Hurwitz's theoremHurwitzsTheorem2004-04-12 00:48:062005-03-07 05:01:59Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}Define the ball at $z_0$ of radius $r$ as $B(z_0,r) = \{ z \in G : \lvert z-z_0 \rvert < r \}$ and $D(z_0,r) = \{ z \in G : \lvert z-z_0 \rvert \leq r \}$ is the closed ball at $z_0$ of radius $r$.
\begin{thm}[Hurwitz]
Let $G \subset {\mathbb{C}}$ be a region and suppose the sequence
of holomorphic functions $\{ f_n \}$ converges uniformly on compact subsets
of $G$ to a holomorphic function $f$. If $f$ is not identically zero,
$D(z_0,r) \subset G$ and $f(z) \not= 0$ for $z$ such that
$\lvert z-z_0 \rvert = r$, then there exists an $N$ such that for all $n \geq N$
$f$ and $f_n$ have the same number of zeros in $B(z_0,r)$.
\end{thm}
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 \PMlinkescapetext{limit}
function is not identically zero and furthermore the \PMlinkescapetext{limit} function is not
zero on the boundary of some ball,
then eventually
the functions of the sequence have the same number of zeros inside this ball as does the \PMlinkescapetext{limit} 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 \PMlinkname{convex subsets}{ConvexSet} but it is most
useful to \PMlinkescapetext{state} for balls.
An immediate consequence of this theorem is this useful corollary.
\begin{cor}
If $G$ is a region and a sequence of holomorphic functions $\{ f_n \}$ converges
uniformly 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.
\end{cor}
\begin{thebibliography}{9}
\bibitem{Conway:complexI}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable I}}.
Springer-Verlag, New York, New York, 1978.
\end{thebibliography}