Notation for transforming angle to an interval such as $(-\pi, \pi]$

Question

Is there a standard or at least often used way of denoting the operation of transforming an angle into a specific interval?

For example, I would like to denote in the equation $$ \alpha = \beta - \gamma $$ that after subtraction the resulting angle value is transformed to the interval $(-\pi, \pi]$.

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Edit: Right now I am thinking of using a notation like this, and just explaining what I mean by that in the text. $$ \alpha = (\beta - \gamma)_{\in (-\pi, \pi]} $$

But as far as I know this is not standard. (Would using a different type of brackets be more appropriate?)

Answer 1

You could say $\alpha \in (-\pi,\pi]$ such that $\alpha = \beta - \gamma +2n\pi,$ $n$ an integer.

(That is the transformation you had in mind, was it not? Adding/subtracting some multiple of $2\pi$?)

Answer 2

If you are speaking of the equivalent angle in that range, then the map $$f(\theta) = \pi - (\theta \operatorname{mod} 2\pi)$$ will produce the desired result. More generally, $$b - (\theta \operatorname{mod} (b-a))\in (a,b]$$ for $a<b$ because $$0\leq\theta\operatorname{mod} (b-a)<b-a$$

$$b-(b-a)<b-(\theta\operatorname{mod} (b-a))\leq b-0$$

$$a<b-(\theta\operatorname{mod} (b-a))\leq b$$