UWEC CERCA 2025

Have you ever been walking your dog and the leash got tangled around a lamp post? Are you wondering what this has to do with Complex Analysis? Don’t worry, we are here to tell you!  Our goal is to explore and explain Rouché’s Theorem through an accessible analogy and mathematical proof. The theorem states: If f and h are each functions that are analytic inside and on a simple closed contour C and if the strict inequality h(z) < f(z) holds at each point on C, then f and f + h must have the same total number of zeros (counting multiplicities) inside C. Therefore, this theorem is typically used to find the location of zeros of a complicated analytic function. In our presentation, we will explain Rouché's Theorem using this dog-walking analogy, discuss why this theorem is valuable, and prove the theorem.