notable point and tangent

The graph begins at the point $(1, 0)$. There the denominator of the derivative tends to $0$, so the slope becomes infinite and the tangent line is vertical. A small increase of $x$ above $1$ makes $y$ rise steeply.

Limits and growth rate

For large $x$, since $\sqrt{x^2 - 1} \approx x$, we have $\operatorname{arcosh} x \approx \ln(2x)$, growing logarithmically and slowly. It has no horizontal asymptote.

Specific values

For example $\operatorname{arcosh} 1 = 0$ and $\operatorname{arcosh} 2 = \ln(2 + \sqrt{3}) \approx 1.317$.

Applications

It arises as the value of the integral $\displaystyle\int \frac{dx}{\sqrt{x^2 - 1}}$ and appears in geometry involving hyperbolas and in physics such as electromagnetism and heat conduction.