1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 303

-37r/2 37r/2 57r/2 77r/2

Fig. 5.44

with G;+i on top of Gil it is clear from (5.25-3) that over z = +1, and
also over z = -1, there are infinitely many branch points of order 1, while
z = oo is a logarithmic branch point that does not belong to the surface.
In view of this and the preliminary discussion made above concerning the
mapping defined by w = sin z, we conclude that the connections between
G; are to be established as shown schematically in Fig. 5.45, which depicts
cross sections of the branch lines indicated.

Thus on the cut (-oo, -1) the upper edge of G2k+i is glued to the lower

edge of G 2 k+ 2 , while the lower edge of G 2 k+ 1 is glued to the upper edge of
G 2 k+2· On the cut ( +1, +oo) the upper edge of G 2 k is glued to the lower
of G2k+i, while the lower edge of G2k is glued to the upper of G2k+i· The
branch point z = -1 is to be added to each G 2 k+ 1 , and the point z = +1
to each G 2 k. The reader will notice that G2k is conne.cted with G 2 k+ 1
and with G 2 k_ 1 , although along different branch lines. Similarly, G 2 k+1
is connected with G 2 k and G 2 k+2 along different branch lines. Hence the
Riemann surface just constructed is connected, so we may go from a sheet

Gs

G2 x

G1 x

Ga x

G -1 x

G -2 x

L u L u

(-oc, -1) (+1, +oo)

Fig. 5.45