# Definition:Inverse Tangent/Arctangent

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## Definition

### Real Numbers

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

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 $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ 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 $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

### 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.