Functions. Limits and Continuity. Arcs and Curves 167t E [a,,8] and>. E [0,1]. Then there is a p > 0 and subintervals [t;_ 1 ,t;],
[Aj- 1 , Aj] such thatlf(t,>.)-zol > p and
fort E (t;-1, t;], >. E [Aj-1, >.J-]. Let B;, Bi be the arguments correspondingto >. = Aj-l, >. = Aj, respectively. If we define
f(t;, Aj) - zo
<Yi=
J(t;, Aj-1) - zowe see, as before, that lo-; - ll < 1, and we can choose W; =Argo-; with
IW;I <^1 krr. Since
Bi= B; + '1!; - '1!;-1 + 2k7r
we conclude, as in the case of (3.16-2), that k = 0. It follows that L::~=l Bi =
l:~=l B;, so the winding number is the same for the curves corresponding
to the values Aj-l, Aj of>.. Hence the curves corresponding to >. = 0 and
. = 1 have the same winding number with respect to z 0 •
3.17 HOMOLOGY. THE CONNECTIVITY OF
A REGION
Definition 3.31 A cycle C (in particular, a closed contour) with graph
contained in an open set A is said to be homologous to zero in A, denoted
C ,..., 0 in A, if for every point z 0 ¢ A we have
Dc(zo) = 0
i.e., if the winding number of C with respect to every point in the
complement of A is zero.
Two chains C 1 and C2 (in particular, two cycles) are said to be homol-
ogous on an open set A, and we write C 1 ,..., C2 in A, if C 1 + (-C 2 ) is a
cycle homologous to zero in A. i.e., if C 1 + ( -C 2 ) ,..., 0 in A.
Clearly, the homology relation is reflexive, symmetric, and transitive.
Examples 1. In Fig. 3.26 the open set A is the region bounded by the
simple curves r 1 and r 2. For the contour C shown in the figure we have
C ,..., 0 in A. However, for the contour C' we have C' f 0 in A ( C' is not
homologous to zero in A).
- In Fig. 3.27 the open set A is the region bounded by the simple curve
r with points Z1 and Z2 deleted. The contour C1 winds once in the positive
direction about each of the points z 1 and z 2 • On the other hand, the curve
/l winds once in the positive direction around z1 (but does not wind about
z 2 ). Similarly, /2 winds once also in the positive direction around z2 (but