Definition:Inverse Tangent/Complex/Definition 2

Definition
Let $z \in \C$ be a complex number.

The inverse tangent of $z$ is the multifunction defined as:
 * $\tan^{-1} \left({z}\right) := \left\{{\dfrac 1 {2 i} \ln \left({\dfrac {i - z} {i + z}}\right) + 2 k \pi: k \in \Z}\right\}$

where $\ln$ denotes the complex natural logarithm as a multifunction.

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

Also see

 * Equivalence of Definitions of Complex Inverse Tangent Function


 * Definition:Complex Arctangent


 * Definition:Inverse Hyperbolic Tangent/Complex/Definition 2