Definition:Real Area Hyperbolic Function
Definition
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.