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        "title": "Zur Theorie der eindeutigen analytischen Functionen (ActaMath6,229–244)",
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        "description": "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.",
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          "citation": "Zur Theorie der eindeutigen analytischen Functionen (ActaMath6,229–244)",
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      "statement": "# Runge: rational approximation of single-valued analytic functions\n\nRunge 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.\n\n## Scientific scope\n\nThis 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nCarl Runge\n\nSource-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.\n\n\n\n## Original sources\n\n- [Zur Theorie der eindeutigen analytischen Functionen (ActaMath6,229–244)](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/4802-11511_2006_Article_BF02400416.pdf)\n",
      "summary": "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.",
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    "id": "runge-holomorphic-rational-approximation-1885",
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    "title": "Runge: rational approximation of single-valued analytic functions",
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    "what_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.",
    "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.",
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