AWhat changed
Runge constructs rational approximations to a single-valued analytic function on a plane domain, converging uniformly on compact subsets away from its poles. Cauchy integrals on a nested square-grid exhaustion are replaced by finite rational sums; a pole-moving argument then relocates unwanted poles outside the approximation region. In particular, a holomorphic function on a domain can be locally uniformly approximated by rational functions without poles in that domain.
Scientific scope
Runge constructs rational approximations to a single-valued analytic function on a plane domain, converging uniformly on compact subsets away from its poles. Cauchy integrals on a nested square-grid exhaustion are replaced by finite rational sums; a pole-moving argument then relocates unwanted poles outside the approximation region. In particular, a holomorphic function on a domain can be locally uniformly approximated by rational functions without poles in that domain. This is complex rational approximation, not Runge's 1895numericalODEscheme. The approximation is local-uniform on interior compact sets, not uniform on an arbitrary entire open domain or its boundary, and does not assert polynomial approximation regardless of complementary holes. The source treats meromorphic functions using principal parts, rather than declaring poles regular. Its Section 2 prescribed-domain/prescribed-pole constructions serve as context, not separately counted or attributed solely to Runge: the original editor explicitly notes related prior Mittag-Leffler work. Later Mergelyan and Riemann-surface generalizations remain separate.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is complex rational approximation, not Runge's 1895numericalODEscheme. The approximation is local-uniform on interior compact sets, not uniform on an arbitrary entire open domain or its boundary, and does not assert polynomial approximation regardless of complementary holes. The source treats meromorphic functions using principal parts, rather than declaring poles regular. Its Section 2 prescribed-domain/prescribed-pole constructions serve as context, not separately counted or attributed solely to Runge: the original editor explicitly notes related prior Mittag-Leffler work. Later Mergelyan and Riemann-surface generalizations remain separate.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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