Definition:Inverse Tangent/Arctangent

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Definition

Real Numbers

Arctangent Function

From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From the same source, we also have that:

$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$


Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$.


This function is called arctangent of $x$ and is written $\arctan x$.

Thus:

The domain of $\arctan x$ is $\R$
The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.


Complex Plane

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.