Mittag-Leffler theorem
E480874
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Mittag-Leffler theorem canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T4927284 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Mittag-Leffler theorem Context triple: [Weierstrass factorization theorem, isRelatedTo, Mittag-Leffler theorem]
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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D.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
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E.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Mittag-Leffler theorem Target entity description: The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
D.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
-
E.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in complex function theory
ⓘ
theorem in complex analysis ⓘ |
| allowsConstructionOf | meromorphic functions with prescribed principal parts ⓘ |
| appearsIn | advanced textbooks on complex analysis ⓘ |
| appliesTo |
Riemann surfaces
ⓘ
open subsets of the complex plane ⓘ |
| assumes |
a discrete set of poles
ⓘ
compatibility conditions on principal parts for global existence ⓘ |
| canBeFormulatedUsing |
cohomology of sheaves
ⓘ
divisors and line bundles ⓘ |
| characterizes | meromorphic functions by their principal parts ⓘ |
| concerns |
meromorphic functions
ⓘ
poles of meromorphic functions ⓘ principal parts of Laurent series ⓘ |
| contrastsWith |
Weierstrass factorization theorem which prescribes zeros instead of poles
ⓘ
linked to:
Weierstrass factorization theorem
|
| field | complex analysis ⓘ |
| formalizedIn | sheaf cohomology as vanishing of H^1 for certain sheaves on the Riemann sphere ⓘ |
| generalizes | partial fraction decompositions in the complex plane ⓘ |
| hasConsequence | any admissible principal part data can be realized by a meromorphic function ⓘ |
| hasVersion |
classical version on the complex plane
ⓘ
sheaf-theoretic formulation ⓘ version on Riemann surfaces ⓘ |
| historicalPeriod | late 19th century mathematics ⓘ |
| implies | existence of meromorphic functions with given poles and principal parts ⓘ |
| involves | series that converge normally on compact subsets away from poles ⓘ |
| isAnalogOf | Weierstrass factorization theorem for zeros vs poles ⓘ |
| namedAfter | Gösta Mittag-Leffler ⓘ |
| partOf | classical function theory ⓘ |
| relatedTo |
Riemann–Roch theorem
ⓘ
Runge's theorem ⓘ
linked to:
Runge approximation theorem
Weierstrass factorization theorem ⓘ theory of divisors on Riemann surfaces ⓘ |
| requires |
Laurent series theory
ⓘ
basic topology of the complex plane ⓘ knowledge of meromorphic functions ⓘ |
| typeOf |
existence theorem
ⓘ
representation theorem ⓘ |
| usedIn |
Nevanlinna theory
ⓘ
complex analytic geometry ⓘ construction of meromorphic functions with prescribed singularities ⓘ global analysis on complex manifolds ⓘ theory of Riemann surfaces ⓘ value distribution theory ⓘ |
| usedToShow | existence of meromorphic functions with given divisor of poles ⓘ |
| uses |
Laurent series expansions
ⓘ
series of meromorphic functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Mittag-Leffler theorem Description of subject: The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.