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