What is recorded here?
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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Scientific scope
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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People and systems
Gösta Mittag-Leffler
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English catalogue narrative
Mittag-Leffler: constructing analytic functions with prescribed isolated singular parts
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
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.
Reading status
This 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.
People and systems
Gösta Mittag-Leffler
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.