Legendre’s relation for elliptic integrals
E695822
Legendre’s relation for elliptic integrals is a fundamental identity connecting complete elliptic integrals of the first and second kinds, playing a key role in the theory and applications of elliptic functions.
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
formula in analysis
ⓘ
mathematical identity ⓘ result in the theory of elliptic integrals ⓘ |
| appearsIn | Legendre’s Traité des fonctions elliptiques NERFINISHED ⓘ |
| category |
elliptic integral identity
ⓘ
special function identity ⓘ |
| connects |
complete elliptic integrals of the first kind
ⓘ
complete elliptic integrals of the second kind ⓘ |
| domainCondition | 0 < k < 1 ⓘ |
| expresses | algebraic relation between complete elliptic integrals ⓘ |
| field |
elliptic functions
ⓘ
elliptic integrals ⓘ mathematical analysis ⓘ special functions ⓘ |
| hasCanonicalForm | K(k)E(k') + E(k)K(k') - K(k)K(k') = \frac{\pi}{2} ⓘ |
| hasConstantTerm | \pi/2 ⓘ |
| hasGeneralization |
Legendre-type relations for hypergeometric functions
ⓘ
Legendre-type relations for incomplete elliptic integrals ⓘ |
| hasProperty |
involves product combinations of K and E
ⓘ
symmetric in k and k' up to sign conventions ⓘ |
| involves |
complementary modulus
ⓘ
complete elliptic integrals with complementary modulus ⓘ |
| isFundamentalIn |
classical theory of elliptic functions
ⓘ
theory of elliptic integrals ⓘ |
| isUsedIn |
asymptotic analysis of elliptic integrals
ⓘ
numerical analysis of elliptic integrals ⓘ theory of theta functions ⓘ |
| namedAfter | Adrien-Marie Legendre NERFINISHED ⓘ |
| relatedTo |
Jacobi elliptic functions
NERFINISHED
ⓘ
Legendre normal form of elliptic integrals NERFINISHED ⓘ modular parameter of elliptic integrals ⓘ |
| usedFor |
computing periods of elliptic functions
ⓘ
deriving identities for elliptic functions ⓘ studying modular transformations ⓘ theory of elliptic curves ⓘ transformations of elliptic integrals ⓘ |
| usesNotation |
E(k) for complete elliptic integral of the second kind
ⓘ
K(k) for complete elliptic integral of the first kind ⓘ k' = \sqrt{1-k^2} for complementary modulus ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.