Equivalence of Definitions of Real Area Hyperbolic Cosecant
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Theorem
The following definitions of the concept of Real Area Hyperbolic Cosecant are equivalent:
Definition 1
The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:
- $\forall x \in \R_{\ne 0}: \map \arcsch x := y \in \R: x = \map \csch y$
where $\map \csch y$ denotes the hyperbolic cosecant function of $y$.
Definition 2
The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:
- $\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$
where:
- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Proof
Definition 1 implies Definition 2
| \(\ds x\) | \(=\) | \(\ds \csch y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \sinh y\) | Definition of Hyperbolic Cosecant | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\paren {\dfrac 1 x}^2 + 1} }\) | Definition of Real Inverse Hyperbolic Sine | ||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }\) |
If $x \ge 0$, then:
| \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }\) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } x}\) | Positive Square Root is Positive |
If $x < 0$, then:
| \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }\) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {-x} }\) | Positive Square Root is Positive |
That is:
- $y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$
$\Box$
Definition 2 implies Definition 1
| \(\ds y\) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac 1 {x^2} + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \arsinh \dfrac 1 x\) | Definition of Real Area Hyperbolic Sine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \sinh y\) | Definition of Hyperbolic Sine | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \csch y\) | Definition of Hyperbolic Cosecant |
$\Box$
Therefore:
| \(\text {(1)}: \quad\) | \(\ds x = \csch y\) | \(\implies\) | \(\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }\) | Definition 1 implies Definition 2 | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }\) | \(\implies\) | \(\ds x = \csch y\) | Definition 2 implies Definition 1 | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x = \csch y\) | \(\iff\) | \(\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }\) |
$\blacksquare$