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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Jacobi theta function	3

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 ⓘ

Referenced by (6)

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

Jacobi triple product → givesProductRepresentationOf → Jacobi theta functions ⓘ

this entity surface form: Jacobi theta function

Jacobi triple product → givesSeriesRepresentationOf → Jacobi theta functions ⓘ

this entity surface form: Jacobi theta function

Jacobi → knownFor → Jacobi theta functions ⓘ

subject surface form: Carl Gustav Jacob Jacobi

Carl → notableWork → Jacobi theta functions ⓘ

subject surface form: Carl Gustav Jacob Jacobi

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

Ramanujan theta function → relatedTo → Jacobi theta functions ⓘ

this entity surface form: Jacobi theta function