Jacobi theta functions
E621095
Jacobi theta functions are special functions in complex analysis and number theory that encode modular and elliptic properties, playing a central role in the theory of elliptic functions, modular forms, and various applications in mathematical physics.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobi theta function | 3 |
| Jacobi theta functions canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833542 — 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 theta functions Context triple: [arithmetic–geometric mean identities, relatedConcept, Jacobi theta functions]
-
A.
Ramanujan theta function
The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.
-
B.
Jacobi elliptic functions
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.
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C.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
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E.
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.
- 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 theta functions Target entity description: Jacobi theta functions are special functions in complex analysis and number theory that encode modular and elliptic properties, playing a central role in the theory of elliptic functions, modular forms, and various applications in mathematical physics.
-
A.
Ramanujan theta function
The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.
-
B.
Jacobi elliptic functions
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.
-
C.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
D.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex-analytic functions
ⓘ
modular objects ⓘ special functions ⓘ |
| dependsOnVariable |
complex parameter τ
ⓘ
complex variable z ⓘ |
| domainOfτ | upper half-plane ⓘ |
| encodes |
elliptic properties
ⓘ
modular properties ⓘ |
| field |
complex analysis
ⓘ
mathematical physics ⓘ number theory ⓘ |
| hasParameter | nome q = e^{iπτ} ⓘ |
| hasProperty |
entire in z for fixed τ in upper half-plane
ⓘ
holomorphic in τ in upper half-plane ⓘ |
| hasSeriesExpansion | Fourier series in q ⓘ |
| hasVariant |
Jacobi theta function θ₁
NERFINISHED
ⓘ
Jacobi theta function θ₂ NERFINISHED ⓘ Jacobi theta function θ₃ NERFINISHED ⓘ Jacobi theta function θ₄ NERFINISHED ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi NERFINISHED ⓘ |
| relatedTo |
Dedekind eta function
NERFINISHED
ⓘ
Jacobi elliptic functions NERFINISHED ⓘ Riemann theta function NERFINISHED ⓘ Weierstrass elliptic functions NERFINISHED ⓘ modular group SL(2,ℤ) NERFINISHED ⓘ |
| satisfies |
Jacobi triple product identity
NERFINISHED
ⓘ
heat equation ⓘ quasi-periodicity relations ⓘ |
| transformsUnder | modular transformations ⓘ |
| usedIn |
Poisson summation formula applications
ⓘ
conformal field theory ⓘ construction of modular forms of half-integral weight ⓘ partition theory ⓘ q-series ⓘ representation theory of affine Lie algebras ⓘ statistical mechanics ⓘ string theory ⓘ theory of elliptic curves ⓘ theory of elliptic functions ⓘ theory of modular forms ⓘ theory of theta characteristics ⓘ theta functions of lattices ⓘ theta inversion formulas ⓘ |
| usedToConstruct | elliptic functions with given periods ⓘ |
| usedToExpress |
Green functions on tori
ⓘ
characters of conformal field theories ⓘ partition functions of lattice models ⓘ solutions of certain differential equations ⓘ |
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 theta functions Description of subject: Jacobi theta functions are special functions in complex analysis and number theory that encode modular and elliptic properties, playing a central role in the theory of elliptic functions, modular forms, and various applications in mathematical physics.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jacobi theta function
this entity surface form:
Jacobi theta function
subject surface form:
Carl Gustav Jacob Jacobi
this entity surface form:
Jacobi theta function