Selberg integral

E246700

mathematical integral multidimensional integral result in analysis special function identity

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.

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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	Euler beta integral ⓘ linked to: Beta function
hasVariant	discrete Selberg integral ⓘ linked to: 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 ⓘ Gamma function reflection formula ⓘ linked to: Euler’s reflection formula
namedAfter	Atle Selberg ⓘ
parameter	alpha ⓘ beta ⓘ gamma ⓘ
proposedBy	Atle Selberg ⓘ
relatedTo	Barnes G-function ⓘ Dyson integral ⓘ Jack polynomials ⓘ Jacobi ensemble ⓘ Macdonald polynomials ⓘ Mehta integral ⓘ linked to: Selberg integral Selberg trace formula ⓘ Vandermonde determinant ⓘ linked to: Vandermonde matrix beta function ⓘ beta-ensembles ⓘ gamma function ⓘ linked to: 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

Referenced by (4)

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

Atle Selberg → knownFor → Selberg integral ⓘ

Atle Selberg → notableWork → Selberg integral ⓘ

Selberg integral → relatedTo → Mehta integral ⓘ

linked to: Selberg integral

Selberg integral → hasVariant → discrete Selberg integral ⓘ