Equivalence of Definitions of Hyperbolic Secant
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Theorem
The following definitions of the concept of Hyperbolic Secant are equivalent:
Definition 1
The hyperbolic secant function is defined on the complex numbers as:
- $\sech: X \to \C$:
- $\forall z \in X: \sech z := \dfrac 2 {e^z + e^{-z} }$
where:
- $X = \set {z: z \in \C, \ e^z + e^{-z} \ne 0}$
Definition 2
The hyperbolic secant function is defined on the complex numbers as:
- $\sech: X \to \C$:
- $\forall z \in X: \sech z := \dfrac 1 {\cosh z}$
where:
- $\cosh$ is the hyperbolic cosine
- $X = \set {z: z \in \C, \ \cosh z \ne 0}$
Proof
\(\ds \forall z \in \set {z \in \C: \ e^z + e^{-z} \ne 0}: \, \) | \(\ds \) | \(\) | \(\ds \frac 2 {e^z + e^{-z} }\) | Definition 1 of Hyperbolic Secant | ||||||||||
\(\ds \forall z \in \set {z \in \C: \ \frac {e^z + e^{-z} } 2 \ne 0}: \, \) | \(\ds \) | \(=\) | \(\ds 1 / \frac {e^z + e^{-z} } 2\) | |||||||||||
\(\ds \forall z \in \set {z \in \C: \ \cosh z \ne 0}: \, \) | \(\ds \) | \(=\) | \(\ds 1 / \cosh z\) | Definition 2 of Hyperbolic Secant |
$\blacksquare$