Question 2.4.1: Image of a Circular Arc under w = z². Find the image of the ......

Let C be the circular arc defined by |z| = 2, 0 ≤ arg(z) ≤ π/2, shown in color in Figure 2.18(a), and let C^{\prime} denote the image of C under w = z². Since each point in C has modulus 2 and since the mapping w = z² squares the modulus of a point, it follows that each point in C^{\prime} has modulus 2² = 4. This implies that the image C^{\prime} must be contained in the circle |w| = 4 centered at the origin with radius 4. Since the arguments of the points in C take on every value in the interval [0, π/2] and since the mapping w = z² doubles the argument of a point, it follows that the points in C^{\prime} have arguments that take on every value in the interval [2 · 0, 2 · (π/2)] = [0, π]. That is, the set C^{\prime} is the semicircle defined by |w| = 4, 0 ≤ arg(w) ≤ π. In conclusion, we have shown that w = z² maps the circular arc C shown in color in Figure 2.18(a) onto the semicircle C^{\prime} shown in black in Figure 2.18(b).