Hurwitz’s theorem

Define the ball at $z_0$ of radius $r$ as and $D(z_0,r)=\{z\in G:|z-z_0|\le r\}$ is the closed ball at $z_0$ of radius $r$.

Theorem (Hurwitz).

Let $G\mathrm{\subset }\mathrm{C}$ be a region and suppose the sequenceof holomorphic functions $\mathrm{\{}{f}_n\mathrm{\}}$ converges uniformly on compact subsetsof $G$ to a holomorphic function $f$. If $f$ is not identically zero,$D\mathit{}\mathrm{(}{z}_{\mathrm{0}}\mathrm{,}r\mathrm{)}\mathrm{\subset }G$ and $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\ne }\mathrm{0}$ for $z$ such that$\mathrm{|}z\mathrm{-}{z}_{\mathrm{0}}\mathrm{|}\mathrm{=}r$, then there exists an $N$ such that for all $n\mathrm{\ge }N$$f$ and $f_n$ have the same number of zeros in $B\mathit{}\mathrm{(}{z}_{\mathrm{0}}\mathrm{,}r\mathrm{)}$.

What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always converges to a holomorphic function but that’s another theorem altogether), the function is not identically zero and furthermore the function is notzero on the boundary of some ball,then eventuallythe functions of the sequence have the same number of zeros inside this ball as does the function.

Do note the requirement for $f$ not being identically zero. For example the sequence $f_n(z):=\frac{1}{n}$ converges uniformly on compact subsets to$f(z):=0$, but $f_n$ have no zeros anywhere, while $f$ is identically zero.

Also in general this result holds for bounded convex subsets (http://planetmath.org/ConvexSet) but it is mostuseful to for balls.

An immediate consequence of this theorem is this useful corollary.

Corollary.

If $G$ is a region and a sequence of holomorphic functions $\mathrm{\{}{f}_n\mathrm{\}}$ convergesuniformly on compact subsets of $G$ to a holomorphic function $f$, and furthermore if $f_n$ never vanishes (is not zero for any point in $G$), then$f$ is either identically zero or also never vanishes.