Equivalence of Definitions of Hyperbolic Tangent

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Theorem

The following definitions of the concept of Hyperbolic Tangent are equivalent:

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:

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


Proof

\(\displaystyle \forall z \in \set {z \in \C: \ e^{2 z} + 1 \ne 0}: \ \ \) \(\displaystyle \) \(\) \(\displaystyle \frac {e^{2 z} - 1} {e^{2 z} + 1}\) Definition 3 of Hyperbolic Tangent
\(\displaystyle \forall z \in \set {z \in \C: \ e^z + e^{-z} \ne 0}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle \frac {e^z \paren {e^z - e^{-z} } } {e^z \paren {e^z + e^{-z} } }\)
\(\displaystyle \forall z \in \set {z \in \C: \ e^z + e^{-z} \ne 0}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle \frac {e^z - e^{-z} } {e^z + e^{-z} }\) Definition 1 of Hyperbolic Tangent
\(\displaystyle \forall z \in \set {z \in \C: \ \frac {e^z + e^{-z} } 2 \ne 0}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {\dfrac {e^z - e^{-z} } 2} } {\paren {\dfrac {e^z + e^{-z} } 2} }\)
\(\displaystyle \forall z \in \set {z \in \C: \ \cosh z \ne 0}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh z} {\cosh z}\) Definition 2 of Hyperbolic Tangent

$\blacksquare$