Definition:Hyperbolic Tangent

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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.


Sources