Definition:Inverse Cotangent/Arccotangent
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Definition
Real Numbers
From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.
From the same source, we also have that:
- $\cot x \to + \infty$ as $x \to 0^+$
- $\cot x \to - \infty$ as $x \to \pi^-$
Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.
This function is called arccotangent of $x$ and is written $\arccot x$.
Thus:
Complex Plane
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 denoted as
The symbol used to denote the arccotangent function is variously seen as:
- $\arccot$
- $\operatorname {acot}$
- $\operatorname {actn}$