1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 225

The preceding proof is more general than that indicated for Theorem

5.4 in the sense that it holds in C*; i.e., any one of the four points under

consideration could be oo.

5.6 Fixed Points of the Bilinear Transformation

In what follows it is convenient to assume that in

w = Lz = az+b

ez+d

the coefficients satisfy the condition ad - be = 1. This prescription implies
no loss of generality since any other case can be reduced to this one upon
division of the numerator and denominator of the fraction by .,/ad - be
(either root may be used).
By considering the w-sphere as superimposed to the z-sphere, we wish
to investigate which points of the z-sphere (if any) are left invariant or
fixed under the bilinear transformation.
The condition Lz = z leads to the quadratic
ez^2 + (d - a)z - b = 0 (5.6-1)
Thus if e f= 0, we obtain the roots
a-d+.,/H
Z1 = --2-c-- and

where

a-d-.,/H

2e

H = (d - a)^2 + 4bc =(a+ d)^2 - 4

If H f= 0, we have two distinct finite invariant points z 1 and z 2 , and

only those, since in Section 5.5 we saw that oo is not carried unto itself

when e f= 0. If H = 0, i.e., if a+ d = ±2, there is only one fixed point,

namely z 1 = z 2 = (a - d)/2c. However, this point will be regarded as a

double point.

In the case c = 0, equation (5.6-1) reduces to

(d-a)z-b=O

and if a f= d, we have the finite fixed point z 1 = bf ( d-a), and, in addition,

the point oo is fixed, since the transformation becomes linear, i.e., w =

(a/d)z + (b/d). We note that c = 0 and ad-be= 1 imply that ad= 1 and

so a f= 0, d f= 0, and if a f= d, then H f= 0.

If e = 0, a = d, b f= O, we have a = d = ±1, H = 0, and oo is the only

fixed point, but it will be regarded as double, since z 1 = b/(d - a) -t oo

as d -t a.