Definition:Inverse Tangent/Complex

Definition

Let $S$ be the subset of the complex plane:

$S = \C \setminus \set {0 + i, 0 - i}$

Definition 1

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

$\forall z \in S: \inv \tan z := \set {w \in \C: \map \tan w = z}$

where $\map \tan w$ is the tangent of $w$.

Definition 2

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

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

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

Arctangent

The principal branch of the complex inverse tangent function is defined as:

$\map \arctan z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {i - z} {i + z} }$

where $\Ln$ denotes the principal branch of the complex natural logarithm.

Also see

• Equivalence of Definitions of Complex Inverse Tangent Function

• Definition:Complex Inverse Sine
• Definition:Complex Inverse Cosine
• Definition:Complex Inverse Cotangent
• Definition:Complex Inverse Secant
• Definition:Complex Inverse Cosecant

• Results about the inverse tangent function can be found here.