Definition:Area Hyperbolic Secant
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Definition
Complex
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 $1 - z^2$.
Real
The principal branch of the real inverse hyperbolic secant function is defined as:
- $\forall x \in S: \map \arsech x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {1 - x^2}$ specifically denotes the positive square root of $x^2 - 1$
That is, where $\map \arsech x \ge 0$.
Symbol
The symbol used to denote the area hyperbolic secant function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic secant function is $\arsech$.
A variant symbol used to denote the area hyperbolic secant function is $\operatorname {asech}$.