Question 2.3.3: Image of a Circle under Magnification. Find the image of the......
Image of a Circle under Magnification
Find the image of the circle C given by |z| = 2 under the linear mapping M(z) = 3z.
Since M is a magnification with magnification factor of 3, each point on the circle |z| = 2 will be mapped onto a point with the same argument but with modulus magnified by 3. Thus, each point in the image will have modulus 3 · 2 = 6. The image points can have any argumen t since the points z in the circle |z| = 2 can have anyargumen t. Therefore, the image C^{\prime } is the circle |w| = 6 that is centered at the origin and has radius 6. In Figure 2.13 we illustrate this mapping in a single copyof the complex plane. Under the mapping M(z) = 3z, the circle C shown in color in Figure 2.13 is mapped onto the circle C^{\prime } shown in black in Figure 2.13.
