10.2 • BILlNEAR TRANSFORMATIONS 407y- i
w = S(z)v- i
Figure 10 .6 The bilinear mapping w = S (z) = [(l -i) z + 2] / [(1 + i) z + 2).
- Corollary 10 .1 (The implicit formula with a point at infinity) In Equation
(10-18), the point at infinity can be introduced as one of the prescribed points
in either the z plane or the w plane.
Proof:
Case 1 If z3 = oo, then we can write •o-za z - z3 - z,-oo z - oo = 1 and substitute this
expression into Equation (10-18) to obtain
Z - Zt W - Wt W2 - W3
---=Case 2 If W3 = oo, then we can write ":;:;:; = u:;:;: = l and substitute this
expression into Equation ( 10-18) to obtain
Z - Z1 Z2 - Z3 W - W1
-----=--- (10-21)Equation (10-21) is sometimes used to map the crescent-shaped region that
lies between the tangent circles onto an infinite strip.- EXAMPLE 10.7 Find the bilinear transformation that maps the crescent-
shaped region that lies inside the disk lz - 21 < 2 and outside the circle lz -ll =
1 onto a horizontal strip.
Solution For convenience we choose z 1 = 4, z 2 = 2 + 2i, and z3 = 0 and t he
image values w 1 = 0, w 2 = 1, and w 3 = oo, respectively. The ordered triplez1, z2, and zs gives the circle lz - 21 = 2 a positive orientation and the disk
lz - 21 < 2 has a left orientation. The image points wl, w2, and w3 all lie on
the extended u-axis, and they determine a left orientat ion for the upper half-
plane Im(w) > 0. Therefore, we can use the second implicit formula (Equation
(10- 21 )) to write
z-42+2i- O w-0
z-02+2i-4 = 1-0'