Under the transformation wz1z1 Find the image of 1 z r 1 2

Solution

: Image under f : z 7 1/z of (a) {z : 0 < arg(z) < /6} = {zrei : 0 < < /6} In this case z 1 = r 1 e i : 0 < < /6 so image is the wedge {z : |z| > 0, 0 > arg(z) > /6} (b) D(0; 2) Soln: D(0; 2) = {z = rei|r < 2} 1/z = 1/rei : 1/r > 1/2 so image is {z : |z| > 1/2} (c) {z : 0 < Im(z) < 1} Soln: {z|0 < Im(z) < 1} = {z = x + iy|0 < y < 1} 1 z = z¯ zz¯ = x iy x 2 + y 2 The line z R is sent to R \\ {0}. (The point is sent to 0.) The line z = x + i (where (Im)(z) = 1) contains i, 1 + i, 1 + i, and is sent to the circline containing 1/i, 1/(1 + i) = (1 i)/2 and 1/(1 + i) = (1i) 2 . These three points are = i, 1/ 2e i/4 , 1/ 2e 3i/4 . This is the circle with centre i/2 and radius 1/2 Under f(z) = 1/z, the point i/2 is sent to 2i. This point is outside the circle described above. So the strip {z : 0 < Im(z) < 1} is sent under f(z) = 1/z to the region below the real axis and outside the circle {z : |z + i/2| = 1/2.