Definition:Inverse Tangent/Complex
< Definition:Inverse Tangent(Redirected from Definition:Complex Inverse Tangent)
Jump to navigation
Jump to search
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
- 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.