1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 287

y z

Go

x

v

y z

Ha

G1

H1

x u

z H2

y

G2

e + 4'71'

x

Fig. 5.27

At the exceptional points z = 0 and z = oo, all n branches of w = * \fZ
assume the same values, namely, w = 0 and w = oo, respectively. As z
describes a circle lzl = r once in the positive direction about the origin
(or about the point oo in the negative directiont), a value Wk goes into
w k+l; i.e., a rotation of 27r radians about 0 (or oo) will change a branch of
the function into the next (and Wn-l back into w 0 ). For this reason it is
said that the points 0 and oo are points of ramification or critical points of

the function. Since the equation wn = z relating z and w is an algebraic

equation of order n, it is said, more specifically, that 0 and oo are critical
algebraic points of order n - 1.

tif one considers the spherical representation, the origin maps into the south pole,
oo into the north pole, and a parallel circle that is described counterclockwise
with respect to the south pole is seen as described clockwise from the north pole.