Question 2.4.4: Image of a Circular Wedge under w = z³. Determine the image......
Image of a Circular Wedge under w = z³
Determine the image of the quarter disk defined by the inequalities |z| ≤ 2, 0 ≤ arg(z) ≤ π/2, under the mapping w = z³.
Let S denote the quarter disk and let S^{\prime} denote its image under w = z³. Since the moduli of the points in S vary from 0 to 2 and since the mapping w = z³ cubes the modulus of a point, it follows that the moduli of the points in S^{\prime} vary from 0³ = 0 to 2³ = 8. In addition, because the arguments of the points in S vary from 0 to π/2 and because the mapping w = z³ triples the argument of a point, we also have that the arguments of the points in S^{\prime} vary from 0 to 3π/2. Therefore, S^{\prime} is the set given by the inequalities |w| ≤ 8, 0 ≤ arg(w) ≤ 3π/2, shown in gray in Figure 2.24(b). In summary, the set S shown in color in Figure 2.24(a) is mapped onto the set S^{\prime} shown in grayin Figure 2.24(b) by w = z³.
