1550251515-Classical_Complex_Analysis__Gonzalez_

244 Chapter^5

Let z 1 and z 2 be any points and C the "line" determined by them. This

line intersects the line at infinity in two ideal points. Call zi the ideal point

of the half-ray with origin z 1 and containing z 2 , and z2 the ideal point of
the half-ray with origin z 2 and containing z 1 (Fig. 5.17).

We know that the cross ratio (z 1 , z 2 , zi, zi) is real and invariant un-

der the transformation (5.11-1). However, it does not satisfy the required
properties of a distance. For example, it does not vanish for z1 = z2. Also,
for a point z 3 E C such that z 2 lies between z 1 and z 3 , we have

(5.11-2)

so that the additivity of distances for points on the same line would not
be satisfied either. But (5.11-2) points out to the possibility of using

ln( z 1 , z 2 , zi, zi) as a suitable definition for the distance between z 1 and

z 2 • This could be done provided that the cross ratio has a value greater
than 1 whenever z 1 -:f z 2 , since the logarithms of positive real numbers less
than 1 are negative and those of nonpositive real numbers are undefined
in the real system.
To see that this is indeed the case, we note that there is a transforma-

tion (5.11-1) that maps the points z;, zi, z 2 , zi into the points w2 = O,

W1 = i, w 2 = ir, wi = oo, respectively, for some r > 1, and by the
invariance of the cross ratio we have

(5.11-3)

Hence it is admissible to set

(5.11-4)

this definition satisfying all the properties of a distance function, namely:

1. d(z1,z2) ~ 0.

2. d(z 1 ,z 2 ) = 0 iff z 1 = z 2.

c

Fig. 5.l'l