## Search This Blog

THE BEAUTY OF MATHEMATICS- Collected from The Internet and Various Books to enrich The students and Teachers. SUPPORT with YOUR COMMENTS ...

## Friday, October 14, 2011

### 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.