# Definition:Hyperbolic Tangent

## Definition

### Definition 1

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

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

where:

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

### Definition 2

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

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

where:

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

### Definition 3

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

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

where:

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

## Also denoted as

• The notation $\operatorname{th} z$ is also found for $\tanh z$.

## Also see

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

## Linguistic Note

The usual symbol tanh for hyperbolic tangent is awkward to pronounce.

Some pedagogues say it as tansh, and some as than (where the th is voiceless as in thin, for example).

Others prefer the mouthful which is hyperbolic tan