Definition:Inverse Hyperbolic Cosine/Complex/Definition 2
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Definition
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.
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 defined as
In expositions of the inverse hyperbolic functions, it is frequently the case that the $2 k \pi i$ constant is ignored, in order to simplify the presentation.
It is also commonplace to gloss over the multifunctional nature of the complex square root, and report this definition as:
- $\forall z \in \C: \map {\cosh^{-1} } z := \map \ln {z + \sqrt {z^2 - 1} }$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.56$: Inverse Hyperbolic Functions
- Weisstein, Eric W. "Inverse Hyperbolic Cosine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicCosine.html