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