1550251515-Classical_Complex_Analysis__Gonzalez_

290

S' 0

00 00

I

----!--....\

u

L U

0 0

Fig. 5.:n

S' 1

Chapter 5

.Ja(Z=-z 1 ), where a is a constant and z 1 is the simple zero of the given
polynomial. Here the ramification points are z 1 and oo, and it suffices to
take two z-planes cut from z 1 to oo and to connect the edges of the cuts
crosswise, exactly as we did for the function w = '1i in Section 5.20. In
fact, the two functions do not differ essentially since we may pass from one
to the other by the linear transformation a(z - z 1 ) = z'. Under this trans-
formation the critical point z 1 will be carried into the origin while the point

oo is left invariant. If the spherical representation is preferred, we have ini-

tially two cut spheres So and S 1 (Fig. 5.33), and after suitable topological
deformations we are back to the hemispheres of Fig. 5.31. Hence the Rie-
mann surface can again be represented by a single sphere. Of course, this
distribution of the points of the two original spheres on this single sphere
is to be made in accordance with the indicated transformations.
For the case m = 2 we have w = *va(z - z 1 )(z - z 2 ), and if z 1 # z 2 ,
the situation is similar to the one considered before, the only difference
being that the ramification points are now z 1 and z 2 , which are both finite.
To see that this is indeed the case and to investigate how the cuts of the

00

s \

I

I

I

---

---I ......

I I I I I I 0

Fig. 5.32