Theorem

Mergelyan's Theorem

Analysis

Mergelyan's Theorem is a result on the approximation of functions by polynomials in complex analysis, proved by the Armenian mathematician Sergei Mergelyan in 1951. It gives a sharp topological condition on a compact subset of the complex plane under which every function continuous on that set and holomorphic on its interior can be uniformly approximated as closely as desired by polynomials.

Facts

Statement

For a compact subset K of the complex plane whose complement is connected, every function continuous on K and holomorphic on the interior of K can be uniformly approximated on K by polynomials. 1

Proof Year

1951 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 Mergelyan's theorem (Wikipedia)

Sources

1. Mergelyan's theorem (Wikipedia)

Statement section
every continuous function f, such that the restriction of f to the interior of K is holomorphic, can be approximated uniformly on K with polynomials.
Lead paragraph
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
In Branch: Complex Analysis, Lead sentence
is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.

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Mergelyan's Theorem

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In Branch