Definition:Area Hyperbolic Function
Definition
Complex Area Hyperbolic Function
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 $z^2 + 1$.
Real Area Hyperbolic Function
Real Area Hyperbolic Sine
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 $x^2 + 1$.
Real Area Hyperbolic Cosine
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 $\map \arcosh x \ge 0$.
Real Area Hyperbolic Tangent
The inverse hyperbolic tangent $\artanh: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$
where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Real Area Hyperbolic Cotangent
The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$
where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Real Area Hyperbolic Secant
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 $\map \arsech x \ge 0$.
Real Area Hyperbolic Cosecant
The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:
- $\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$
where:
- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.