Weierstrass elliptic functions

E110610

Weierstrass elliptic function class of special functions elliptic functions

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.

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All labels observed (5)

Label	Occurrences
Weierstrass elliptic functions canonical	3
Weierstrass ℘-function	2
Weierstrass sigma function (σ)	1
Weierstrass ζ-function	1
Weierstrass σ-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]

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
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 ζ-function ⓘ linked to: Weierstrass elliptic functions Weierstrass σ-function ⓘ linked to: Weierstrass elliptic functions Weierstrass ℘-function ⓘ linked to: Weierstrass elliptic functions
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