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.
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StatementGiven 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 Classification
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1. Runge's Theorem (Wikipedia)
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In complex analysis, Runge's theorem (also known as Runge's approximation theorem) 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
Proved By: Carl Runge, Lead
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in 1885
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