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

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Pole pushing along a chain of discs

Definition

Let $K \subseteq C$ be compact. A pole-pushing chain from $a_{0}$ to $a_{m}$ relative to $K$ is a finite list of closed discs ${\overline{D}}_{1}, \dots, {\overline{D}}_{m}$ such that

${\overline{D}}_{j} \cap K = \emptyset$ for every $j$ ;
$a_{j - 1}, a_{j} \in D_{j}$ for every $j = 1, \dots, m$ .

Given $ε > 0$ , pushing the pole of $(z - a_{0})^{- 1}$ along that chain to $a_{m}$ means finding a rational function $r$ with the following properties:

$r$ is holomorphic on a neighbourhood of $K$ ;
$r$ has at most one finite pole, namely at $a_{m}$ ;
$\sup_{z \in K} ∣ r (z) - (z - a_{0})^{- 1} ∣ < ε$ .

If the terminal point is declared to be $\infty$ , the third clause is kept and the second is replaced by "the approximant is a polynomial".

Depends on

Runge pole sets for rational approximation on a compact set

Used by

Runge's pole-pushing lemma Lemma

Dependency tree · one level

1 result within one dependency step 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, §9.2 before Lemma 9.2.2 (standard reference, not scraped)
M. Weber, Complex Analysis, §4.4 before Lemma 4.4.4 (standard reference, not scraped)