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Runge's theorem

Runge's theorem, also known as Runge's approximation theorem, is a fundamental result in that asserts the uniform approximability of holomorphic functions on compact subsets of the by rational functions with poles in prescribed locations in the complement. Specifically, if K \subset \mathbb{C} is compact, f is holomorphic on a neighborhood of K, and P \subset \hat{\mathbb{C}} \setminus K contains at least one point from each of \hat{\mathbb{C}} \setminus K (where \hat{\mathbb{C}} denotes the extended ), then for every \epsilon > 0, there exists a rational function r(z) with poles only in P such that \sup_{z \in K} |f(z) - r(z)| < \epsilon. A special case occurs when the complement \mathbb{C} \setminus K is connected, allowing approximation by polynomials alone. Named after the German mathematician (1856–1927), the theorem was first proved in his 1885 paper "Zur Theorie der eindeutigen analytischen Funktionen," published in Acta Mathematica. Runge's work built on earlier ideas from on polynomial approximation in the real domain, extending them to the complex setting while addressing the constraints imposed by holomorphy, such as the inability to approximate functions with "holes" in their domains using entire functions like polynomials. The proof relies on to represent the function via contours, followed by approximations using geometric series expansions to shift poles and ensure uniform convergence on K. The theorem's significance lies in its role as a cornerstone for approximation theory in several complex variables and its applications to problems like solving Dirichlet problems on non-smooth domains or constructing Riemann mappings. It has been generalized in various directions, including to several complex variables by Oka and others, and to rational approximation on real manifolds, influencing fields from numerical analysis to conformal mapping. For instance, when K is the closure of a simply connected domain bounded by a Jordan curve, the theorem implies the Weierstrass approximation theorem for holomorphic functions inside the domain.

Statement

General theorem

Runge's theorem provides a fundamental result in complex analysis concerning the uniform approximation of holomorphic functions by rational functions on compact sets. The theorem is set in the extended complex plane, denoted \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}, which is the one-point compactification of the complex plane, topologically equivalent to the . The complement \hat{\mathbb{C}} \setminus K of a compact set K \subset \hat{\mathbb{C}} consists of connected components, which are the maximal open connected subsets of this complement. The general statement of Runge's theorem is as follows: Let K \subset \hat{\mathbb{C}} be a compact set, let f be a function holomorphic on some open neighborhood U of K, and let A \subset \hat{\mathbb{C}} \setminus K be a set containing exactly one point in each connected component of \hat{\mathbb{C}} \setminus K. Then there exists a sequence of rational functions (r_n) whose poles lie only in A such that r_n converges uniformly to f on K. This uniform convergence means that for every \varepsilon > 0, there exists N such that for all n > N and all z \in K, |r_n(z) - f(z)| < \varepsilon. The restriction of poles to A ensures that the approximating rational functions avoid introducing singularities inside or on K, while the choice of exactly one point per connected component of the complement allows the approximations to capture the necessary behavior in each "hole" or unbounded region of the complement. This setup guarantees the existence of such approximations without requiring the complement to be connected, distinguishing the general theorem from its polynomial corollary.

Polynomial approximation corollary

A key special case of Runge's theorem arises when the compact set has a connected complement in the extended complex plane, allowing approximation by rather than more general rational functions. Specifically, if K \subset \mathbb{C} is compact and \mathbb{C} \setminus K is connected, then for any function f holomorphic in a neighborhood of K and any \varepsilon > 0, there exists a p such that |f(z) - p(z)| < \varepsilon for all z \in K. This polynomial approximation corollary follows from the general theorem by selecting the set of poles A = \{\infty\}, as the connectedness of the complement ensures no bounded components require finite poles. Polynomials, being rational functions with their sole pole at infinity, thus suffice for uniform approximation on such sets, simplifying the construction and highlighting the role of topological connectivity in the plane. A classic example is the closed unit disk \overline{\mathbb{D}} = \{ z \in \mathbb{C} : |z| \leq 1 \}, whose complement \mathbb{C} \setminus \overline{\mathbb{D}} is connected. Any function holomorphic in a neighborhood of \overline{\mathbb{D}}, such as f(z) = e^z, can therefore be uniformly approximated on \overline{\mathbb{D}} by polynomials, enabling practical computations in complex analysis.

