10.2 • BILINEAR TRANSFORMATIONS 405- EXAMPLE 10.4 Construct the bilinear transformation w = S (z) that
maps the points z 1 = -i, Z2 = 1, and z3 = i onto the points w 1 = -1, w2 = 0,
and W3 = 1, respectively.
Solution We use the implicit formula (Equation (10-18)) and writez+il- i w+l0-1 w+l
Expanding this equation, we obtain(1 + i) zw + (1 -i)w + (1 + i) z + (1 - i) = (- 1 + i) zw + (- 1 - i)w
+ (1 - i) z + (1 + i). (10-20)Then, collecting terms involving w and zw on the left results in2w + 2zw = 2i - 2iz,
from which we obtain w (1 + z) = i (1 - z). Therefore, the desired bilinear trans-
formation isw --S()_i(l-z) z -.
l + z• EXAMPLE 10 .5 Find the bilinear transformation w = S (z) that maps the
points z 1 = - 2, Z2 = - 1 - i, and za = 0 onto w1 = -1, w2 = 0, and wa = 1,
respectively.Solution Again, we use the implicit formula and writez-(-2) - 1-i-O
z - 0 -l-i - (- 2)w - (-1) 0 - 1