TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Locally uniform convergence preserves the total multiplicity near an isolated zero

Statement

Let $Ω \subseteq C$ be open, let $f_{n} : Ω \to C$ be holomorphic, and suppose $f_{n} \to f$ locally uniformly on $Ω$ . Let $a \in Ω$ be an isolated zero of $f$ of multiplicity $m \geq 1$ . Then there is $r > 0$ such that $\overline{D (a, r)} \subseteq Ω$ , the function $f$ has no zero on $∣ z - a ∣ = r$ , and for all sufficiently large $n$ the function $f_{n}$ has exactly $m$ zeros in $D (a, r)$ counted with multiplicity.

Facts & Assumptions

Given: Holomorphic functions $f_{n}$ on an open set $Ω$ converging locally uniformly to $f$ , and an isolated zero $a$ of $f$ of multiplicity $m$ .

[L1]

A strict boundary perturbation preserves the total zero multiplicity in the disc (Small perturbations preserve the total local zero multiplicity).

Proof

technique · direct

1.1

By the isolated-zero hypothesis, choose $r > 0$ with $\overline{D (a, r)} \subseteq Ω$ such that $a$ is the only zero of $f$ in $\overline{D (a, r)}$ . Then $∣ f ∣$ is a positive continuous function on the circle $∣ z - a ∣ = r$ , so it has a positive minimum there. Call that minimum $η$ .

given

2.1

Because $f_{n} \to f$ locally uniformly and the circle $∣ z - a ∣ = r$ is compact, there is $N$ such that $∣ f_{n} (z) - f (z) ∣ < η \leq ∣ f (z) ∣ (∣ z - a ∣ = r, n \geq N) .$ Applying [L1] on that circle shows that for every $n \geq N$ , the function $f_{n}$ has the same total zero multiplicity in $D (a, r)$ as $f$ , namely $m$ .

step 1.1L1

3.1

Step 2.1 is exactly the claimed persistence of the local multiplicity.

step 2.1∎

Depends on

Small perturbations preserve the total local zero multiplicity

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

J. Lebl, Guide to Cultivating Complex Analysis, §5.4, Theorem 5.4.6 (standard reference, not scraped)
R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)