SWhat 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.
Scientific scope
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. 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
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.
- Sur les suites infinies de fonctions (1907 original)
- Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine (1912 original)
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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