# Definition:Hyperbolic Cotangent/Definition 3

## Definition

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:

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