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    "title": "Mittag-Leffler: constructing analytic functions with prescribed isolated singular parts",
    "summary": "Mittag-Leffler's1884 memoir constructs a single-valued analytic function with prescribed singular parts at a countable isolated setQ, regular away fromQand its accumulation setQprime. Subtracting suitable finite Taylor/rational correction terms from each prescribed part makes their series locally uniformly convergent. Finite polynomial principal parts give the familiar meromorphic-function construction; entire functions of1/(z−a)allow isolated essential singular parts as well. Section 1also treats the stated connected-domain boundary formulation and describes the remaining freedom by addition of a regular function.",
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        "title": "Sur la représentation analytique des fonctions monogènes uniformes d'une variable indépendante (ActaMath4,1–79)",
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        "description": "Accumulation points ofQare excluded; a set merely countable but not isolated is insufficient, and local-uniform convergence is away from the excluded singular set. A finite singular-part polynomial produces a pole, while a transcendental entire function of1/(z−a)generally produces an essential singularity: these cases are not interchangeable. The meromorphic function is not unique without further normalization; on the whole plane two solutions with the same principal parts differ by an entire function. This record covers Section 1, not all 79 pages, later sheaf-cohomological formulations, interpolation with arbitrary zero controls or every general Riemann surface.",
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      "statement": "# Mittag-Leffler: constructing analytic functions with prescribed isolated singular parts\n\nMittag-Leffler's1884 memoir constructs a single-valued analytic function with prescribed singular parts at a countable isolated setQ, regular away fromQand its accumulation setQprime. Subtracting suitable finite Taylor/rational correction terms from each prescribed part makes their series locally uniformly convergent. Finite polynomial principal parts give the familiar meromorphic-function construction; entire functions of1/(z−a)allow isolated essential singular parts as well. Section 1also treats the stated connected-domain boundary formulation and describes the remaining freedom by addition of a regular function.\n\n## Scientific scope\n\nAccumulation points ofQare excluded; a set merely countable but not isolated is insufficient, and local-uniform convergence is away from the excluded singular set. A finite singular-part polynomial produces a pole, while a transcendental entire function of1/(z−a)generally produces an essential singularity: these cases are not interchangeable. The meromorphic function is not unique without further normalization; on the whole plane two solutions with the same principal parts differ by an entire function. This record covers Section 1, not all 79 pages, later sheaf-cohomological formulations, interpolation with arbitrary zero controls or every general Riemann surface.\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\nGösta Mittag-Leffler\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- [Sur la représentation analytique des fonctions monogènes uniformes d'une variable indépendante (ActaMath4,1–79)](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/4767-11511_2006_Article_BF02418410.pdf)\n",
      "summary": "Mittag-Leffler's1884 memoir constructs a single-valued analytic function with prescribed singular parts at a countable isolated setQ, regular away fromQand its accumulation setQprime. Subtracting suitable finite Taylor/rational correction terms from each prescribed part makes their series locally uniformly convergent. Finite polynomial principal parts give the familiar meromorphic-function construction; entire functions of1/(z−a)allow isolated essential singular parts as well. Section 1also treats the stated connected-domain boundary formulation and describes the remaining freedom by addition of a regular function.",
      "title": "Mittag-Leffler: constructing analytic functions with prescribed isolated singular parts"
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    "title": "Mittag-Leffler: constructing analytic functions with prescribed isolated singular parts",
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    "what_changed": "Mittag-Leffler's1884 memoir constructs a single-valued analytic function with prescribed singular parts at a countable isolated setQ, regular away fromQand its accumulation setQprime. Subtracting suitable finite Taylor/rational correction terms from each prescribed part makes their series locally uniformly convergent. Finite polynomial principal parts give the familiar meromorphic-function construction; entire functions of1/(z−a)allow isolated essential singular parts as well. Section 1also treats the stated connected-domain boundary formulation and describes the remaining freedom by addition of a regular function.",
    "scope": "Mittag-Leffler's1884 memoir constructs a single-valued analytic function with prescribed singular parts at a countable isolated setQ, regular away fromQand its accumulation setQprime. Subtracting suitable finite Taylor/rational correction terms from each prescribed part makes their series locally uniformly convergent. Finite polynomial principal parts give the familiar meromorphic-function construction; entire functions of1/(z−a)allow isolated essential singular parts as well. Section 1also treats the stated connected-domain boundary formulation and describes the remaining freedom by addition of a regular function. Accumulation points ofQare excluded; a set merely countable but not isolated is insufficient, and local-uniform convergence is away from the excluded singular set. A finite singular-part polynomial produces a pole, while a transcendental entire function of1/(z−a)generally produces an essential singularity: these cases are not interchangeable. The meromorphic function is not unique without further normalization; on the whole plane two solutions with the same principal parts differ by an entire function. This record covers Section 1, not all 79 pages, later sheaf-cohomological formulations, interpolation with arbitrary zero controls or every general Riemann surface.",
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