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Normal convergence of holomorphic products

Definition

Let $Ω \subseteq C$ be open, and let $f_{n} : Ω \to C$ be holomorphic functions for $n \geq 0$ .

The product

$\prod_{n \geq 0} f_{n} (z)$

is normally convergent on $Ω$ if for every compact set $K \subseteq Ω$ there is an integer $N$ such that $f_{n}$ has no zero on $K$ for all $n \geq N$ and

$\sum_{n \geq N} \sup_{z \in K} ∣ f_{n} (z) - 1 ∣ < \infty .$

Equivalently, after discarding finitely many factors that may contribute zeros, the deviations from $1$ are absolutely summable uniformly on each compact set. This is the product analogue of normal convergence for holomorphic series.

Depends on

Complex infinite-product convention extending the published real definition

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Matthias Weber, Complex Analysis, Ch. 3 §3.2 (standard reference, not scraped)