Question

Find the Bilinear transformation which maps the points z=0,1, infinity onto the points w=0,i,2i

Answer 1

$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$

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

$-iw=zi(w-2i)$

$w=\frac{2iz}{z+1}$