Hurwitz – Notes

definition (Locally uniform convergence):

Say a sequence $f_k\to f$ locally uniformly on $\Omega$ if $f_k\to f$ uniformly on every compact subset. Equivalently, $f_k \to f$ uniformly on any bounded subset of strictly positive distance to ${{\partial}}\Omega$ .

definition (Univalent):

A function $f:\Omega\to {\mathbb{C}}$ is univalent iff $f$ is holomorphic and injective. These are exactly the conformal maps of $\Omega$ to other domains.

theorem (Hurwitz):

Suppose $\left\{{f_k}\right\}$ is a sequence of holomorphic functions on $\Omega$ converging locally uniformly to $f$ on $\Omega$ and $f$ has a zero of order $n$ at $z_0$ . Then there exists some $\delta$ such that for $k\gg 1$ , $f_k$ has exactly $n$ zeros in the disc $\left\{{{\left\lvert {z-z_0} \right\rvert} < \delta}\right\}$ , counted with multiplicity. Moreover, these zeros converge to $z_0$ as $k\to \infty$ .

slogan:

The zeros of the sequence converge to the zeros of the limit.

:qa

If $\left\{{f_k}\right\}$ are univalent functions on $\Omega$ converging normally to $f$ , then either

$f$ is univalent, or
$f$ is constant.