Jacobi elliptic functions
E182748
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobi elliptic functions canonical | 5 |
| Jacobi's elliptic functions treatise | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615212 — 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: Jacobi elliptic functions Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi elliptic functions]
-
A.
Weierstrass 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.
-
B.
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.
-
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.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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E.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
- 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: Jacobi elliptic functions Target entity description: Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
A.
Weierstrass 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.
-
B.
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.
-
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.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
elliptic functions
ⓘ
family of special functions ⓘ |
| appearIn |
solutions of the Korteweg–De Vries equation
ⓘ
solutions of the nonlinear Schrödinger equation ⓘ solutions of the sine-Gordon equation ⓘ |
| are | doubly periodic in the complex plane ⓘ |
| coreSubset | sn, cn, dn ⓘ |
| dependOn |
complex variable
ⓘ
elliptic modulus ⓘ parameter m ⓘ |
| field |
algebraic geometry
ⓘ
complex analysis ⓘ differential equations ⓘ elliptic function theory ⓘ mathematical physics ⓘ |
| generalizes | trigonometric functions ⓘ |
| hasMember |
cd(z,m)
ⓘ
cn(z,m) ⓘ cs(z,m) ⓘ dc(z,m) ⓘ dn(z,m) ⓘ ds(z,m) ⓘ nc(z,m) ⓘ nd(z,m) ⓘ ns(z,m) ⓘ sc(z,m) ⓘ sd(z,m) ⓘ sn(z,m) ⓘ |
| haveLimit |
hyperbolic functions as m → 1
ⓘ
trigonometric functions as m → 0 ⓘ |
| introducedBy | Carl Gustav Jacob Jacobi ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| periodicity | real and imaginary fundamental periods ⓘ |
| relatedTo |
Weierstrass elliptic functions
ⓘ
elliptic integrals ⓘ theta functions ⓘ |
| satisfy |
addition theorems
ⓘ
algebraic relations ⓘ nonlinear differential equations ⓘ |
| timeOfIntroduction | 19th century ⓘ |
| usedIn |
classical mechanics
ⓘ
general relativity ⓘ integrable systems ⓘ nonlinear wave equations ⓘ pendulum equation ⓘ quantum mechanics ⓘ soliton theory ⓘ theory of elliptic curves ⓘ theory of elliptic integrals ⓘ |
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: Jacobi elliptic functions Description of subject: Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Carl Gustav Jacob Jacobi
subject surface form:
Carl Gustav Jacob Jacobi
this entity surface form:
Jacobi's elliptic functions treatise