1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 221

Theorem 5.4 The bilinear transformation has the following properties:

1. Circles are transformed into circles (with the convention of considering
   the straight lines as members of the family of circles).


2. Angles are preserved in magnitude and orientation (i.e., the transfor-
   mation is directly isogonal).


3. The cross ratio of four distinct points is left invariant.


4. The set of all bilinear transformations form a group under composition.

Proof If c =/:-0, properties 1,_2, and 3 follow at once from the correspond-

ing properties of the elementary transformations into which the bilinear

transformation decomposes. If c = 0, the transformation becomes linear

and all three properties also hold.

To prove property 4, let

Then

Ti(z) = az + b

cz+d

and

TiT 2 (z) = (aa + b1)z + (a/3 + b8)

(ca+ d1)z + (c/3 + d8)

which is again a bilinear function. If we let

A= aa + b1,

we see that

B = a/3 + b8, C =ca+ d1,

I: ~II~ ~l=I~ ~I

D = cf3 +d8

so ad - be =/:- 0 and a8 - /31 =/:- 0 imply that AD - BC =/:-0. The identical

transformation is I(z) = z, and the inverse of Ti(z) is

T-l(z) = -dz+ b

l CZ - a
The associative law also holds, so we conclude that the set of all bilinear
transformations with nonzero determinants form a group.
A better insight into the behavior of the bilinear transformation is gained
by replacing the z-and w-planes by the corresponding extended planes (or,
by the corresponding Riemann spheres), and then considering the mapping
of one sphere into the other as given by w = w( z) = ( az + b) / ( cz + d) and
its inverse z = z(w) = (-dw + b)/(cw - a). However, first we need to
extend the definition of the function to the point oo; i.e., we must define
w( oo). This is accomplished by setting

w(.!.) =W(z)= a+bz

z c+dz