Question

find the bilinear transformation which maps the point z=1,i,-1 of the z plane into the points w= 2,i,-2​

Answer 1

SOLUTION

TO DETERMINE

The bilinear transformation which maps the point z = 1 , i , - 1 of the z plane into the points w = 2 , i , - 2

EVALUATION

Let the points 1 , i , - 1 , z of the z plane map into the points 2 , i , - 2 , w

Since the cross - ratio remains unchanged under a bilinear transformation

$\displaystyle\sf{ \frac{(1 - i)( - 1 - z)}{(1 - z)( - 1 - i)} = \frac{(2 - i)( - 2 - w)}{(2 - w)( - 2 - i)} }$

$\displaystyle\sf{ \implies \: \frac{(1 - i)( 1 + z)}{(1 - z)( 1 + i)} = \frac{(2 - i)( 2 + w)}{(2 - w)( 2 + i)} }$

$\displaystyle\sf{ \implies \: w = \frac{2i - 6z}{iz - 3} }$

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