Definition:Inverse Hyperbolic Tangent/Complex/Definition 2

Definition
The inverse hyperbolic tangent is a multifunction $\tanh^{-1}: \C \to \C$ defined as:


 * $\forall x \in \C: \tanh^{-1} \left({x}\right) = \dfrac 1 2 \ln \left({\dfrac {1 + x} {1 - x} }\right)$

where $\ln$ is the complex natural logarithm function.

As $\ln$ is a multifunction it follows that $\tanh^{-1}$ is likewise a multifunction.

Also see

 * Equivalence of Definitions of Inverse Hyperbolic Tangent


 * Definition:Inverse Hyperbolic Sine
 * Definition:Inverse Hyperbolic Cosine
 * Definition:Inverse Hyperbolic Cotangent
 * Definition:Inverse Hyperbolic Secant
 * Definition:Inverse Hyperbolic Cosecant