Equivalence of Definitions of Real Area Hyperbolic Cotangent
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Theorem
The following definitions of the concept of Real Area Hyperbolic Cotangent are equivalent:
Definition 1
The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \arcoth x := y \in \R: x = \coth y$
where $\coth y$ denotes the hyperbolic cotangent function.
Definition 2
The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$
where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Proof
Definition 1 implies Definition 2
| \(\ds x\) | \(=\) | \(\ds \coth y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \tanh y\) | Definition of Hyperbolic Cotangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \artanh \dfrac 1 x\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \map \ln {\dfrac {1 + \frac 1 x} {1 - \frac 1 x} }\) | Definition of Real Area Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }\) |
$\Box$
Definition 2 implies Definition 1
| \(\ds y\) | \(=\) | \(\ds \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \map \ln {\dfrac {1 + \frac 1 x} {1 - \frac 1 x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \artanh \dfrac 1 x\) | Definition of Real Area Hyperbolic Tangent | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 x\) | \(=\) | \(\ds \tanh y\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \coth y\) | Definition of Real Area Hyperbolic Cotangent |
$\Box$
Therefore:
| \(\text {(1)}: \quad\) | \(\ds x = \coth y\) | \(\implies\) | \(\ds y = \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }\) | Definition 1 implies Definition 2 | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds y = \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }\) | \(\implies\) | \(\ds x = \coth y\) | Definition 2 implies Definition 1 | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x = \coth y\) | \(\iff\) | \(\ds y = \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }\) |
$\blacksquare$