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 ${\displaystyle K}$ with closed contour ${\displaystyle \partial K}$, if |g(z)| < |f(z)| on ${\displaystyle \partial K}$, then f and f + g have the same number of zeros inside ${\displaystyle K}$, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour ${\displaystyle \partial K}$ is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.