292 Chapter^5We may note that the bilinear transformation z' = (z-z 1 )/(z-z 2 ) will
carry the point z 1 into 0 and the point z 2 into oo, and so, proceeding as
before, we see that in the spherical representation the Riemann surface is
again topologically equivalent to a single sphere.
A significant change occurs when we consider the case m = 3, namely,
the function w = *Va(z - z 1 )(z - z 2 )(z - z 3 ) with all three roots distinct.
Here the finite critical points are z 1 , z 2 , and z 3 , so that a circuit about
any of these points (described once in the positive direction) will increase
argw by 71", and will change the branch of w, while a circuit about any two
of these points will increase arg w by 271" and will not change the initial
value of w. However, a circuit about all three points z1, z2, z 3 will increase
arg w by 371", and again the branch of w will be changed. Since this last
circuit is equivalent to a circuit about oo, we see that in this case (as in
the case m = 1) the point oo must be regarded also as a critical point.
Thus to construct the corresponding Riemann surface, we take two copies
of the z-plane (or of the z-sphere) and make two cuts, one along a line
joining any two of the finite critical points, say z 1 and z 2 , and another
along a line joining z 3 and oo, then proceed to glue crosswise the edges
of corresponding cuts. In Fig. 5.35 those cuts are illustrated using the
spherical representation.
By continuous deformations the cuts on both spheres can be stretched
into circular holes. Then if we pull out the edges of the cuts and rotate
the spheres, so that corresponding tubes will face each other, the deformed
spheres will look as in Fig. 5.36. In the position of this figure the lower
edge of each cut (labeled "L") now faces the upper edge (labeled "U") of
the corresponding cut on the other sphere.
Hence, if we do some more stretching, the ends of the tubes can be glued
together to form the surface shown in Fig. 5.37. This is the Riemann surface
for the function w = *Va(z - z 1 )(z - z2)(z - z 3 ), which is topologically00Fig. 5.35