1550251515-Classical_Complex_Analysis__Gonzalez_

226 Chapter^5

Finally, when c = 0, a= d, b = 0, equation (5.6-1) is satisfied identically.
In this case the transformation reduces to w = z, and all points are left
invariant.
Summing up, we have:

c#O { ;:~

L has two finite distinct fixed points

L has a double finite fixed point

c=O {H#O

L has two distinct fixed points, one finite,

the other oo

H = O,b # 0

c = O, b = 0, a = d

L has oo as a double fixed point

L leaves all points fixed

5. 7 Multiplier of the Bilinear Transformation

First suppose that c # 0 and H # 0. Then the finite fixed points z 1 and
z 2 , and the point oo, are carried by w = Lz into the points z1, z2, and a/c,

respectively. If w denotes the image of a given z, we have

(5.7-1)

by the in.variance of the cross ratio under L. Equation (5.7-1) gives

w-z1 (a/c)-z1 z-z1

w - z 2 : (a/ c) - z 2 = z - z 2

or
W - Z1 Z - Z1
--=M--

w -Z2 Z - Z2

(5.7-2)

where M = (a - cz1)/( a - cz2). This number Mis called the multiplier of

the bilinear transformation, and it plays an important role in the further
study of the transformation. Equation (5.7-2) gives, in terms of the fixed
points and the multiplier, a convenient alternative form of the bilinear
transformation. From (5.7-2) it follows that the multiplier of the inverse
transformation is 1/M.
Also, if the same transformation is now applied to w and we let μ = Lw,
we have
μ-Z1 W - Z1
--=M--
μ - Z2 W - Z2

(5.7-3)