Proof

Cauchy integral approximation

In the proof of Runge's theorem, the initial approximation step relies on Cauchy's integral formula to express a holomorphic function f defined in a neighborhood of a compact set K \subset \mathbb{C} as an integral over a suitable contour, which is then discretized to yield rational approximants. Specifically, select a rectifiable Jordan curve \Gamma in the domain of holomorphy of f that encloses K and lies entirely in the complement of the bounded components of \mathbb{C} \setminus K, ensuring \Gamma winds once positively around K. For any w \in K, f(w) = \frac{1}{2\pi i} \int_{\Gamma} \frac{f(z)}{z - w} \, dz. This representation holds by Cauchy's integral theorem, as f is holomorphic inside and on \Gamma, and the kernel $1/(z - w) has a simple pole at w inside \Gamma. To approximate this integral, parametrize \Gamma and divide it into n subarcs of equal length, with endpoints z_k for k = 0, \dots, n, and arc lengths \Delta z_k = z_k - z_{k-1}. The Riemann sum approximation is then R_n(w) = \frac{1}{2\pi i} \sum_{k=1}^n \frac{f(z_k)}{z_k - w} \Delta z_k, which defines a rational function with simple poles at the points z_k on \Gamma, hence outside K. Each term in the sum is a scaled basic rational function with a prescribed pole location. The uniform convergence of these Riemann sums to the integral on K follows from the continuity of f and the kernel on the compact set \Gamma \times K, ensuring the integrand f(z)/(z - w) is bounded and uniformly continuous there. As the mesh size of the partition (maximum |\Delta z_k|) tends to zero, \sup_{w \in K} |f(w) - R_n(w)| \to 0, providing arbitrarily close uniform approximations by such rational functions on K. For finite n sufficiently large, the error can be made smaller than any prescribed \epsilon > 0. This step establishes the approximability by rationals with poles off K, forming the foundation for further refinements in the proof.

Pole shifting technique

The pole shifting technique is a key step in the proof of Runge's theorem, enabling the relocation of from an auxiliary \Gamma surrounding the compact set K to prescribed points in the set A \subset \hat{\mathbb{C}} \setminus K, while ensuring uniform convergence of the resulting rational approximants on K. This method relies on expanding the kernel functions from the Cauchy integral representation using , allowing the original on \Gamma to be "shifted" inward toward A without introducing singularities inside K. The validity of the expansion depends on choosing shift points w_0 \in A sufficiently close to the original z_0 \in \Gamma, specifically satisfying |z_0 - w_0| < \dist(w_0, K), to guarantee convergence on K. To shift a single pole from z_0 on \Gamma to w_0 \in A, the reciprocal kernel $1/(z - z_0) is rewritten via the geometric series expansion: \frac{1}{z - z_0} = \frac{1}{z - w_0} \sum_{n=0}^\infty \left( \frac{z_0 - w_0}{z - w_0} \right)^n, which holds for z \in K under the distance condition above, as the series terms satisfy |(z_0 - w_0)/(z - w_0)| < 1 uniformly on K. Truncating this infinite series at a finite order N yields a rational function r_N(z) with a single pole at w_0 and the error term |1/(z - z_0) - r_N(z)| \to 0 uniformly on K as N \to \infty, preserving the approximation properties from the initial Cauchy integral setup. This truncation produces rational approximants whose poles lie exclusively in A, directly supporting the theorem's conclusion for functions holomorphic in a neighborhood of K. For compact sets K whose complement \hat{\mathbb{C}} \setminus K has multiple connected components, one point w_j \in A \cap U_j is selected per component U_j to ensure the shifted poles intersect every unbounded component, as required for the approximation to extend holomorphically. The series expansions are applied component-wise, with the shift distances controlled iteratively along paths connecting z_0 to w_j within each U_j, using intermediate points p_k where each step |p_{k+1} - p_k| < \mathrm{dist}(K, p_k) to maintain uniform convergence on K. This multi-step shifting guarantees that the overall rational approximant converges uniformly without singularities on K, adapting the single-pole technique to the general case. In the special case where approximation by polynomials is desired—corresponding to a "pole at infinity"—the technique first shifts poles to a distant point w_0 with |w_0| > 2 \sup_{z \in K} |z| using the above method. The resulting kernel is then expanded as \frac{1}{z - w_0} = -\frac{1}{w_0} \sum_{n=0}^\infty \left( \frac{z}{w_0} \right)^n, valid since |z/w_0| < 1/2 on K, and truncation to finite degree produces polynomials that approximate the original integral uniformly on K. This handles the infinity component directly, reducing the rational case to the polynomial corollary when \mathbb{C} \setminus K is connected.

