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.
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| partOf | classical function theory ⓘ |
| relatedTo |
Riemann–Roch theorem
NERFINISHED
ⓘ
Runge's theorem NERFINISHED ⓘ Weierstrass factorization theorem NERFINISHED ⓘ 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
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.