Statement of the theorem: Suppose and
are holomorphic in an open set containing a closed, simple curve
and its interior. If
for all
, then
and
have the same number of zeros in the interior of
.
An application: Let . All five zeros of
are inside the disc
and exactly one zero is inside the disc
Proof: Let and let
. First, note that
and
are both holomorphic in
. For
,
. By Rouché’s theorem, the number of zeros of
inside the disc of radius 2 (which is 5, counting multiplicities) is equal to the number of zeros of
inside the disc. For
,
. Again, by Rouché’s theorem, the number of zeros of
inside the disc of radius 1 (one) is equal to the number of zeros of
inside the disc.