1550251515-Classical_Complex_Analysis__Gonzalez_

288 Chapter^5

By using an ingenious device introduced by B. Riemann, it is possible

to consider a multiple-valued function as being single-valued on a certain

surface (the so-called Riemann surface) consisting of a finite or infinite

number of planes or sheets (alternatively, of spheres) connected in a suitable

way, the number of sheets and the manner in which they are connected

depending on the particular function.

For instance, in the case of w = * :r:{i, we have established a one-to-

one correspondence between then cut z-planes G 0 , G1, ... , Gn-l and the

punctured w-plane. The idea of Riemann is simply this: to stack the cut

planes Gk at small distances upon each other in such a way that the Gk+l

plane (k < n -1) will be on top of the Gk with corresponding values of z

put in the same position, with the lower edge of the kth-plane connected

(glued) to the upper edge of the (k + l)th plane (k < n - 1), while the

lower edge of the Gn-l plane is to be regarded as connected to the upper

edge of the Go plane. In this manner the glued edges correspond to the

same ray on the w-plane, and a complete circuit about one of the branch

points will result in passing from one sheet to the next of the Riemann

surface with a corresponding continuous passing from a branch Wk to the

next Wk+l (with Wn-l going into w 0 ). If now the critical points 0 and oo

are added to the surface (at the corresponding ends of the cuts), as well as

to the w-plane, the construction of the Riemann surface will be completed.

In Fig. 5.28 the connection between the edges of the sheets of the Riemann

surface for w = * :r:fi is shown schematically for the cases n = 2 and n = 3

on a cross section of the cuts.

Although in the construction described above the cuts have been made

from 0 to oo along the positive x-axis, those cuts could have been made

along the negative x-axis, or, for that matter, along any simple line con-

necting 0 to oo. It should be noted that the ideal surfaces just constructed

cannot be realized physically (e.g., with paper models), since the last con-

nection, depicted in dashed lines in Fig. 5.28, would have to go across the

preceding connections. Rough models have been made in a cobweb style

with wires and interlacing threads, as shown schematically in Fig. 5.29.

The preceding observation could very well make the beginner somewhat

uneasy about the whole idea of the Riemann surface. This feeling can

G2
G1 G1 '
'
Go
L u
Go
L u

n = (^2) n = 3
Fig. 5.28