AWhat 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.
Scientific 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
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.
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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