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