compute the 'intersection' of regions of the complex plane defined algebraically
compute the 'set difference' between regions of the complex plane defined algebraically
compute the 'union' between regions of the complex plane defined algebraically
&Intersect(A, B, ..., removereim)
&Minus(A, B, ..., removereim)
&Union(A, B, ..., removereim)
A, B, ...
regions of the complex plane defined algebraically, using equations, inequations, and possibly boolean functions
(optional) passed as the last argument, to replace, when possible, occurrences of ℜ⁡z and ℑ⁡z by algebraic expressions not involving Re or Im.
The &Intersect, &Minus, and &Union commands respectively return the intersection, set difference, and union of the regions of the complex plane passed as arguments. As with all infix operators, you can invoke them using the functional form, e.g. `&Intersect`(A, B), or using the infix form A &Intersect B.
In the context of the &Intersect, &Minus and &Union commands, a region of the complex plane is algebraically represented by a single expression, involving relations, RealRanges and ComplexRanges, and possibly nested algebraic expressions representing regions constructed using And, Or, `and` and `or`. To determine whether a region is properly represented from the syntactic point of view you can use the type MathematicalFunctions:-Region as explained in the examples. You can also display the type entering TypeTools:-GetType(MathematicalFunctions:-Region).
Note that inequations always assume that both sides are real, in that there is no meaning for an inequation between complex objects, in view of their two-dimensional character. So for instance z≤0 is equivalent to ℑ⁡z=0∧ℜ⁡z≤0, and you may prefer to see the output expressed using the former syntax, without ℜ and ℑ. For this purpose, pass the optional argument removereim as the last argument. When using the infix notation, due to the precedence of infix &... operators, you need to enclose the keyword removereim together with the last argument. See the example.
Consider first an easy case, say the two regions consisting of the following two segments of the real axis
R__1,R__2 ≔ 22<x,1<x
Verify that R1, and R2 are correctly entered regions
For R1, and R2 given above, the following results can be verified with ease:
R__1 &Union R__2
R__1 &Minus R__2
R__1 &Intersect R__2
The two segments of the real line to the left of -1 and to the right of +1 expressed using Re and Im
R__1,R__2 ≔ And⁡ℑ⁡z=0,ℜ⁡z<−1,And⁡ℑ⁡z=0,1<ℜ⁡z
The union of R1 and R2 is naturally given by
Sometimes the presence of Re or Im clutters the output; in such cases use the option removereim. (Note the parenthesis enclosing removereim after the last argument - this is important in connection with the precedence of infix & operators.)
R__1 &Union R__2,removereim
The MathematicalFunctions[&Intersect], MathematicalFunctions[&Minus] and MathematicalFunctions[&Union] commands were introduced in Maple 2015.
For more information on Maple 2015 changes, see Updates in Maple 2015.
The removereim option was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
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