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 sequence of holomorphic functions $\mathrm{\{}{f}_n\mathrm{\}}$ converges uniformly on compact subsets of $G$ 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 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 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 most useful 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 $\mathrm{\{}{f}_n\mathrm{\}}$ 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.

Title	Hurwitz’s theorem
Canonical name	HurwitzsTheorem
Date of creation	2013-03-22 14:17:55
Last modified on	2013-03-22 14:17:55
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30C15
Related topic	CompositionAlgebraOverAlgebaicallyClosedFields
Related topic	CompositionAlgebrasOverMathbbR
Related topic	CompositionAlgebrasOverFiniteFields
Related topic	CompositionAlgebrasOverMathbbQ