artanh

artanh, often written as atanh, is the inverse function of the hyperbolic tangent. The hyperbolic tangent maps the real line onto the interval (-1, 1), so artanh is real-valued for real inputs x in (-1, 1). Its complex extension has branch points at ±1.

On its real domain, artanh x can be defined by the formula artanh x = (1/2) ln((1+x)/(1−x)) for x in (-1, 1). This expression leads to the associated power series artanh x = x + x^3/3 + x^5/5 + … for |x| < 1.

Artanh is an odd function, satisfying artanh(−x) = −artanh(x). Its derivative is d/dx artanh x = 1/(1 − x^2) for |x| < 1. As the inverse of tanh on the real line, it satisfies tanh(artanh x) = x for x in (−1, 1).

In the complex plane, artanh z can be written as (1/2)[ln(1+z) − ln(1−z)], with the logarithm introducing branch cuts. The principal value typically involves branch cuts along (−∞, −1] and [1, ∞).

Applications of artanh arise in solving equations involving hyperbolic tangents, in certain integration problems, and in various areas of mathematics and physics. It is part of the family of inverse hyperbolic functions, alongside arsinh, arccosh, and arccoth.