artanhx

artanhx, commonly written arctanh(x), is the inverse hyperbolic tangent function. It returns the value y such that tanh(y) = x. It is defined for real x with |x| < 1 and extended to complex values via analytic continuation. A convenient real formula is artanh(x) = 1/2 ln((1+x)/(1−x)).

Domain and range: For real arguments, artanh maps the open interval (-1, 1) onto the entire real line. As x approaches 1 from below, artanh(x) tends to +∞, and as x approaches −1 from above, artanh(x) tends to −∞. The function has vertical asymptotes at x = ±1.

Derivative and series: The derivative is d/dx artanh(x) = 1/(1 − x^2) for |x| < 1. The function admits a power series representation artanh(x) = x + x^3/3 + x^5/5 + … = ∑_{n=0}^∞ x^{2n+1}/(2n+1) for |x| < 1.

Inverse relation and symmetry: artanh is the inverse of tanh, so y = artanh(x) iff tanh(y) = x. The function is odd: artanh(−x) = −artanh(x).

Complex extension: In the complex plane, artanh is analytic with a branch cut along (−∞, −1] and [1, ∞), reflecting the multivalued nature of the logarithm used in its definition.

See also: tanh, arctanh, inverse hyperbolic functions.