Definition:Area Hyperbolic Sine
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Definition
Complex
The principal branch of the complex inverse hyperbolic sine function is defined as:
- $\forall z \in \C: \map \Arsinh z := \map \Ln {z + \sqrt {z^2 + 1} }$
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 inverse hyperbolic sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:
- $\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number
- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.