CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-28

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Small perturbations preserve the total local zero multiplicity

Statement

Let $f$ be holomorphic on a neighbourhood of the closed disc $\overline{D (a, r)}$ , and suppose $f$ has no zero on the circle $∣ z - a ∣ = r$ . Let $m$ be the total multiplicity of the zeros of $f$ in $D (a, r)$ . If $g$ is holomorphic on a neighbourhood of $\overline{D (a, r)}$ and

$∣ g (z) - f (z) ∣ < ∣ f (z) ∣ (∣ z - a ∣ = r),$

then $g$ has exactly $m$ zeros in $D (a, r)$ , counted with multiplicity.

In particular, if $a$ is an isolated zero of $f$ of order $m$ and the disc is chosen so that $a$ is the only zero of $f$ in $\overline{D (a, r)}$ , then every such perturbation $g$ has exactly $m$ zeros in $D (a, r)$ counted with multiplicity.

Facts & Assumptions

Given: A holomorphic function $f$ on a neighbourhood of $\overline{D (a, r)}$ , a holomorphic function $g$ on the same neighbourhood, and $∣ g - f ∣ < ∣ f ∣$ on $∣ z - a ∣ = r$ .

[L1]

Rouché's theorem gives equal zero counts inside a closed contour when the strict boundary inequality holds (Rouche's theorem in the classical strict-inequality form).

Proof

technique · direct

1.1given

Let $γ (t) = a + r e^{i t}$ for $0 \leq t \leq 2 π$ . The hypothesis says $∣ g - f ∣ < ∣ f ∣$ on $γ^{*}$ , and $f$ has no zero there.

2.1step 1.1L1

Applying [L1] to the contour $γ$ shows that $f$ and $g$ have the same weighted zero count inside $∣ z - a ∣ < r$ . Because both are holomorphic, there is no pole term, so that weighted count is exactly the total multiplicity of the interior zeros. Hence $g$ has as many zeros in the disc, counted with multiplicity, as $f$ does.

3.1step 2.1∎

The isolated-zero specialization is the case where that total multiplicity for $f$ is the single local order $m$ at $a$ .

Depends on

Used by

Locally uniform convergence preserves the total multiplicity near an isolated zero Theorem

Dependency tree · two levels

9 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 (standard reference, not scraped)
R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)