1550251515-Classical_Complex_Analysis__Gonzalez_

168 Chapter3

@,

A

Qc

r1

Fig. 3.26

not about z 1 ). Hence the cycle C 2 = 11 + 12 winds once in the positive
direction around each of the points z1 and z2, and we have C1 + (-C2) "'0
in A, or C 1 "' C2 in A.
Later (Theorem 7.16) we prove that if C 1 and C 2 are two closed contours
that are homotopic in A, they are homologous in A.
As an alternative to Definition 3.29, we have

Definition 3.32 A region R (not the whole of C) is said to be simply
connected if for each cycle C in R we have C "' 0 in R. The complex plane
C is also defined to be simply connected. In other terms, the winding
number of any cycle in C with respect to oo is considered to be zero.

Theorem 3.18 A convex open set is a simply connected region.

Proof This is of course trivially true if the convex open set is C. Suppose
that the convex open set A is a proper subset of C. Clearly, such a set is
a region, since by definition of convexity any two points in the set can be
connected by a line segment all whose points lie in the set.

Let z 0 E A'. Then there is a ray L: z = z 0 + bt, 0 :::; t < +oo, with

initial point at zo, which does not intersect A. Otherwise, the two opposite
rays z = Zo + bt and z = zo - bt, 0:::; t < +oo, will intersect A, and hence

r

Fig. 3.27