Jacobi elliptic functions

E182748
elliptic functions family of special 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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All labels observed (2)

Label Occurrences
Jacobi elliptic functions canonical 5
Jacobi's elliptic functions treatise 1

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 ⓘ

Referenced by (6)

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

Carl Gustav Jacob Jacobi → notableWork → Jacobi elliptic functions ⓘ
Weierstrass elliptic functions → relatedConcept → Jacobi elliptic functions ⓘ
Jacobi → knownFor → Jacobi elliptic functions ⓘ
subject surface form: Carl Gustav Jacob Jacobi
Jacobi operator → relatedTo → Jacobi elliptic functions ⓘ
Carl → notableWork → Jacobi elliptic functions ⓘ
subject surface form: Carl Gustav Jacob Jacobi
Carl → notableWork → Jacobi elliptic functions ⓘ
subject surface form: Carl Gustav Jacob Jacobi
this entity surface form: Jacobi's elliptic functions treatise