Definition:Inverse Tangent/Complex/Definition 2

Definition
Let $S$ be the subset of the complex plane:
 * $S = \C \setminus \left\{{0 + i, 0 - i}\right\}$

The inverse tangent is a multifunction defined on $S$ as:


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

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

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

Also see

 * Equivalence of Definitions of Complex Inverse Tangent Function


 * Definition:Complex Arctangent


 * Definition:Inverse Hyperbolic Tangent/Complex/Definition 2