Equivalence of Definitions of Real Area Hyperbolic Sine
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Theorem
The following definitions of the concept of Real Area Hyperbolic Sine are equivalent:
Definition 1
The inverse hyperbolic sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:
- $\forall x \in \R: \map \arsinh x := y \in \R: x = \map \sinh y$
where $\map \sinh y$ denotes the hyperbolic sine function.
Definition 2
The inverse hyperbolic sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:
- $\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number
- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.
Proof
Definition 1 implies Definition 2
Let $x = \sinh y$.
Let $z = e^y$.
Then:
| \(\ds x\) | \(=\) | \(\ds \frac {e^y - e^{- y} } 2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 x\) | \(=\) | \(\ds e^y - e^{- y}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 x e^y\) | \(=\) | \(\ds e^{2 y} - 1\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds z^2 - 2 x z - 1\) | \(=\) | \(\ds 0\) | Power of Power | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \frac {2 x \pm \sqrt {\paren {-2 x}^2 + 4} } 2\) | Quadratic Formula | ||||||||||
| \(\ds \) | \(=\) | \(\ds x \pm \sqrt {x^2 + 1}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds e^y\) | \(=\) | \(\ds x \pm \sqrt {x^2 + 1}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \map \ln {x \pm \sqrt {x^2 + 1} }\) |
If $x \ge 0$, then:
| \(\ds \sqrt {x^2 + 1}\) | \(>\) | \(\ds \sqrt {x^2}\) | Square Root is Strictly Increasing | |||||||||||
| \(\ds \) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sqrt {x^2 + 1}\) | \(>\) | \(\ds x\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x - \sqrt {x^2 + 1}\) | \(<\) | \(\ds 0\) |
If $x < 0$, then:
| \(\text {(1)}: \quad\) | \(\ds -\sqrt {x^2 + 1}\) | \(<\) | \(\ds 0\) | Positive Square Root is Positive | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds x\) | \(<\) | \(\ds 0\) | Assumption | ||||||||||
| \(\ds x - \sqrt {x^2 + 1}\) | \(<\) | \(\ds 0\) | $(1) + (2)$ |
Since the natural logarithm of a negative number is not defined:
- $y = \map \ln {x + \sqrt {x^2 + 1} }$
$\Box$
Definition 2 implies Definition 1
Let $y = x + \sqrt {x^2 + 1}$.
| \(\ds \map \sinh {\map \ln {x + \sqrt {x^2 + 1} } }\) | \(=\) | \(\ds \map \sinh {\ln y}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{\ln y} - e^{-\ln y} } 2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {y - \frac 1 y} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {y^2 - 1} {2 y}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {x + \sqrt {x^2 + 1} }^2 - 1} {2 x + 2 \sqrt {x^2 + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^2 + 2 x \sqrt {x^2 + 1} + \paren {x^2 + 1} - 1} {2 x + 2 \sqrt {x^2 + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 x^2 + 2 x \sqrt {x^2 + 1} } {2 x + 2 \sqrt {x^2 + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {2 x + 2 \sqrt {x^2 + 1} } x} {2 x + 2 \sqrt {x^2 + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x\) |
$\Box$
Therefore:
| \(\text {(1)}: \quad\) | \(\ds x = \sinh y\) | \(\implies\) | \(\ds y = \map \ln {x + \sqrt {x^2 + 1} }\) | Definition 1 implies Definition 2 | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds y = \map \ln {x + \sqrt {x^2 + 1} }\) | \(\implies\) | \(\ds x = \sinh y\) | Definition 2 implies Definition 1 | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x = \sinh y\) | \(\iff\) | \(\ds y = \map \ln {x + \sqrt {x^2 + 1} }\) |
$\blacksquare$