DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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Runge approximation on a plane domain

Definition

Let $Ω \subseteq C$ be a plane domain, let $P \subseteq \hat{C} ∖ Ω$ , and let $f : Ω \to C$ be holomorphic.

One says that $f$ is Runge-approximable on $Ω$ with poles in $P$ when there is a sequence of rational functions $(r_{n})$ such that every finite pole of $r_{n}$ lies in $P \cap C$ , the only possible pole at infinity also lies in $P$ , and

$r_{n} \to f locally uniformly on Ω .$

If $\hat{C} ∖ Ω$ is connected and $P = {\infty}$ , this is simply polynomial approximation on $Ω$ .

Remarks

The compact-set theorem supplies the local pieces of this definition. The next theorem upgrades them to a single sequence on the whole domain by choosing an exhaustion.

Depends on

Runge polynomial approximation when the complement is connected

Used by

Runge approximation on plane domains Theorem

Dependency tree · two levels

2 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, Lemma 9.2.5 and Corollary 9.2.6 (standard reference, not scraped)
M. Weber, Complex Analysis, Theorem 4.4.6 (standard reference, not scraped)