Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPlDC FUNCTIONS 215
singularities at a 1 , ••• , an' Theorem 10.35, applied to the function 9 and the
open set 0 0 , shows that
ig(Z) dz = o. (3)
Hence
i i 1 n 1 n
1t1 r i = 1 1t1 r k = 1
-2. f(z) dz = L -2. Qk(Z) dz = L Res (Qk; ak) Indr (ak),
and sincefand Qk have the same residue at ak' we obtain (2).
IIII
We conclude this chapter with two typical applications of the residue
theorem. The first one concerns zeros of holomorphic functions, the second is the
evaluation of a certain integral.
10.43 Theorem Suppose y is a closed path in a region 0, such that Indy (ex) = 0
for every ex not in O. Suppose also that Indy (ex) = 0 or 1 for every ex E 0 - y*,
and let 01 be the set of all ex with Indy (ex) = 1.
For any f E H(O) let N J be the number of zeros off in 01' counted according
to their multiplicities.
(a) Iff E H(O) andfhas no zeros on y* then
where r = f 0 y.
(b) If also 9 E H(O) and
1 i f'(z)
N J = 21ti y f(z) dz = Indr (0) (1)
If(z) - g(z) I < If(z) I for all z E y* (2)
Part (b) is usually called Rouche's theorem. It says that two holomorphic
functions have the same number of zeros in 01 if they are close together on the
boundary of 01' as specified by (2).
PROOF Put qJ = f'1f, a meromorphic function in O. If a E 0 and f has a zero
of order m = m(a) at a, then f(z) = (z - arh(z), where hand Ilh are holomorphic
in some neighborhood V of a. In V - {a},
qJ(z) = f'(z) = ~ + h'(z).
f(z) z - a h(z)
(3)
Thus
Res (qJ; a) = m(a). (4)