artanh function

Martin McBride, 2020-02-03
Tags artanh
Categories hyperbolic functions

The artanh function is a hyperbolic function. It is the inverse of the tanh, and is also known as the inverse hyperbolic tangent function.

Equation and graph

The artanh function is defined as the inverse of tanh, ie if:

$$ x = \tanh y $$


$$ y = \operatorname{artanh} x $$

There is a also a formula for finding artanh directly:

$$ \operatorname{artanh} x =\frac12\ln\left(\frac{1+x}{1-x}\right) $$

Here is a graph of the function:

The function is only valid for -1 < x < 1.

artanh as inverse of tanh

This animation illustrates the relationship between the tanh function and the artanh function:

The first, blue, curve is the tanh function.

The grey dashed line is the line $y=x$.

The second, red, curve is the artanh function. As with any inverse function, it is identical to the original function reflected in the line $y=x$.

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