RealRange(a, b) represents a real interval, that is, a segment of the real line specified by the values of its two extremes a, b, which can be any two real numbers (possibly $\mathrm{\infty }$) satisfying $a\le b$. The interval is closed, i.e., includes its extremes, unless they are represented by Open(a) (and/or Open(b)), in which case the interval is open with respect to a (and/or b).

Similarly ComplexRange($\mathrm{\alpha }$, $\mathrm{\beta }$) represents a complex interval, that is, a rectangle in the complex plane specified by two points $\mathrm{\alpha }$ = a + b I and $\mathrm{\beta }$ = c + d I, where {a, b, c, d} represent real numbers. This complex interval is closed and so includes the four line segments $\left(a\,c\right)$, $\left(b\,d\right)$ and the other two parallel segments delimiting the rectangle, unless any of its four real extremes $\left\{a\,b\,c\,d\right\}$ is entered like, for instance, Open(a), in which case the interval is "open" with respect to that point. In the presence of more Open extremes the interval may be open with respect to one or more delimiting segments.

A rapid picture of what is and what is not included in a given complex range is obtained by converting it to a real range or converting it to a relation - see the examples.

Note: in Maple, by convention, when you say, for instance, $z\le 1$, it is implicitly assumed that $\mathrm{\Im }\left(z\right)=0$.

$\mathrm{RealRange}\left(0\,0\right)$

$0$

$\mathrm{ComplexRange}\left(0\,0\right)$

$\mathrm{ComplexRange}\left(0\,0\right)$

$\mathrm{convert}\left(\,\mathrm{RealRange}\right)$

$0$

$\mathrm{CR}≔\mathrm{ComplexRange}\left(-1-\mathrm{I}\,\mathrm{Open}\left(0\right)+\mathrm{I}\right)$

$\mathrm{CR}≔\mathrm{ComplexRange}\left(\mathrm{-1}-\mathrm{I}\,\mathrm{Open}\left(0\right)+\mathrm{I}\right)$

$\mathrm{convert}\left(\mathrm{CR}\,\mathrm{RealRange}\right)$

$\left[\mathrm{-1}\,0\right),\mathrm{I}\left[\mathrm{-1}\,1\right]$

$z::\mathrm{CR}$

$z::\mathrm{ComplexRange}\left(\mathrm{-1}-\mathrm{I}\,\mathrm{Open}\left(0\right)+\mathrm{I}\right)$

$\mathrm{convert}\left(\,\mathrm{RealRange}\right)$

$\mathrm{\Re }\left(z\right)::\left[\mathrm{-1}\,0\right),\mathrm{\Im }\left(z\right)::\left[\mathrm{-1}\,1\right]$

$z\mathbf{in}\mathrm{CR}$

$z\in \mathrm{ComplexRange}\left(\mathrm{-1}-\mathrm{I}\,\mathrm{Open}\left(0\right)+\mathrm{I}\right)$

$\mathrm{convert}\left(\,\mathrm{RealRange}\right)$

$\mathrm{\Re }\left(z\right)\in \left[\mathrm{-1}\,0\right),\mathrm{\Im }\left(z\right)\in \left[\mathrm{-1}\,1\right]$

$\mathrm{ComplexRange}\left(\mathrm{\alpha }\,\mathrm{\beta }\right)$

$\mathrm{ComplexRange}\left(\mathrm{\alpha }\,\mathrm{\beta }\right)$

$\mathrm{convert}\left(\,\mathrm{RealRange}\right)$

$\mathrm{ComplexRange}\left(\mathrm{\alpha }\,\mathrm{\beta }\right)$

$\mathrm{CR}≔z\mathbf{in}\mathrm{ComplexRange}\left(-1-\mathrm{I}\,\mathrm{Open}\left(1\right)+\mathrm{I}\right)$

$\mathrm{CR}≔z\in \mathrm{ComplexRange}\left(\mathrm{-1}-\mathrm{I}\,\mathrm{Open}\left(1\right)+\mathrm{I}\right)$

$\mathrm{convert}\left(\mathrm{CR}\,\mathrm{relation}\right)$

$\mathrm{-1}\le \mathrm{\Re }\left(z\right)\wedge \mathrm{\Re }\left(z\right)<1\wedge \mathrm{-1}\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le 1$

$\mathrm{convert}\left(\mathrm{CR}\&comma;\mathrm{RealRange}\right)$

$\mathrm{\Re }\left(z\right)\in \left[\mathrm{-1}\&comma;1\right),\mathrm{\Im }\left(z\right)\in \left[\mathrm{-1}\&comma;1\right]$

$\mathrm{map}\left(\mathrm{convert}\&comma;\left[\right]\&comma;\mathrm{relation}\right)$

$\left[\mathrm{-1}\le \mathrm{\Re }\left(z\right)\wedge \mathrm{\Re }\left(z\right)<1\&comma;\mathrm{-1}\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le 1\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\mathrm{arccot}\right)$

$\left[\mathrm{arccot}\left(z\right)\&comma;z\in \mathrm{ComplexRange}\left(-\mathrm{\infty }\mathrm{I}\&comma;\mathrm{-I}\right)\vee z\in \mathrm{ComplexRange}\left(\mathrm{I}\&comma;\mathrm{\infty }\mathrm{I}\right)\right]$

$\mathrm{z\_CR}≔z\mathbf{in}\mathrm{ComplexRange}\left(0\&comma;1\right)$

$\mathrm{z\_CR}≔z\in \mathrm{ComplexRange}\left(0\&comma;1\right)$

$\mathrm{convert}\left(\mathrm{z\_CR}\&comma;\mathrm{RealRange}\right)$

$z\in \left[0\&comma;1\right]$

$\mathrm{convert}\left(\mathrm{z\_CR}\&comma;\mathrm{relation}\right)$

$0\le z\wedge z\le 1$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\arcsin \right)$

$\left[\arcsin \left(z\right)\&comma;z\le \mathrm{-1}\vee 1\le z\right]$