Definition:Inverse Hyperbolic Cotangent/Real/Definition 2

Definition
Let $S$ denote the union of the unbounded open real intervals:
 * $S := \left({-\infty \,.\,.\, -1}\right) \cup \left({1 \,.\,.\, +\infty}\right)$

The inverse hyperbolic cotangent $\coth^{-1}: S \to \R$ is a real function defined on $S$ as:


 * $\forall x \in S: \coth^{-1} \left({x}\right) := \dfrac 1 2 \operatorname{ln} \left({\dfrac {x + 1} {x - 1} }\right)$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

Also known as
The inverse hyperbolic cotangent function is also known as the hyperbolic arccotangent function.

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Cotangent