Definition:Inverse Hyperbolic Cotangent/Real/Definition 1
Definition
Let $S$ denote the union of the unbounded open real intervals:
- $S := \openint \gets {-1} \cup \openint 1 \to$
The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \arcoth x := y \in \R: x = \coth y$
where $\coth y$ denotes the hyperbolic cotangent function.
Symbol
The symbol used to denote the area hyperbolic cotangent function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cotangent function is $\arcoth$.
A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {acoth}$.
A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {actnh}$.
Also known as
The real inverse hyperbolic cotangent is also known as the (real) area hyperbolic cotangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Note that as the real hyperbolic cotangent $\coth$ is injective, its inverse is properly a function on its domain.
Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.
Some sources refer to it as hyperbolic arccotangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cotangent.