1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 223

we see that the simple ratio is a bilinear function of the third point. Thus,

in accordance with the definition in (5.5-3), we have

w( oo) = ( zi, z 2 , oo) = 1
If z1 = oo but z2 f= oo,z-f= oo,z 2 f= z, we define
( oo, Zz, Z) = 00
and if z2 = oo but z1 f= oo, z f= oo, z 1 f= z, we define
(z1, oo, z) = 0
The cross ratio of four distinct points, one of which is the point oo, can
now be determined easily in terms of the corresponding simple ratios. Thus

(z1,z2,z3,oo) = (zi,z2,z3): (z1,z2,oo) = (z1,z2,z3)

(z1,z2,oo,z4) = (zi,z2,oo): (z1,z2,z4) = 1: (zi,z2,z4)

To determine the value of the cross ratio when the point oo is in the first
or second position, we note that for finite points we have

(~.~.~.~)=(~.~.~.~)

so that the cross ratio does not vary if the first pair of points is interchanged
with the second pair (keeping the order of the points within each pair).
Now, agreeing by definition to hold the property true for the case of any
one point being oo, we have

(z1,oo,z3,z4) = (z3,z4,z1,oo) = (z3,z4,z1)

and

(oo,z2,z3,z4) = (z3,z4,oo,z2) = 1: (z3,z4,z2)
All these results can also be justified by a limiting process. For instance,
in the last case we may begin be writing, for four finite points,

z3 - z z4 - z (z3 / z) - 1. z2 - z3

(z,z2,z3,z4) = : ---= ( / ).

Z3 - Z2 Z4 - Z2 Z4 Z - 1 Z2 - Z4

and letting z ~ oo, we get, as before,

(oo,z2,z3,z4) = 1: (z3,Z4,z2)

Let z2, z 3 , z 4 be three distinct points on the Riemann sphere. Then
there exists a bilinear transformation T, and only one, which carries those
three points into the points 1, 0, oo, respectively. In fact, if none of the
given points is oo, T will be given by

Tz = z - Z3

Z-Z4

(5.5-4)