Definition:Area Hyperbolic Function

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 $z^2 + 1$.

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.