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    "title": "Montel: compactness and omitted-value criteria for holomorphic families",
    "summary": "An unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.",
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        "description": "Convergence is local uniform convergence inside a complex domain, not uniform convergence up to every boundary or convergence of the whole original sequence. The omitted values must be shared and distinct; allowing infinity as a subsequential limit is essential. For meromorphic functions the convergence definition must be changed and a three-value condition is different; it is not asserted here from the holomorphic statement. The 1907 bounded proof treats simple connectivity first and then finite-connectivity/covering extension. The 1912 theorem is recorded in its holomorphic setting; modular-function/logarithm lifting must be applied on suitable simply connected subdomains, not assumed globally single-valued on every multiply connected domain. Earlier Arzela/Ascoli, Weierstrass, Stieltjes/Vitali and Picard/Borel/Schottky/Landau inputs remain distinct.",
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      "statement": "# Montel: compactness and omitted-value criteria for holomorphic families\n\nAn unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.\n\n## Scientific scope\n\nConvergence is local uniform convergence inside a complex domain, not uniform convergence up to every boundary or convergence of the whole original sequence. The omitted values must be shared and distinct; allowing infinity as a subsequential limit is essential. For meromorphic functions the convergence definition must be changed and a three-value condition is different; it is not asserted here from the holomorphic statement. The 1907 bounded proof treats simple connectivity first and then finite-connectivity/covering extension. The 1912 theorem is recorded in its holomorphic setting; modular-function/logarithm lifting must be applied on suitable simply connected subdomains, not assumed globally single-valued on every multiply connected domain. Earlier Arzela/Ascoli, Weierstrass, Stieltjes/Vitali and Picard/Borel/Schottky/Landau inputs remain distinct.\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\nPaul Montel\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 les suites infinies de fonctions (1907 original)](https://www.numdam.org/item/10.24033/asens.580.pdf)\n- [Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine (1912 original)](https://www.numdam.org/article/ASENS_1912_3_29__487_0.pdf)\n",
      "summary": "An unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.",
      "title": "Montel: compactness and omitted-value criteria for holomorphic families"
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    "what_changed": "An unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.",
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