atanhx

atanhx, also known as the inverse hyperbolic tangent or arc hyperbolic tangent, is a mathematical function. It is the inverse function of the hyperbolic tangent, tanh(x). The notation atanhx is commonly used, though arctanhx is also frequently seen.

The domain of atanhx is the open interval (-1, 1). For any value x within this interval, atanhx(x) returns a real number. The range of atanhx is all real numbers. Mathematically, atanhx(x) is defined as half the natural logarithm of the ratio of 1+x to 1-x. That is, atanhx(x) = (1/2) * ln((1+x)/(1-x)).

This function has applications in various fields, including mathematics, physics, and engineering. For instance, it appears in solutions to differential equations and in the study of hyperbolic geometry. The graph of atanhx is symmetric with respect to the origin, indicating it is an odd function. As x approaches 1 from the left, atanhx(x) approaches positive infinity, and as x approaches -1 from the right, atanhx(x) approaches negative infinity.