Real and Complex Analysis (Rudin)

111 REAL AND COMPLEX ANALYSIS
We conclude that
Indy (z) = Ind h (z) = Ind h (w)
= Inde (w) + Indy (w) = 1 + Indy (w).
The first of these equalities follows from (2), since h = Y -+- f The second holds
because z and w lie in D(a; r), a connected set which does not intersect h*.
The third follows from (1), since h -+- g = C -+- y, and the fourth is a consequence
of Theorem 10.11. This completes the proof.
IIII
We now turn to a brief discussion of another topological concept that is
relevant to Cauchy's theorem.
10.38 Homotopy Suppose Yo and YI are closed curves in a topological space X,
both with parameter interval I = [0, 1]. We say that Yo and YI are X-homotopic if
there is a continuous mapping H of the unit square 12 = I x I into X such that
H(s, 0) = Yo(s), H(s, 1) = YI(S), H(O, t) = H(I, t) (1)
for all s E I and tEl. Put Yt(s) = H(s, t). Then (1) defines a one-parameter family
of closed curves Yt in X, which connects Yo and YI' Intuitively, this means that Yo
can be continuously deformed to YI' within X.
If Yo is X -homotopic to a constant mapping Y I (i.e., if yr consists of just one
point), we say that Yo is null-homotopic in X. If X is connected and if every closed
curve in X is null-homotopic, X is said to be simply connected.
For example, every convex region a is simply connected. To see this, let Yo be
a closed curve in a, fix z I E a, and define
H(s, t) = (1 - t)yo(s) + tz I (0 ~ s ~ 1, 0 ~ t ~ 1). (2)
Theorem 10.40 will show that condition (4) of Cauchy's theorem 10.35 holds
whenever r 0 and r I are a-homotopic closed paths. As a special case of this, note
that condition (1) of Theorem 10.35 holdsfor every closed path r in a ifa is simply
connected.
10.39 Lemma If Yo and YI are closed paths with parameter interval [0,1], ifrx
is a complex number, and if
I YI(S) - Yo(s) I < I rx - Yo(s) I (0 ~ s ~ 1) (1)
PROOF Note first that (1) implies that rx ¢: y~
define Y = (YI - rx)f(yo - rx). Then
and rx ¢: yr. Hence one can
r:=~_ y~
Y YI - rx Yo - rx
(2)