Let A ={ z E C : Im (z+2/z-2)≥ 1}.
(a) Sketch the set A in the complex plane.
(b) Is z = −2i a boundary point of A? Provide reasons for your answer.
(c) Is this set open, closed, both or neither? Provide reasons for your answer.
At first, we set: with real We receive: .
From the latter considerations we receive that: . In order to plot the graph, we need to plot the curve: . The graph is below:
a). All points that are inside of the circle drawn above belong to set . It can be obtained from the inequality: .
b). Substitute the point into equation of the circle: . Thus, it is a boundary point. It can be also check in a formal way: any neighborhood of the point contains points that belong to and points that do not belong to .
c). The set is closed because it contains all boundary points. We can consider the set , which is determined by the inequality. For any point that belongs to there is also a small neighborhood that belongs to . Namely, points: also belong to for small values of complex ( is small). Therefore, the set is opened and, respectively, the set is closed.
Comments