1550251515-Classical_Complex_Analysis__Gonzalez_

222 Chapter 5

and then defining

w(oo) = W(O) = a/c (5.5-3)

Thus if c f 0, the image of oo is the finite point a/ c, and if c = 0 (in

which case a f 0), we have w( oo) = oo. That is, for the special case of

the linear function w = (a/d)z + (b/d), the image of oo is oo. Similarly,
for the inverse function we have

-d

z(oo) = -

c

We also note that the point z 1 = -b/a, called the zero of w(z), maps

into the origin on the w-sphere and, conversely, z(O) = -b/a. Similarly,

w(O) = b/d and z(b/d) = 0. On the other hand, the point z2 = -d/c, called

the pole of w(z), maps into the point oo on the w-sphere and, conversely,

z(oo) = -d/c (as seen above).

This is illustrated in Fig. 5.5. Since the bilinear function, as well as its
inverse, are single-valued on the Riemann sphere, we conclude that this
function defines a one-to-one mapping from one sphere onto the other.
Elsewhere (Selected Topics, Theorem 2.9) we prove that if a function is
one-to-one and has a pole in C*, the function is bilinear with a nonzero
determinant.
Because the definition of the bilinear function has been extended to the
point oo, we should examine what meaning is to be attached to the simple
ratio (and hence to the cross ratio) when one of the points involved is oo.
Letting

0

z-sphere

Fig. 5.5

w(z) =

0

w-sphere