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 (3)
| Label | Occurrences |
|---|---|
| Legendre’s relation for elliptic integrals canonical | 2 |
| Legendre normal form of elliptic integrals | 1 |
| Legendre’s relation | 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 ⓘ |
| 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 ⓘ |
| relatedTo |
Jacobi elliptic functions
ⓘ
Legendre normal form of elliptic integrals ⓘ
linked to:
Legendre’s relation for elliptic integrals
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Legendre
linked to: Legendre’s relation for elliptic integrals
Legendre’s relation for elliptic integrals
→
relatedTo
→
Legendre normal form of elliptic integrals
ⓘ
linked to: Legendre’s relation for elliptic integrals