Definition:Inverse Hyperbolic Tangent/Real
Definition
Let $S$ denote the open real interval:
- $S := \openint {-1} 1$
Definition 1
The inverse hyperbolic tangent $\artanh: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map \artanh x := y \in \R: x = \map \tanh y$
where $\map \tanh y$ denotes the hyperbolic tangent function.
Definition 2
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.
Graph of Inverse Hyperbolic Tangent
The graph of the real inverse hyperbolic tangent function appears as:
Also known as
The real inverse hyperbolic tangent is also known as the (real) area hyperbolic tangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Note that as the real hyperbolic tangent $\tanh$ is injective, its inverse is properly a function on its domain.
Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.
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
- Definition:Real Inverse Hyperbolic Sine
- Definition:Real Inverse Hyperbolic Cosine
- Definition:Real Inverse Hyperbolic Cotangent
- Definition:Real Inverse Hyperbolic Secant
- Definition:Real Inverse Hyperbolic Cosecant
- Results about the inverse hyperbolic tangent can be found here.
Sources
- Weisstein, Eric W. "Inverse Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicTangent.html