Theorem

Runge's Theorem

Analysis

Runge's Theorem states that a function holomorphic on an open subset of the complex plane can be approximated uniformly on any compact subset by rational functions whose poles lie in a prescribed set meeting every bounded component of the complement, and that ordinary polynomials suffice whenever that complement is connected. Named for Carl Runge, it is a foundational approximation result of complex analysis underlying later work on function theory on open sets.

Facts

Statement

Given a holomorphic function on a compact set, one can approximate it as well as desired by rational functions whose poles lie only at chosen points, one in each hole of the set. 1

Proof Year

1885 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Complex Analysis

Source Runge's Theorem (Wikipedia)

Sources

1. Runge's Theorem (Wikipedia)

Wikimedia Foundation

Lead section, figure caption
Given a holomorphic function f on the blue compact set and a point in each of the holes, one can approximate f as well as desired by rational functions having poles only at those three points.
Lead section, naming sentence
It is named after the German mathematician Carl Runge who first proved it in 1885.
In Branch: Complex Analysis, Lead sentence
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Ru

Switch Atlas

The Inner World

The Civil World

The Natural World

Runge's Theorem

Has Statement Form

In Branch