y = \operatorname{artanh} x

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

$\operatorname{artanh} x$ , the inverse hyperbolic tangent or area hyperbolic tangent, is the inverse of $\tanh x$ ¹. Since $\tanh$ maps the whole real line 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 $y$ with $x = \tanh y$ . Written with a logarithm it takes the following form.

\operatorname{artanh} x = \frac{1}{2}\ln\frac{1 + x}{1 - x}

The expression makes sense when $\dfrac{1 + x}{1 - x} > 0$ , that is for $-1 < x < 1$ .

Domain and range

The domain is the open interval $(-1, 1)$
The range is all real numbers
It increases monotonically
It is an odd function

The endpoints $x = \pm 1$ are excluded, which is the counterpart of $\tanh$ never quite reaching the values $\pm 1$ .

Symmetry and monotonicity

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

\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^2}

Inside the domain the denominator is positive, so the derivative is always positive and the function increases monotonically.

Asymptotes

Approach	$\operatorname{artanh} x$
$x \to 1^{-}$	$\to +\infty$
$x \to -1^{+}$	$\to -\infty$

The lines $x = 1$ and $x = -1$ are vertical asymptotes. Being the inverse of the bounded $\tanh$ , it grows without bound inside a narrow interval.

Notable values

$x$	$\operatorname{artanh} x$
$0$	$0$
$0.5$	$\dfrac{1}{2}\ln 3 \approx 0.5493$
$0.9$	$\approx 1.4722$
$0.99$	$\approx 2.6467$

Near the origin $\operatorname{artanh} x \approx x$ , but the curve rears up as it nears the ends. The divergence is only logarithmic, however: moving $x$ from $0.99$ to $0.999$ raises the value only to about $3.8$ .

Series expansion

The expansion about the origin has only odd powers and converges for $|x| < 1$ .

\operatorname{artanh} x = x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots

The coefficients are the same as those of $\arctan x$ ; only the alternation of signs is missing.

Applications

Fisher's $z$ transformation in statistics, applied to a correlation coefficient²
The addition of velocities in special relativity, through rapidity
The result of the integral $\int \frac{dx}{1 - x^2}$

In terms of rapidity, the addition of velocities becomes plain addition once passed through $\operatorname{artanh}$ . It can be read as a change of coordinates that stretches a bounded world of velocities out onto an unbounded line.

Inverse hyperbolic functions, Wikipedia
Fisher transformation, Wikipedia

Inverse Hyperbolic Tangent y=artanh⁡xy = \operatorname{artanh} xy=artanhx