294 Chapter^5Proceeding in the same manner, we see that the cases m = 2p + 1 and
m = 2p + 2 (p = 0, 1, 2,: .. ) lead to the same type of Riemann surface,
namely, a sphere with p handles. The number p is called the genus of the
surface. Thus an ordinary sphere is of genus O, while a torus is of genus 1.
5.22 THERIEMANNSURFACEOFw = {/Z+v'z 2
As still another illustration of the construction of the Riemann surface cor-
responding to a multiple-valued function, we consider the example stated
in the heading. Since the first term on the right-hand side has in gen-
eral three values and the second term has two, by combining those values
in pairs we see that w is a six-valued function of z. Hence the Riemann
surface for this function will consist of six sheets, and the question is to
determine how those sheets are to be connected. Here the critical points
are z = O, z = 2 and also z = oo, since a simple circuit about both points
0 and 2 (or, equivalently, about oo) changes a value of w into another, as
we shall. see momentarily.
If we cut all six replicas of the z-plane along the real axis from 0 to 2,
and also along the negative real axis from 0 to oo, the notation for each of
the six branches of w can be made specific by its value at some point not
lying on the cuts. Choosing z = 3, we define
wo(3) = ?-'3 + 1, w1 (3) = 11?-'3 + 1,
wa(3) = ?-'3 -1, w4(3) = 11?-'3 -1,where 17 = e^2 7T:i/^3 •
w2(3) = 172 ?-'3 + 1
ws(3) = 172 .q'3° - 1
A circuit about z = O, described once in the positive direction, which
does not enclose z = 2, will change the subindices of the branches of w
according to the substitution
(012)(345) (5.22-1)that is, wo will go into w 1 , w 1 into w 2 , and w 2 into w 0 • Similarly, the other
three branches will permute cyclically.
On the other hand, a circuit about z = 2, described in the same manner
(and not enclosing z = 0), will change the branches as indicated by the
substitution
(03)(14)(25) (5.22-2)Hence a circuit about both points 0 and 2 will change the branches as
indicated by the product of the substitutions (5.22-1) and (5.22-2), namely,
(0 1 2 3 4 5)
4 5 3 1 2 0 (^5 ·^22 -^3 )