Definition:Inverse Hyperbolic Cotangent

Complex Plane

Let $S$ be the subset of the complex plane:

$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

Definition 1

The inverse hyperbolic cotangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\coth^{-1} } z := \set {w \in \C: z = \map \coth w}$

where $\map \coth w$ is the hyperbolic cotangent function.

Definition 2

The inverse hyperbolic cotangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\coth^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.

Real Numbers

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.

Also known as

The principal branch of the inverse hyperbolic cotangent is also known as the area hyperbolic cotangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

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

• Definition:Inverse Hyperbolic Sine
• Definition:Inverse Hyperbolic Cosine
• Definition:Inverse Hyperbolic Tangent
• Definition:Inverse Hyperbolic Secant
• Definition:Inverse Hyperbolic Cosecant

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