Definition:Inverse Cotangent/Complex
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Definition
Let $S$ be the subset of the complex plane:
- $S = \C \setminus \left\{{0 + i, 0 - i}\right\}$
Definition 1
The inverse cotangent is a multifunction defined on $S$ as:
- $\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{w \in \C: \cot \left({w}\right) = z}\right\}$
where $\cot \left({w}\right)$ is the cotangent of $w$.
Definition 2
The inverse cotangent is a multifunction defined on $S$ as:
- $\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{\dfrac 1 {2 i} \ln \left({\dfrac {z + i} {z - i}}\right) + k \pi: k \in \Z}\right\}$
where $\ln$ denotes the complex natural logarithm as a multifunction.
Arccotangent
The principal branch of the complex inverse cotangent function is defined as:
- $\map \arccot z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {z + i} {z - i} }$
where $\Ln$ denotes the principal branch of the complex natural logarithm.