Beta function

E621094

Euler integral of the first kind special function

The Beta function is a special function in mathematics, closely related to the Gamma function, that arises in calculus, probability theory, and complex analysis, particularly in evaluating integrals and expressing various identities.

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All labels observed (3)

Label	Occurrences
Beta function canonical	1
Euler beta function	1
Euler beta integral	1

How this entity was disambiguated

This entity first appeared as the object of triple T6833508 — 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: Beta function
Context triple: [arithmetic–geometric mean identities, usesConcept, Beta function]

A. Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
B. Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
C. Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
D. Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
E. Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
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: Beta function
Target entity description: The Beta function is a special function in mathematics, closely related to the Gamma function, that arises in calculus, probability theory, and complex analysis, particularly in evaluating integrals and expressing various identities.

A. Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
B. Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
C. Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
D. Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
E. Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	Euler integral of the first kind ⓘ special function ⓘ
appearsIn	Euler reflection formula for Gamma function ⓘ linked to: Euler’s reflection formula evaluation of definite integrals ⓘ
category	special functions of mathematical physics ⓘ
definedFor	complex numbers x and y with positive real parts ⓘ
domain	{(x,y)∈ℂ² : Re(x)>0, Re(y)>0} (for integral definition) ⓘ
extendedBy	analytic continuation ⓘ
generalizedBy	incomplete beta function ⓘ
hasAlternativeName	Euler beta function ⓘ linked to: Beta function
hasAsymptoticRelation	via asymptotics of Gamma function for large parameters ⓘ
hasConnection	B(x,y)=Γ(x)Γ(y)/Γ(x+y) implies Γ(x)Γ(1-x)=π/sin(πx) ⓘ
hasDefinitionIntegral	B(x,y)=∫₀¹ t^{x-1}(1-t)^{y-1} dt ⓘ
hasIdentity	B(x,y)=2∫₀^{π/2} (sin θ)^{2x-1}(cos θ)^{2y-1} dθ ⓘ B(x,y)=Γ(x)Γ(y)/Γ(x+y) ⓘ B(x,y)=∫₀^∞ t^{x-1}/(1+t)^{x+y} dt (alternative form) ⓘ
hasProperty	B(x,y)=B(y,x) ⓘ holomorphic in x and y where Γ(x),Γ(y),Γ(x+y) are finite ⓘ meromorphic function of two complex variables ⓘ
hasRecurrence	B(x+1,y)=x/(x+y)·B(x,y) ⓘ B(x,y+1)=y/(x+y)·B(x,y) ⓘ
hasSeriesExpansion	B(x,y)=∑_{n=0}^∞ (-1)^n C(y-1,n)/(n+x) (under suitable conditions) ⓘ
hasSymbol	B(x,y) ⓘ
introducedBy	Leonhard Euler ⓘ
namedAfter	Greek letter Beta ⓘ
normalizes	Beta distribution density ⓘ
relatedTo	Beta distribution ⓘ Dirichlet integrals ⓘ linked to: Dirichlet beta function Gamma function ⓘ binomial coefficients ⓘ hypergeometric functions ⓘ incomplete beta function ⓘ
specialValue	B(1,1)=1 ⓘ B(1,y)=1/y ⓘ B(1/2,1/2)=π ⓘ B(x,1)=1/x ⓘ
symmetricIn	x and y ⓘ
usedIn	Bayesian statistics ⓘ calculus ⓘ combinatorics ⓘ complex analysis ⓘ mathematical physics ⓘ order statistics ⓘ probability theory ⓘ random matrix theory ⓘ statistics ⓘ
usedToExpress	integrals involving powers of sine and cosine ⓘ integrals of rational functions of polynomials ