Definition:Complex Area Hyperbolic Function
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Definition
Complex Area Hyperbolic Sine
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 $z^2 + 1$.
Complex Area Hyperbolic Cosine
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 $z^2 - 1$.
Complex Area Hyperbolic Tangent
The principal branch of the complex inverse hyperbolic tangent function is defined as:
- $\forall z \in \C: \map \Artanh z := \dfrac 1 2 \, \map \Ln {\dfrac {1 + z} {1 - z} }$
where $\Ln$ denotes the principal branch of the complex natural logarithm.
Complex Area Hyperbolic Cotangent
The principal branch of the complex inverse hyperbolic cotangent function is defined as:
- $\forall z \in \C: \map \Arcoth z := \dfrac 1 2 \map \Ln {\dfrac {z + 1} {z - 1} }$
where $\Ln$ denotes the principal branch of the complex natural logarithm.
Complex Area Hyperbolic Secant
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 $1 - z^2$.
Complex Area Hyperbolic Cosecant
The principal branch of the complex inverse hyperbolic cosecant function is defined as:
- $\forall z \in \C_{\ne 0}: \map \Arcsch z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.