Question #337302

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.


1
Expert's answer
2022-05-05T10:54:28-0400

At first, we set: z=x+iyz=x+iy with real x,y.x,y. We receive: z+2z2=x+iy+2x+iy2=x+iy+2x+iy2=(x+iy+2)(x2iy)(x2)2+y2=(x2)(x+2)+y2+i(y(x2)y(x+2))(x2)2+y2\frac{z+2}{z-2}=\frac{x+iy+2}{x+iy-2}=\frac{x+iy+2}{x+iy-2}=\frac{(x+iy+2)(x-2-iy)}{(x-2)^2+y^2}=\frac{(x-2)(x+2)+y^2+i(y(x-2)-y(x+2))}{(x-2)^2+y^2}.

From the latter considerations we receive that: Im(z+2z2)=4y(x2)2+y2Im\left(\frac{z+2}{z-2}\right)=\frac{-4y}{(x-2)^2+y^2}. In order to plot the graph, we need to plot the curve: 4y(x2)2+y2=1\frac{-4y}{(x-2)^2+y^2}=1. The graph is below:



a). All points that are inside of the circle drawn above belong to set AA. It can be obtained from the inequality: 4y(x2)2+y21\frac{-4y}{(x-2)^2+y^2}\geq1.

b). Substitute the point into equation of the circle: 4222+22=1\frac{4\cdot2}{2^2+2^2}=1. Thus, it is a boundary point. It can be also check in a formal way: any neighborhood of the point z=2iz=-2i contains points that belong to AA and points that do not belong to AA.

c). The set is closed because it contains all boundary points. We can consider the set C/A\mathbb{C}/{A}, which is determined by the inequalityIm(z+2z2)<1Im\left(\frac{z+2}{z-2}\right)<1. For any point that belongs to C/A\mathbb{C}/{A} there is also a small neighborhood that belongs to C/A\mathbb{C}/{A}. Namely, points: z+εz+\varepsilon also belong to C/A\mathbb{C}/{A} for small values of complex ε\varepsilon(ε|\varepsilon| is small). Therefore, the set C/A\mathbb{C}/{A} is opened and, respectively, the set AA is closed.


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