TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Runge approximation on plane domains

Statement

Let $Ω \subseteq C$ be a plane domain, let $P \subseteq \hat{C} ∖ Ω$ meet every connected component of $\hat{C} ∖ Ω$ , and let $f : Ω \to C$ be holomorphic. Then $f$ is Runge-approximable on $Ω$ with poles in $P$ .

Facts & Assumptions

Given: A plane domain $Ω$ , a pole set $P$ meeting every component of $\hat{C} ∖ Ω$ , and a holomorphic function $f$ on $Ω$ .

[L1]

Every compact set inside an open Euclidean set has a compact Jordan neighbourhood still inside that open set (A compact subset of an open Euclidean set has a compact Jordan neighborhood inside that open set).

[L2]

Runge approximation on one compact set holds once the pole set meets every component of its complement (Runge approximation with a prescribed pole set).

[L3]

Local-uniform approximation on a plane domain means uniform approximation on each compact set in an exhaustion (Runge approximation on a plane domain).

Proof

technique · direct

1.1

Choose an increasing exhaustion $K_{1} \subseteq K_{2} \subseteq \dots$ of compact subsets of $Ω$ with $⋃_{n} K_{n} = Ω$ and $K_{n} \subseteq int K_{n + 1}$ . By recursively applying [L1] and filling every complementary component of the chosen Jordan neighbourhood that lies entirely in $Ω$ , we may also require that every connected component of $\hat{C} ∖ K_{n}$ meets $P$ .

givenL1construct

2.1

Apply [L2] to each $K_{n}$ with tolerance $2^{- n}$ . This gives a rational function $r_{n}$ with poles in $P$ and $\sup_{z \in K_{n}} ∣ f (z) - r_{n} (z) ∣ < 2^{- n} .$

L2step 1.1choose

3.1

Fix a compact set $K \subseteq Ω$ . Choose $N$ with $K \subseteq K_{N}$ . Then for every $n \geq N$ one has $K \subseteq K_{n}$ , so step 2.1 gives $\sup_{z \in K} ∣ f (z) - r_{n} (z) ∣ \leq 2^{- n}$ . Hence $r_{n} \to f$ uniformly on $K$ . Since $K$ was arbitrary, [L3] gives local-uniform convergence on $Ω$ .

step 2.1L3algebra∎

Depends on

Used by

Choice bookkeeping for Runge and Mittag-Leffler Remark

Dependency tree · two levels

15 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, Corollary 9.2.6 (standard reference, not scraped)
M. Weber, Complex Analysis, Theorem 4.4.6 (standard reference, not scraped)