Definition:Inverse Hyperbolic Secant/Complex
Definition
Definition 1
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.
Definition 2
The inverse hyperbolic secant is a multifunction defined as:
- $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
where:
- $\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
- $\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$
- $\ln$ denotes the complex natural logarithm as a multifunction.
Principal Branch
The principal branch of the complex inverse hyperbolic secant function is defined as:
- $\forall z \in \C: \map \Arsech z := \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
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.
In the complex plane, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic secant strictly for the principal branch.
Also see
- Definition:Complex Inverse Hyperbolic Sine
- Definition:Complex Inverse Hyperbolic Cosine
- Definition:Complex Inverse Hyperbolic Tangent
- Definition:Complex Inverse Hyperbolic Cotangent
- Definition:Complex Inverse Hyperbolic Cosecant
- Results about the inverse hyperbolic secant can be found here.
Sources
- Weisstein, Eric W. "Inverse Hyperbolic Secant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicSecant.html