Definition:Inverse Tangent/Complex/Definition 2

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

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


 * $\forall z \in S: \tan^{-1} \paren z := \set {\dfrac 1 {2 i} \ln \paren {\dfrac {i - z} {i + z} } + k \pi: k \in \Z}$

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

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

Also see

 * Equivalence of Definitions of Complex Inverse Tangent Function


 * Definition:Complex Arctangent


 * Definition:Inverse Hyperbolic Tangent/Complex/Definition 2