Transformation w = a z

Let w = a z, where a ≠ 0


If a = │a│ e^(i α) and, z = │z │ e^(i θ), then


w = │a│ │z│ e^i(θ + α )


The image of z is obtained by rotating the vector z through the angle α and magnifying or contracting the length of z by the factor │a│.


Thus the transformation w = a z is referred to as a rotation or magnification.



Example 1:


Find the image of the region y > 1 under the map w = ( 1 – i ) z


Solution:


Let w = u + i v ; z = x + i y


Given w = ( 1 – i ) z

i.e., z = 1/2 ( 1 + i) w      [ since ( 1 – i ) ( 1 + i) = 2]


i.e., x + i y = 1/2 ( 1 + i) (u + i v )

i.e., x = (u- v )/2 ; y = (u+v)/2


Hence the region y >1 is mapped on the region u + v > 2 in w –plane.

Example 2 :

Determine the region R of the w plane into which the triangular region D enclosed by the lines
x = 0, y = 0, x + y = 3 is transformed under the transformation w = 2z.


Solution:
Let w = u +i v; z = x + i y

Given, w =2 z

i.e., u +i v = 2 (x + i y)

i.e., u = 2 x ; v = 2 y


When x = 0, u = 0

The line x = 0 is transformed into the line u = 0 in the w – plane.


When y = 0, v = 0

The line y = 0 is transformed into the line v = 0 in the w – plane.


When x + y = 3 , we get


u/2 + v/2 = 3


i.e., u + v = 6

The line x + y = 3 is transformed into the line u + v = 6 in the w – plane.