Definition:Inverse Hyperbolic Cosecant

Complex Plane

Definition 1

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$

where $\map \csch w$ is the hyperbolic cosecant function.

Definition 2

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

where:

$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$

$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$

$\ln$ denotes the complex natural logarithm considered as a multifunction.

Real Numbers

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.

Also known as

The principal branch of the inverse hyperbolic cosecant is also known as the area hyperbolic cosecant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccosecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cosecant.

Also see

• Definition:Inverse Hyperbolic Sine
• Definition:Inverse Hyperbolic Cosine
• Definition:Inverse Hyperbolic Tangent
• Definition:Inverse Hyperbolic Cotangent
• Definition:Inverse Hyperbolic Secant

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