Definition:Inverse Hyperbolic Tangent/Real/Definition 2
Definition
Let $S$ denote the open real interval:
- $S := \openint {-1} 1$
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.
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
Sources
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse hyperbolic function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse hyperbolic function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse hyperbolic function