SWhat changed
Picard proves that a nonconstant entire holomorphic function can omit at most one finite complex value. The October 1879 note strengthens this: a transcendental entire function assumes every finite value infinitely often, with at most one exception. The 1880 memoir supplies the detailed modular-function/monodromy proof and local extension: near an isolated essential singularity, a holomorphic function assumes every finite value infinitely often except possibly one; the corresponding meromorphic version permits at most two exceptional values on the sphere. These linked results are staged as one contribution family.
Scientific scope
Picard proves that a nonconstant entire holomorphic function can omit at most one finite complex value. The October 1879 note strengthens this: a transcendental entire function assumes every finite value infinitely often, with at most one exception. The 1880 memoir supplies the detailed modular-function/monodromy proof and local extension: near an isolated essential singularity, a holomorphic function assumes every finite value infinitely often except possibly one; the corresponding meromorphic version permits at most two exceptional values on the sphere. These linked results are staged as one contribution family. Constant entire functions and polynomials must be excluded from the appropriate statements. The local assertion concerns an isolated essential singularity and every punctured neighbourhood, not arbitrary boundary singularities or multivalued analytic functions. In the1880 text the no-accumulating-poles case has at most one finite exception; when infinitely many poles approach the singularity, at most two finite exceptions are allowed, while infinity is attained at those poles. Modular/elliptic-function inputs from Hermite,Fuchs,Dedekind and prior Weierstrass theory remain inherited. The 1879 announcement is distinguished from the fuller 1880 treatment used for this record.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Constant entire functions and polynomials must be excluded from the appropriate statements. The local assertion concerns an isolated essential singularity and every punctured neighbourhood, not arbitrary boundary singularities or multivalued analytic functions. In the1880 text the no-accumulating-poles case has at most one finite exception; when infinitely many poles approach the singularity, at most two finite exceptions are allowed, while infinity is attained at those poles. Modular/elliptic-function inputs from Hermite,Fuchs,Dedekind and prior Weierstrass theory remain inherited. The 1879 announcement is distinguished from the fuller 1880 treatment used for this record.
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 une propriété des fonctions entières
- Sur les fonctions entières
- Mémoire sur les fonctions entières
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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