# Hurwitz

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 $${{\partial}}\Omega$$.

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

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 $$k\to \infty$$.

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.