For questions on Rouché's theorem, which relates the number of roots of two holomorphic functions $f,g$ in some bounded domain $D$ given that $|g|<|f|$ on $\partial D$.
Rouché's theorem states: if $\partial D$ is a simple, closed curve with interior $D$ and $f,g$ are holomorphic on $D$ with $|g|<|f|$ on $\partial D$, then $f$ and $f-g$ have the same number of roots on $D$, counting multiplicities. This is particularly useful for finding the number of roots of polynomials: for example, if we consider $p(z) = z^6+z^3-20z+13$, on $\partial D = \{w : |w|=3\}$, then $p$ has six roots on $\overline{D}$, since $|z^3-20z+13|\le 100$ and $|z^6|=729$ on $\partial D$.
Michael Artin used Rouché's theorem to give a proof of the Fundamental Theorem of Algebra.
Consider using the tags complex-analysis, holomorphic-functions, or similar in conjunction with this tag.
