Equivalence of Definitions of Real Area Hyperbolic Secant
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Theorem
The following definitions of the concept of Real Area Hyperbolic Secant are equivalent:
Definition 1
The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map {\sech^{-1} } x := y \in \R_{\ge 0}: x = \map \sech y$
where $\map \sech y$ denotes the hyperbolic secant function.
Definition 2
The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map {\sech^{-1} } x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$
Proof
Definition 1 implies Definition 2
| \(\ds x\) | \(=\) | \(\ds \sech y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \cosh y\) | Definition of Hyperbolic Secant | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\paren {\dfrac 1 x}^2 - 1} }\) | Definition of Real Inverse Hyperbolic Cosine | ||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 - x^2} {x^2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 - x^2} } x}\) | as $x > 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}\) |
$\Box$
Definition 2 implies Definition 1
| \(\ds y\) | \(=\) | \(\ds \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 - x^2} {x^2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\paren {\dfrac 1 x}^2 - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \arcosh \dfrac 1 x\) | Definition of Real Area Hyperbolic Cosine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \cosh y\) | Definition of Hyperbolic Cosine | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \sech y\) | Definition of Hyperbolic Secant |
$\Box$
Therefore:
| \(\text {(1)}: \quad\) | \(\ds x = \sech y\) | \(\implies\) | \(\ds y = \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}\) | Definition 1 implies Definition 2 | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds y = \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}\) | \(\implies\) | \(\ds x = \sech y\) | Definition 2 implies Definition 1 | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x = \sech y\) | \(\iff\) | \(\ds y = \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}\) |
$\blacksquare$