Definition:Inverse Hyperbolic Sine

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Definition

Complex Plane

Definition 1

The inverse hyperbolic sine is a multifunction defined as:

$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{w \in \C: z = \sinh \left({w}\right)}\right\}$

where $\sinh \left({w}\right)$ is the hyperbolic sine function.


Definition 2

The inverse hyperbolic sine is a multifunction defined as:

$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 + 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 + 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$

where:

$\sqrt{\left|{z^2 + 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$
$\arg \left({z^2 + 1}\right)$ denotes the argument of $z^2 + 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.


Hyperbolic Arcsine

The principal branch of the complex inverse hyperbolic sine function is defined as:

$\forall z \in \C: \map {\Sinh^{-1} } z := \map \Ln {z + \sqrt {z^2 + 1} }$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.


Real Numbers

Definition 1

The inverse hyperbolic sine $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as:

$\forall x \in \R: \map {\sinh^{-1} } x := y \in \R: x = \map \sinh y$

where $\map \sinh y$ denotes the hyperbolic sine function.


Definition 2

The inverse hyperbolic sine $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as:

$\forall x \in \R: \map {\sinh^{-1} } x := \map \ln {x + \sqrt {x^2 + 1} }$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number
$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.


Also see

  • Results about the inverse hyperbolic sine can be found here.