1550251515-Classical_Complex_Analysis__Gonzalez_

228 Chapter 5

w-z1 =M(z-z1)

where M = a/d = a^2 • It is an easy matter to check that this multiplier

again satisfies (5. 7-5).

In the case c "# 0, H = 0 the transformation has a double finite fixed

point z 1 = (a - d)/2c, and the relationship between z - Z1 and w - w1

is given by

1 1

---= --±c

W - Z1 Z - Z1

In fact, we have

az + b az 1 + b z - z1

w-z1= --- =

CZ + d CZ1 + d (CZ + d)( CZ1 + d)

But

a-d a+d

CZ1 + d = --+ d = --= ±1

2 2

since H = 0 implies that a + d = ±2. Hence

1 =±cz+d

W -Z1 Z - Z1

= ± c( Z - Z1) + CZ1 + d

z-z1

1

= ±c+--

z -Z1

(5.7-7)

Note If the image of the point z = p is w = 0, and that of the point

z = q is w = oo, the bilinear transformation can be written as

z-p

w=k--

z-q

where the value of the constant k can be determined if the image of a third

point is given.

Example Suppose that the points 0, oo, i are given as the images of the

points 1, -1, oo, respectively. Then k = i, and the transformation is

. z-l
w=i--
z+l

5.8 Classification of the Bilinear Transformations

First we consider the case where the transformation has two finite distinct

fixed points z1 and z2. As shown in Section 5.6, this case occurs when c # 0