Definition:Inverse Cotangent/Arccotangent
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 the arccotangent of $x$ and is written $\arccot x$.
Thus:
- The domain of the arccotangent is $\R$
- The image of the arccotangent is $\openint 0 \pi$.
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.
Terminology
There exists the popular but misleading notation $\cot^{-1} x$, which is supposed to denote the inverse cotangent function.
However, note that as $\cot x$ is not an injection, it does not have a well-defined inverse.
The $\arccot$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.
Therefore it is preferred to the notation $\cot^{-1} x$, which (as pointed out) can be confusing and misleading.
Sometimes, $\operatorname {Cot}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccot$.
However this can also be confusing due to the visual similarity between that and the lowercase $\text c$.
Some sources hyphenate: arc-cotangent.
Symbol
The symbol used to denote the arccotangent function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccotangent function is $\arccot$.
A variant symbol used to denote the arccotangent function is $\operatorname {acot}$.
A variant symbol used to denote the arccotangent function is $\operatorname {actn}$.