Generalizations

Mergelyan's theorem

Mergelyan's theorem provides a significant advancement in the theory of polynomial approximation in complex analysis. It states that if K \subset \mathbb{C} is a compact set whose complement \mathbb{C} \setminus K is connected, and f: K \to \mathbb{C} is continuous on K and holomorphic on the interior \hat{K} of K, then for every \varepsilon > 0, there exists a p such that |f(z) - p(z)| < \varepsilon for all z \in K. This result was proved by Sergei Nikolaevich Mergelyan in his 1951 paper. The theorem strengthens the polynomial approximation corollary of Runge's theorem, which applies to functions holomorphic in an open neighborhood of K, by relaxing the holomorphy condition to mere continuity up to the boundary of K while still requiring the complement's connectivity. This key improvement allows for the uniform approximation of a broader class of functions, namely those in the algebra A(K) of continuous functions on K that are holomorphic in \hat{K}, using polynomials dense in this space. Without the connected complement condition, such approximation may fail, as counterexamples exist for disconnected complements. Mergelyan's proof builds directly on Runge's theorem by first extending f holomorphically to a suitable neighborhood of K, enabling the application of Runge's approximation techniques. One approach involves convolving f with approximate identity kernels, such as suitable mollifiers, to smooth it across the boundary while preserving holomorphy in the interior, followed by approximation via the Cauchy-Green formula or Mergelyan's lemma for kernel estimates. Alternatively, solving Dirichlet problems on annular regions around components of K facilitates the holomorphic extension, ensuring the approximants remain polynomials after invoking Runge's result. These methods highlight the theorem's constructive nature, though the original proof emphasizes the density of polynomials in A(K) for connected complements.

Oka-Weil theorem

The Oka-Weil theorem generalizes to several complex variables. It states that if K is a compact subset of a Stein manifold X, and f is holomorphic on a neighborhood of K, then f can be uniformly approximated on K by holomorphic functions on X that are rational with poles outside K. This was proved by Kiyoshi Oka in the 1940s and independently by in 1952. The theorem extends the approximation to higher-dimensional complex manifolds, requiring the domain to be (a complex analogue of contractible with no holes) to ensure the cohomology conditions for approximation hold. It plays a central role in , influencing approximation of holomorphic maps to .

Runge domains

A Runge domain is an open set U \subset \mathbb{C} such that every function holomorphic on U can be uniformly approximated on every compact subset of U by entire functions, that is, functions holomorphic on the entire complex plane \mathbb{C}. This property extends the approximation capabilities of from individual compact sets to the entire domain U, allowing global approximation properties to hold for the space of holomorphic functions on U with the topology of uniform convergence on compact subsets. A key characterization of Runge domains in \mathbb{C} is topological: U is a Runge domain if and only if its complement in the extended complex plane \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} is connected. Equivalently, the complement \mathbb{C} \setminus U has no bounded connected components. This condition ensures that there are no "holes" in U that would prevent the uniform approximation by entire functions, as bounded connected components in the complement would create separate components in \hat{\mathbb{C}} \setminus U, disconnecting it from the point at infinity. The notion of Runge domains generalizes the core idea of Runge's theorem to open sets by applying the theorem iteratively. Specifically, one constructs an exhaustion of U by a sequence of compact subsets K_n with connected complements in \hat{\mathbb{C}}, approximates the holomorphic function on each K_n using Runge's theorem (via rational functions with poles shifted to infinity to obtain polynomials, which are dense among entire functions on bounded sets), and ensures the approximations converge uniformly on compacts across the exhaustion. Examples of Runge domains include the entire complex plane \mathbb{C} (whose complement is the singleton \{\infty\}, connected) and the open unit disk \{ z \in \mathbb{C} : |z| < 1 \} (whose complement \{ |z| \geq 1 \} is unbounded and connected, attaching to \infty). In contrast, the punctured plane \mathbb{C} \setminus \{0\} is not a Runge domain, as \hat{\mathbb{C}} \setminus (\mathbb{C} \setminus \{0\}) = \{0, \infty\} consists of two disconnected components. Similarly, the plane minus finitely many points fails the condition due to multiple bounded point components in the complement.

