Selberg integral
E246700
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Selberg integral canonical | 2 |
| Mehta integral | 1 |
| discrete Selberg integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2252108 — 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: Selberg integral Context triple: [Atle Selberg, knownFor, Selberg integral]
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Hardy Z-function
The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
- 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: Selberg integral Target entity description: The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Hardy Z-function
The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical integral
ⓘ
multidimensional integral ⓘ result in analysis ⓘ special function identity ⓘ |
| closedForm | product of gamma functions ⓘ |
| dimension | n-dimensional ⓘ |
| domain | unit interval ⓘ |
| field |
combinatorics
ⓘ
mathematical analysis ⓘ orthogonal polynomials ⓘ probability theory ⓘ random matrix theory ⓘ representation theory ⓘ special functions ⓘ |
| generalizationOf |
Beta function
ⓘ
surface form:
Euler beta integral
|
| hasVariant |
Selberg integral
self-linksurface differs
ⓘ
surface form:
discrete Selberg integral
elliptic Selberg integral ⓘ q-Selberg integral ⓘ |
| influenced |
development of random matrix theory
ⓘ
theory of multivariate orthogonal polynomials ⓘ |
| integrandFeature |
absolute value of Vandermonde determinant to a power
ⓘ
pairwise interaction terms ⓘ power singularities at 0 and 1 ⓘ |
| involves |
Beta function identities
ⓘ
Euler’s reflection formula ⓘ
surface form:
Gamma function reflection formula
|
| namedAfter | Atle Selberg ⓘ |
| parameter |
alpha
ⓘ
beta ⓘ gamma ⓘ |
| proposedBy | Atle Selberg ⓘ |
| relatedTo |
Barnes G-function
ⓘ
Dyson integral ⓘ Jack polynomials ⓘ Jacobi ensemble ⓘ Macdonald polynomials ⓘ Selberg integral self-linksurface differs ⓘ
surface form:
Mehta integral
Selberg trace formula ⓘ Vandermonde matrix ⓘ
surface form:
Vandermonde determinant
beta function ⓘ beta-ensembles ⓘ Gamma function ⓘ
surface form:
gamma function
|
| usedIn |
Selberg-type integrals in conformal field theory
ⓘ
combinatorial enumeration problems ⓘ computation of eigenvalue correlation functions ⓘ constant term identities ⓘ evaluation of random matrix partition functions ⓘ normalization constants of beta-ensembles ⓘ orthogonality relations for Jack polynomials ⓘ |
| yearProved | 1944 ⓘ |
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: Selberg integral Description of subject: The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Mehta integral
this entity surface form:
discrete Selberg integral