Definition:Inverse Hyperbolic Secant/Complex/Definition 2

Definition
The inverse hyperbolic secant is a multifunction defined as:


 * $\forall z \in \C: \operatorname{sech}^{-1} \left({z}\right) := \left\{{\ln \left({\dfrac {1 + \sqrt{\left|{1 - z^2}\right|} e^{\left({1 / 2}\right) \arg \left({1 - z^2}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$

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

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

Also defined as
This concept is also reported as:
 * $\tan^{-1} \left({z}\right) := \left\{{\ln \left({\dfrac 1 z + \sqrt{\dfrac 1 {z^2} - 1}}\right) + 2 k \pi: k \in \Z}\right\}$

or:
 * $\tan^{-1} \left({z}\right) := \left\{{ \ln \left({ \dfrac 1 z + \sqrt{\left({ \dfrac 1 z + 1 }\right)} \sqrt{\left({\dfrac 1 z - 1}\right)} }\right) + 2 k \pi: k \in \Z }\right\}$

In the above, the complexity arising from the multifunctional nature of the square root has been omitted for the purpose of simplification.

Also see

 * Equivalence of Definitions of Inverse Hyperbolic Secant


 * Definition:Inverse Secant/Complex/Definition 2