Applications

Approximation of holomorphic functions

Runge's theorem enables the construction of meromorphic functions with prescribed poles by approximating suitable rational functions that incorporate the desired principal parts at those poles, particularly when the complement of the domain has disconnected components allowing poles in each bounded component. Specifically, for a meromorphic function on an open set U containing a compact set K, one can uniformly approximate it on K by rational functions whose poles lie only in prescribed points, one in each bounded component of \mathbb{C} \setminus K, ensuring the approximants capture the local behavior near the poles. This approach underpins the , which guarantees the existence of a meromorphic function in U with exactly the specified poles and principal parts, constructed via a series of such rational approximations that converge uniformly on compact subsets away from the poles. A representative example arises in approximating holomorphic functions on an annular domain, such as the compact set K = \{ z \in \mathbb{C} : 1/2 \leq |z| \leq 2 \}, where the complement has two components: the inner disk |z| < 1/2 and the exterior |z| > 2. For a function f holomorphic in a neighborhood of K, Runge's theorem permits uniform approximation on K by rational functions with a single in the inner disk (e.g., at z = 0) and possibly at infinity for the outer component, facilitating the derivation of expansions on the annulus through such rational . This technique highlights how poles placed strategically in the "hole" allow faithful reproduction of the function's behavior without altering holomorphy on K. In solving boundary value problems, Runge's theorem supports approximations of solutions to Cauchy or Dirichlet problems on domains like annuli by rational functions that satisfy prescribed boundary conditions, leveraging the theorem's ability to place poles off the domain while matching boundary data uniformly. For instance, on an annulus \Omega = \Omega_1 \setminus \Omega_2, holomorphic approximation via rational functions relates to solving mixed \bar{\partial}-problems with boundary conditions, where vanishing cohomology ensures the existence of such approximants that align with the problem's data. Numerically, Runge's theorem provides the theoretical basis for algorithms computing holomorphic extensions or approximations in complex analysis software, such as those employing rational approximation on compact sets with holes, where the error decreases exponentially with degree for analytic functions. Implementations like the exploit this by adaptively selecting poles to achieve , enabling efficient computation of approximations in tools for complex function evaluation and extension.

Connections to functional analysis

Runge's theorem establishes that rational functions with poles outside a compact set K \subset \mathbb{C} are dense in the Banach space A(K) of functions that are continuous on K and holomorphic in its interior, equipped with the supremum norm \|f\|_\infty = \sup_{z \in K} |f(z)|. This density result follows directly from the uniform approximation property of the theorem, allowing any holomorphic function near K to be approximated arbitrarily closely by such rationals on K. On Runge domains—those for which the complement is connected—polynomials form a dense subspace in A(K), providing a polynomial approximation foundation in these functional analytic settings. In spaces of bounded holomorphic functions, such as H^\infty(\Omega) on an \Omega \subset \mathbb{C}, Runge's theorem implies that polynomials (or entire functions on suitable unbounded domains) are dense in the topology of on compact subsets, provided \Omega satisfies the necessary conditions. This extends the approximation to the inductive limit topology on H^\infty(\Omega), where the sup on compacts ensures and facilitates techniques for bounded functions. Such density is pivotal for analyzing the structure of these spaces as uniform algebras. Runge's theorem influences Lavrentiev's theorem, which guarantees uniform polynomial approximation of continuous functions on compact sets of measure zero or nowhere dense in the plane, serving as a precursor to broader density results in approximation theory. It also underpins the Oka-Weil theorem, which generalizes these ideas to several complex variables, asserting that on O(X)-convex compact sets in a Stein manifold X, global holomorphic functions approximate those locally holomorphic near the set, with rational-like approximations in one variable inspiring the multivariable framework. In modern , Runge's theorem enables the approximation of resolvents (\lambda - T)^{-1} for operators T on Banach spaces by rational functions with poles outside the , facilitating the construction of holomorphic functional calculi and decompositions. This rational approximation is central to the study of subnormal operators, where the theory intertwines with the density of rationals in spaces like R^2(K, \mu)—the closure of rational functions with poles off a compact K—to analyze subspaces and embeddings into operators. These connections extend to , supporting Beurling-type theorems for multiplication operators on rational Hardy-like spaces.