Real and Complex Analysis (Rudin)

220 REAL AND COMPLEX ANALYSIS
Next, we let 0 1 be the set of all complex numbers z for which Indr (z) =
0, and we define
If z E 0 ("'\ 0 1 , the definition of 0 1 makes it clear that h1(Z) = h(z). Hence
there is a function q> E H(O u 0 1 ) whose restriction to 0 is h and whose
restriction to 0 1 is h 1•
Our hypothesis (1) shows that 0 1 contains the complement of O. Thus q>
is an entire function. 0 1 also contains the unbounded component of the complement
of r*, since Indr (.~) is 0 there. Hence
(10)
lim q>(z) = lim h 1 (z) = O. (11)
1%1 .... 00 1%1 .... 00
Liouville's theorem implies now that q>(z) = 0 for every z. This proves (8), and
hence (2).
To deduce (3) from (2), pick a E 0 - r* and define F(z) = (z - a)f(z).
Then
1 1 1 1 F(z)
-. f(z) dz = -. -- dz = F(a) . Ind r (a) = 0,
2m r 2m r z - a
because F(a) = O.
Finally, (5) follows from (4) if (3) is applied to the cycle r = r 1 - r o.
This completes the proof. ////
10.36 Remarks
(a) If Y is a closed path in a convex region 0 and if IX ¢ 0, an application of
Theorem 10.14 to f(z) = (z - 1X)-1 shows that Indy (IX) = O. Hypothesis
(1) of Theorem 10.35 is therefore satisfied by every cycle in 0 if 0 is
convex. This shows that Theorem 10.35 generalizes Theorems 10.14 and
10.15.
(b) The last part of Theorem 10.35 shows under what circumstances integration
over one cycle can be replaced by integration over another, without
changing the value of the integral. For example, let 0 be the plane with
three disjoint closed discs Di removed. If r, Y1' Y2' Y3 are positively
oriented circles in 0 such that r surrounds D1 u D2 U D3 and Yi surrounds
Di but not D j for j "# i, then
if(Z) dz = J1 i.t(Z) dz
for every f E H(O).
(c) In order to apply Theorem 10.35, it is desirable to have a reasonably
efficient method of finding the index of a point with respect to a closed
path. The following theorem does this for all paths that occur in practice.
(12)