1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 289

Fig. 5.29

be quickly dispelled by topological mappings of spheres, as follows: Let So
and S1 be the cut Riemann spheres corresponding to the cut planes Go and
G 1 , respectively (Fig. 5.30). Transform the two spheres into hemispheres
by assigning to the point with spherical coordinates (1, (), ,,P) the point with
coordinates (1, %8, ,,P ), and rotate them so that the edges labeled U and
L will face each other (Fig. 5.31). Then it suffices to put together the

two hemispheres S& and Si, gluing the edge L of S& to the edge U of Si

as well as the edge L of Sf to the edge U of S~. The resulting sphere

S (Fig. 5.32) is the Riemann surface of the function w = *Vz· In the

case of the function w = * {/Z a similar construction can be made with
a preliminary deformation of three cut spheres into three spherical lunes,
each lune being one-third of the corresponding sphere.

5.21 THE RIEMANN SURFACE OF w = *~

We wish to illustrate further the concept of the Riemann surface by con-
sidering those associated with the function w = *JP(Z), where P(z)
is a polynomial of degree m. For the case m = 1 we may write w =

I I

I

I

• -1 - - -
  L U I

0

Fig. 5.30

I

I

I

I

00

s,