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

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Runge pole sets for rational approximation on a compact set

Definition

Let $K \subseteq C$ be compact, let $U$ be an open neighbourhood of $K$ , and let $f : U \to C$ be holomorphic. A subset $P \subseteq \hat{C} ∖ K$ is a Runge pole set for $K$ when every connected component of $\hat{C} ∖ K$ meets $P$ .

One says that $f$ is rationally approximable on $K$ with poles in $P$ when for every $ε > 0$ there is a rational function $r$ whose finite poles all lie in $P \cap C$ and whose only possible pole at $\infty$ is also in $P$ , such that

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

Remarks

If $\hat{C} ∖ K$ is connected, then the singleton $P = {\infty}$ is a Runge pole set. In that case the approximants are exactly polynomials.

Used by

Pole pushing along a chain of discs Definition
Runge approximation with a prescribed pole set Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, §9.2 (standard reference, not scraped)
M. Weber, Complex Analysis, §4.4 (standard reference, not scraped)