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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Legendre’s relation for elliptic integrals canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7861130 — 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: Legendre’s relation for elliptic integrals Context triple: [Adrien-Marie Legendre, knownFor, Legendre’s relation for elliptic integrals]
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A.
Gauss transformation for elliptic integrals
The Gauss transformation for elliptic integrals is a classical iterative procedure introduced by Carl Friedrich Gauss that relates and simplifies elliptic integrals via transformations closely connected to the arithmetic–geometric mean.
-
B.
Euler’s reflection formula
Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
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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E.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern 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: Legendre’s relation for elliptic integrals Target entity description: 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.
-
A.
Gauss transformation for elliptic integrals
The Gauss transformation for elliptic integrals is a classical iterative procedure introduced by Carl Friedrich Gauss that relates and simplifies elliptic integrals via transformations closely connected to the arithmetic–geometric mean.
-
B.
Euler’s reflection formula
Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
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.
-
E.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern complex analysis.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Legendre’s relation for elliptic integrals Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.