Arctanh

Arctanh, short for inverse hyperbolic tangent, is the inverse function of the hyperbolic tangent. Since tanh maps the real line onto the open interval (-1, 1), arctanh is defined as a function from (-1, 1) to the real numbers, giving the unique y with tanh(y) = x.

For real x with |x| < 1, arctanh(x) can be expressed as arctanh(x) = (1/2) ln((1 + x)/(1 - x)). Equivalently, arctanh(x) = (1/2)[ln(1 + x) − ln(1 − x)]. The derivative is d/dx arctanh(x) = 1/(1 − x^2) for |x| < 1. The function has the power series expansion arctanh(x) = x + x^3/3 + x^5/5 + …, valid for |x| < 1.

In the complex plane, arctanh(z) may be written as arctanh(z) = (1/2) [log(1 + z) − log(1 − z)]. It has branch points at z = ±1 with common branch cuts along the real axis for |z| ≥ 1, typically taken as the intervals (−∞, −1] and [1, ∞). A useful identity is arctanh(z) = −i arctan(i z) for complex z, relating it to the inverse tangent.

Arctanh appears in various contexts, including integration of rational functions, hyperbolic geometry, and probabilistic and statistical calculations involving hyperbolic functions.