Definition:Inverse Hyperbolic Secant/Complex/Definition 1

Definition

The inverse hyperbolic secant is a multifunction defined as:

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

where $\map \sech w$ is the hyperbolic secant function.

Also known as

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

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

Also see

• Equivalence of Definitions of Complex Inverse Hyperbolic Secant

Sources

• Weisstein, Eric W. "Inverse Hyperbolic Secant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicSecant.html