CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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Runge polynomial approximation when the complement is connected

Statement

Let $K \subseteq C$ be compact and let $\hat{C} ∖ K$ be connected. If $f$ is holomorphic on a neighbourhood of $K$ , then for every $ε > 0$ there is a polynomial $p$ such that

$\sup_{z \in K} ∣ f (z) - p (z) ∣ < ε .$

Facts & Assumptions

Given: A compact set $K$ with connected complement and a holomorphic function $f$ on a neighbourhood of $K$ .

[L1]

Runge approximation holds for every pole set meeting each complementary component (Runge approximation with a prescribed pole set).

Proof

technique · direct

1.1givenL1

Since $\hat{C} ∖ K$ is connected, the singleton $P = {\infty}$ meets its unique complementary component. Applying [L1] with that pole set gives rational approximants whose only possible pole is at $\infty$ .

2.1step 1.1algebra∎

If $r = P / Q$ is such a rational function in lowest terms and $Q$ has positive degree, then a zero of $Q$ would give a finite pole of $r$ . Therefore $Q$ is constant, so $r$ is a polynomial. Hence the approximants from step 1.1 are polynomials.

Depends on

Runge approximation with a prescribed pole set

Used by

Runge approximation on a plane domain Definition
1/z is not uniformly approximable by polynomials on the unit circle Example
Mittag-Leffler on the complex plane Theorem

Dependency tree · two levels

4 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, §9.1 and Corollary 9.2.6 (standard reference, not scraped)
M. Weber, Complex Analysis, Corollary 4.4.5 (standard reference, not scraped)