Definition:Inverse Hyperbolic Cosine
Definition
Complex Plane
Definition 1
The inverse hyperbolic cosine is a multifunction defined as:
- $\forall z \in \C: \map {\cosh^{-1} } z := \set {w \in \C: z = \map \cosh w}$
where $\map \cosh w$ is the hyperbolic cosine function.
Definition 2
The inverse hyperbolic cosine is a multifunction defined as:
- $\forall z \in \C: \map {\cosh^{-1} } z := \set {\map \ln {z + \sqrt {\size {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } + 2 k \pi i: k \in \Z}$
where:
- $\sqrt {\size {z^2 - 1} }$ denotes the positive square root of the complex modulus of $z^2 - 1$
- $\map \arg {z^2 - 1}$ denotes the argument of $z^2 - 1$
- $\ln$ denotes the complex natural logarithm considered as a multifunction.
Real Numbers
Let $S$ denote the subset of the real numbers:
- $S = \set {x \in \R: x \ge 1}$
Definition 1
The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:
- $\forall x \in S: \map {\cosh^{-1} } x := \set {y \in \R: x = \map \cosh y}$
where $\map \cosh y$ denotes the hyperbolic cosine function.
Definition 2
The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:
- $\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x \pm \sqrt {x^2 - 1} }$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {x^2 - 1}$ denotes the square root of $x^2 - 1$
Hence for $x > 1$, $\map {\cosh^{-1} } x$ has $2$ values.
For $x < 1$, $\map {\cosh^{-1} } x$ is not defined.
Principal Branch
Complex Plane
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 $z^2 - 1$.
Real Numbers
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 $\map \arcosh x \ge 0$.
Also known as
The principal branch of the inverse hyperbolic cosine is known as the area hyperbolic cosine, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Some sources refer to it as hyperbolic arccosine, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cosine.
Also see
- Definition:Inverse Hyperbolic Sine
- Definition:Inverse Hyperbolic Tangent
- Definition:Inverse Hyperbolic Cotangent
- Definition:Inverse Hyperbolic Secant
- Definition:Inverse Hyperbolic Cosecant
- Results about the inverse hyperbolic cosine can be found here.