Real and Complex Analysis (Rudin)

204 REAL AND COMPLEX ANALYSIS
By Theorem 10.7, (1) shows that Indy E H(o.). The image of a connected
set under a continuous mapping is connected ([26], Theorem 4.22), and since
Indy is an integer-valued function, Indy must be constant on each component
of 0..
Finally, (2) shows that I Indy (z) I < 1 if I z I is sufficiently large. This
implies that Indy (z) = 0 in the unbounded component of 0..
IIII
Remark: If A(t) denotes the integral in (3), the preceding proof shows that
2n Indy (z) is the net increase in the imaginary part of A(t), as t runs from 0( to
p, and this is the same as the net increase of the argument of y(t) - z. (We
have not defined" argument" and will have no need for it.) If we divide this
increase by 2n, we obtain" the number of times that y winds around z," and
this explains why the term "winding number" is frequently used for the
index. One virtue of the preceding proof is that it establishes the main
properties of the index without any reference to the (multiple-valued) argument
of a complex number.
10.11 Theorem If y is the positively oriented circle with center at a and radius
r, then
if I z - al < r,
if Iz - al > r.
PROOF We take y as in Sec. 10.9(a). By Theorem 10.10, it is enough to
compute Indy (a), and 10.9(2) shows that this equals
-. 1 i --= dz -
r
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