Definition:Inverse Hyperbolic Tangent/Complex/Definition 2
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Definition
Let $S$ be the subset of the complex plane:
- $S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$
The inverse hyperbolic tangent is a multifunction defined on $S$ as:
- $\forall z \in S: \map {\tanh^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$
where $\ln$ denotes the complex natural logarithm considered as a multifunction.
Also known as
The inverse hyperbolic tangent is also known as the area hyperbolic tangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Some sources refer to it as hyperbolic arctangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic tangent.
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.57$: Inverse Hyperbolic Functions
- Weisstein, Eric W. "Inverse Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicTangent.html