Rouché's theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K {\displaystyle K} with closed contour ∂ K {\displaystyle \partial K} , if |g(z)| < |f(z)| on ∂ K {\displaystyle \partial K} , then f and f + g have the same number of zeros inside K {\displaystyle K} , where each zero is counted as many times as its multiplicity. This theorem assumes that the contour ∂ K {\displaystyle \partial K} is simple, that is, without self-intersections.