Rouché's Theorem

Theorem
Let $f$ and $g$ be complex-valued functions which are holomorphic in the interior of some simply connected region $D$.

Let $\left\vert{g \left({z}\right)}\right\vert < \left\vert{f \left({z}\right)}\right\vert$ on the boundary of $D$.

Then $f$ and $f + g$ have the same number of zeroes in the interior of $D$ counted up to multiplicity.