Inverse Hyperbolic Tangent $y = \operatorname{artanh} x$

$\operatorname{artanh} x$ (the inverse hyperbolic tangent, or area hyperbolic tangent) is the inverse of the hyperbolic tangent $\tanh x = \dfrac{\sinh x}{\cosh x} = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}$. Since $\tanh$ maps all real numbers monotonically onto the open interval $(-1, 1)$, the domain of its inverse is $(-1, 1)$.

Definition and closed form

$y = \operatorname{artanh} x$ is the value $y$ satisfying $x = \tanh y$. Using logarithms it can be written as

This expression is valid when $\dfrac{1 + x}{1 - x} > 0$, that is for $-1 < x < 1$.

Domain and range

The domain is the open interval $(-1, 1)$ and the range is all real numbers $(-\infty, \infty)$. The endpoints $x = \pm 1$ are excluded.

Symmetry and monotonicity

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

is positive throughout the domain, so the function is strictly increasing.

Asymptotes and limits

As $x \to 1^-$, $\operatorname{artanh} x \to +\infty$, and as $x \to -1^+$, $\operatorname{artanh} x \to -\infty$. Hence the lines $x = 1$ and $x = -1$ are vertical asymptotes.

Notable points

The graph passes through the origin $(0, 0)$, and near the origin $\operatorname{artanh} x \approx x$. For example, $\operatorname{artanh} 0.5 = \frac{1}{2}\ln 3 \approx 0.5493$.

Relations and applications

$\operatorname{artanh}$ appears in statistics as Fisher's $z$-transformation of the correlation coefficient, and in special relativity as the rapidity used to add velocities. It also arises as the value of the integral $\displaystyle\int \frac{dx}{1 - x^2}$.