# Definition:Hyperbolic Cotangent

## Definition

The hyperbolic cotangent is one of the hyperbolic functions:

### Definition 1

The hyperbolic cotangent function is defined on the complex numbers as:

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {e^z + e^{-z} } {e^z - e^{-z}}$

where:

$X = \set {z : z \in \C, \ e^z - e^{-z} \ne 0}$

### Definition 2

The hyperbolic cotangent function is defined on the complex numbers as:

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {\cosh z} {\sinh z}$

where:

$\sinh$ is the hyperbolic sine
$\cosh$ is the hyperbolic cosine
$X = \set {z : z \in \C, \ \sinh z \ne 0}$

### Definition 3

The hyperbolic cotangent function is defined on the complex numbers as:

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {e^{2 z} + 1} {e^{2 z} - 1}$

where:

$X = \set {z : z \in \C, \ e^{2 z} - 1 \ne 0}$

### Definition 4

The hyperbolic cotangent function is defined on the complex numbers as:

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac 1 {\tanh z}$

where:

$\tanh$ is the hyperbolic tangent
$X = \set {z : z \in \C, \ \sinh z \ne 0}$
where $\sinh$ is the hyperbolic sine.

### Real Hyperbolic Cotangent

On the real numbers it is defined similarly.

The real hyperbolic cotangent function is defined on the real numbers as:

$\coth: \R_{\ne 0} \to \R$:
$\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$

where it is noted that at $x = 0$:

$e^x - e^{-x} = 0$

and so $\coth x$ is not defined at that point.

## Also denoted as

The notation $\operatorname {cth} z$ can also be found for hyperbolic cotangent.

## Also see

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