Montel's theorem

In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal.

The first, and simpler, version of the theorem states that a uniformly bounded family of holomorphic functions defined on an open subset of the complex numbers is normal.

This theorem has the following formally stronger corollary. Suppose that ${\displaystyle {\mathcal {F}}}$ is a family of meromorphic functions on an open set ${\displaystyle D}$. If ${\displaystyle z_{0}\in D}$ is such that ${\displaystyle {\mathcal {F}}}$ is not normal at ${\displaystyle z_{0}}$, and ${\displaystyle U\subset D}$ is a neighborhood of ${\displaystyle z_{0}}$, then ${\displaystyle \bigcup _{f\in {\mathcal {F}}}f(U)}$ is dense in the complex plane.

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