Definition:Hyperbolic Cotangent

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Definition

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:

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


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.


Sources