Weierstrass elliptic functions
E110610
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Weierstrass elliptic functions canonical | 2 |
| Weierstrass ζ-function | 1 |
| Weierstrass σ-function | 1 |
| Weierstrass ℘-function | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T940264 — 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: Weierstrass elliptic functions Context triple: [Karl Weierstrass, notableFor, Weierstrass elliptic functions]
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A.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern complex analysis.
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B.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes
Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes is a seminal mathematical paper by Niels Henrik Abel that develops fundamental results on transcendental functions and helped lay groundwork for modern analysis.
-
E.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
- 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: Weierstrass elliptic functions Target entity description: Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
A.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern complex analysis.
-
B.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes
Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes is a seminal mathematical paper by Niels Henrik Abel that develops fundamental results on transcendental functions and helped lay groundwork for modern analysis.
-
E.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Weierstrass elliptic function
ⓘ
Weierstrass elliptic function ⓘ Weierstrass elliptic function ⓘ class of special functions ⓘ elliptic functions ⓘ |
| are |
doubly periodic
ⓘ
meromorphic ⓘ periodic in two independent complex directions ⓘ |
| centralRoleIn |
the theory of Abelian integrals
ⓘ
the theory of complex tori ⓘ the theory of elliptic curves ⓘ the theory of modular forms ⓘ the uniformization of elliptic curves ⓘ |
| contrastWith | singly periodic trigonometric functions ⓘ |
| derivativeRelation | −ζ′(z) = ℘(z) ⓘ |
| domain | complex plane modulo a lattice ⓘ |
| field |
algebraic geometry
ⓘ
complex analysis ⓘ elliptic curves ⓘ elliptic function theory ⓘ number theory ⓘ |
| hasDoublePoleAt | origin modulo the lattice ⓘ |
| hasOrderOfPoleAtLatticePoints | 2 ⓘ |
| havePeriodLattice | two-dimensional lattice in the complex plane ⓘ |
| includes |
Weierstrass elliptic functions
self-linksurface differs
ⓘ
surface form:
Weierstrass ζ-function
Weierstrass elliptic functions self-linksurface differs ⓘ
surface form:
Weierstrass σ-function
Weierstrass elliptic functions self-linksurface differs ⓘ
surface form:
Weierstrass ℘-function
|
| introducedBy | Karl Weierstrass ⓘ |
| is |
doubly periodic meromorphic function
ⓘ
elliptic function with respect to a lattice Λ ⓘ entire function ⓘ even function ⓘ quasi-periodic meromorphic function ⓘ |
| logarithmicDerivative | σ′(z)/σ(z) = ζ(z) ⓘ |
| maps | complex torus C/Λ to elliptic curve ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| parameterDependsOn | lattice invariants g2 and g3 ⓘ |
| relatedConcept |
Jacobi elliptic functions
ⓘ
j-invariant ⓘ modular invariants g2 and g3 ⓘ |
| satisfiesDifferentialEquation | (℘′(z))^2 = 4℘(z)^3 - g2℘(z) - g3 ⓘ |
| timePeriodOfDevelopment | 19th century ⓘ |
| usedFor |
construction of elliptic curves over C
ⓘ
explicit formulas for periods and quasi-periods ⓘ parametrization of elliptic integrals ⓘ solutions of certain nonlinear differential equations ⓘ uniformization of elliptic curves by complex tori ⓘ |
| usedToParametrize | elliptic curves in Weierstrass form ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weierstrass elliptic functions Description of subject: Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
Weierstrass elliptic functions
→
includes
→
Weierstrass elliptic functions
self-linksurface differs
ⓘ
this entity surface form:
Weierstrass ℘-function
Weierstrass elliptic functions
→
includes
→
Weierstrass elliptic functions
self-linksurface differs
ⓘ
this entity surface form:
Weierstrass ζ-function
Weierstrass elliptic functions
→
includes
→
Weierstrass elliptic functions
self-linksurface differs
ⓘ
this entity surface form:
Weierstrass σ-function