Definition:Inverse Hyperbolic Sine/Complex/Definition 2

Definition
Let $\sinh: \C \to \C$ denote the hyperbolic sine as defined on the set of complex numbers.

The inverse hyperbolic sine is a multifunction $\sinh^{-1}: \C \to \C$ defined as:


 * $\forall z \in \C: \sinh^{-1} \left({x}\right) := \ln \left({z + \sqrt{\left|{z^2 + 1}\right|} e^{\frac i 2 \arg \left({z^2 + 1}\right)} }\right) + 2 k \pi$

where:
 * $\sqrt{\left|{z^2 + 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$
 * $\arg \left({z^2 + 1}\right)$ denotes the argument of $z^2 + 1$
 * $\ln$ denotes the complex natural logarithm considered as a multifunction.

Also see

 * Equivalence of Definitions of Inverse Hyperbolic Sine