Definition:Inverse Hyperbolic Cotangent/Real

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Definition

Let $S$ denote the union of the unbounded open real intervals:

$S := \openint \gets {-1} \cup \openint 1 \to$


Definition 1

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.


Definition 2

The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$

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


Graph of Inverse Hyperbolic Cotangent

The graph of the real inverse hyperbolic cotangent function appears as:


Arcoth.png


Also denoted as

The symbol used to denote the area hyperbolic cotangent function is variously seen as:

  • $\arcoth$
  • $\operatorname {acoth}$
  • $\operatorname {actnh}$


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.


Also see

  • Results about the inverse hyperbolic cotangent can be found here.


Sources

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