1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions

Gs------~

G4-------

G3-------

Gz------

G1-- '\\

Go------~ L ~· U :-------

Fig. 5.39

295

Thus the edges of the cuts made from 0 to 2 are to be joined as follows:

those of G 0 and Ga crosswise, as well as those of G 1 and G 4 , and those of

G 2 and Gs. Figure 5.39 shows the connection between the sheets of the

Riemann surface as they appear in a cross section AB made between 0 and

2; see also Fig. 5.41.

Along the cuts made from 0 to -oo the edges are to be joined as follows

(in view of 5.22-3): the upper edge of G 0 to the lower edge of G 4 , the upper
edge of G 4 to the lower'edge of G 2 , the upper edge of G 2 to the lower edge
of Ga, the upper edge of Ga to the lower edge of G 1 , the upper edge of Gi
to the lower edge of Gs, and finally the upper edge of G~ to the lower edge
of G 0 • Figure 5.40 shows the connection between the sheets as they appear
in a cross section CD made between 0 and -oo.
To check our construction, let z describe a small circle about 0 (leaving
2 outside), in the positive direction, starting at some point z 0 in the lower
G 0 sheet (Fig. 5.41). Since z 0 can be joined to point 3 without crossing the
cuts, we get w 0 (z 0 ). As z crosses the cut be.tween 0 and 2, z goes into Ga
and we begin obtaining values of wa. Then as z crosses the cut between

Gs

G4

G3

Gz

G1

Go

L

Fig. 5.40