Jacobi theta functions

E621095

complex-analytic functions modular objects special functions

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.

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All labels observed (6)

Label	Occurrences
Jacobi theta functions canonical	4
Jacobi theta function	3
Jacobi theta function θ₁	1
Jacobi theta function θ₂	1
Jacobi theta function θ₃	1
Jacobi theta function θ₄	1

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.
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
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 ⓘ linked to: Jacobi theta functions Jacobi theta function θ₂ ⓘ linked to: Jacobi theta functions Jacobi theta function θ₃ ⓘ linked to: Jacobi theta functions Jacobi theta function θ₄ ⓘ linked to: Jacobi theta functions
namedAfter	Carl Gustav Jacob Jacobi ⓘ
relatedTo	Dedekind eta function ⓘ Jacobi elliptic functions ⓘ Riemann theta function ⓘ Weierstrass elliptic functions ⓘ modular group SL(2,ℤ) ⓘ linked to: SL(2,ℤ)
satisfies	Jacobi triple product identity ⓘ linked to: Jacobi triple product 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

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

arithmetic–geometric mean identities → relatedConcept → Jacobi theta functions ⓘ

Carl Gustav Jacob Jacobi → knownFor → Jacobi theta functions ⓘ

subject linked to: Jacobi

Jacobi triple product → givesProductRepresentationOf → Jacobi theta function ⓘ