Runge's theorem

Runge's theorem states that a function holomorphic near a compact subset of the complex plane can be uniformly approximated there by rational functions with poles outside the set, under a condition on its complement.

Uniform convergence: A sequence of functions converges uniformly when one error bound works at every point of the domain. Runge approximation controls the maximum error across the entire compact set.
Rational function: A rational function is a quotient of two polynomials, defined wherever its denominator is nonzero. Runge's approximants are rational functions whose poles lie outside the compact set.
Carl Runge: Carl Runge was a German mathematician known for work in numerical analysis and complex function theory. Runge introduced the approximation theorem that bears his name.
Rational approximation: Rational approximation uses quotients of polynomials to approximate functions on specified domains. Runge's theorem gives a foundational existence result for rational approximation in the complex plane.
Stone–Weierstrass theorem: The Stone–Weierstrass theorem gives conditions under which an algebra of continuous functions is dense in all continuous functions on a compact space. It is a broad algebraic approximation theorem, unlike Runge's holomorphic and planar-topology setting.
Holomorphic function: A complex-valued function is holomorphic on an open set when it is complex differentiable at every point there. The theorem approximates functions that are holomorphic on neighborhoods of the compact set.
Polynomial: A polynomial is a finite sum of powers of a variable with fixed coefficients. Polynomials suffice when the compact set's complement is connected.
Weierstrass approximation theorem: The Weierstrass approximation theorem states that every continuous function on a closed interval can be uniformly approximated by polynomials. It is a real-variable predecessor to approximation results for holomorphic functions.
Polynomial approximation: Polynomial approximation represents a function by polynomials that become close under a chosen error measure. Runge's theorem yields polynomial approximants when the compact set has connected complement.
Hartogs extension theorem: Hartogs' extension theorem states that holomorphic functions in several complex variables often extend across compact sets with no interior. It shows how approximation and extension phenomena differ between one and several complex variables.

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