# Definition:Inverse Hyperbolic Cosecant/Complex/Definition 1

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## Definition

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.

## 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

## Sources

- Weisstein, Eric W. "Inverse Hyperbolic Cosecant." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicCosecant.html