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.