Question 7.2.2: Image of a Circle. Find the image of the unit circle |z| = 1......
Image of a Circle
Find the image of the unit circle |z| = 1 under the linear fractional transformation T(z) = (z + 2)/ (z − 1). What is the image of the interior |z| < 1 of this circle?
The pole of T is z = 1 and this point is on the unit circle |z| = 1.
Thus, from Theorem 7.3 we conclude that the image of the unit circle is a line. Since the image is a line, it is determined by any two points. Because T(−1) = −\frac{1}{2} and T(i) = −\frac{1}{2}−\frac{3}{2} i, we see that the image is the line u = −\frac{1}{2}. To answer the second question we first note that a linear fractional transformation is a rational function, and so it is continuous on its domain. As a consequence, the image of the interior |z| < 1 of the unit circle is either the half-plane u < −\frac{1}{2} or the half-plane u > −\frac{1}{2} . Using z = 0 as a test point, we find that T(0) = −2, which is to the left of the line u = −\frac{1}{2} , and so the image is the half-plane u < −\frac{1}{2}. This mapping is illustrated in Figure 7.13. The circle |z| = 1 is shown in color in Figure 7.13(a) and its image u = −\frac{1}{ 2} is shown in black in Figure 7.13(b).
