Question

Bilinear transformation w=f(x) which transforms the points z=∞,i,0 w= -1,-i,1

Answer 1

Answer:

f(z) = ( 1 - z ) / ( 1 + z )

Step-by-step explanation:

f(z) = ( az + b ) / ( cz + d )

f(0) = 1   =>   b / d = 1   =>   b = d

So  f(z) = ( az + b ) / ( cz + b )

f(i) = -i   =>   ( ai + b ) / ( ci + b ) = -i   =>   ai + b = c - bi   =>  a = -b and c = b.

So f(z) = ( -bz + b ) / ( bz + b ) = ( 1 - z ) / ( 1 + z ).

Check:  f(∞) = ( 1/∞ - 1 ) / ( 1/∞ + 1 ) = ( 0 - 1 ) / ( 0 + 1 ) = -1 / 1 = -1.