Definition:Inverse Cotangent/Complex

From ProofWiki
Jump to navigation Jump to search

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.


Also see