1550251515-Classical_Complex_Analysis__Gonzalez_

166 Chapter^3

each component of ( 'Y )'. This constant has value zero on the unbounded
component of ('Y)' (the exterior of 'Y*).

Proof With the same notations as before, choose zb close enough to zo so

that lzo -zb I < p and lz(t)-zb I > p for all t E [a, ,8]. Then the subdivision

used to define n.,,(zo) also works for n.,,(zb). If we let

z(t;) -zb

ry;=

z(ti) -zo

we obtain l'r/i - 1 I < 1, so that Re 'r/i > 0, and we may choose v; = Arg 'f/i

with Iv; I <^1 krr. The argument e; corresponding to the point zb is related

to B; by the equation

e: = ei + V; - Vi-1 + 2k1r (3.16-2)

(Fig. 3.25), which gives -1 < k < 1. Hence k = 0, and we obtain

n n

L:ei = L:ei or n.,,(zb) = n.,,(zo)

i=l i=l

To show that n.,,(zo) = 0 whenever Zo lies on the unbounded component

of J'*, choose p > 2max lz(t)I, a:::; t:::; /3, and lzol > %P· Then we have

lz(t) - zol:::^3 /2P -^1 /2P = P > lz(t) -z(a)I

for all t E [a,,8]. The last inequality shows that n.,,(z 0 ) can now be ob-
tained without subdividing the interval [a, /3]. Letting t 0 = a and t 1 = ,8

in (3.16-1), we get 6 = 1, so 81 = 0, n.,,(zo) = 0.

Theorem 3.17 The winding number n,,(zo) is invariant under a contin-
uous deformation of 'Y that does not pass through the point z 0 •

Proof Suppose that the homotopy of arcs is defined by the continuous
function f(t,>..), a:::; t:::; /3, 0:::; >..:::; 1, and that f(t,>..)-/:- z 0 for all

Fig. 3.25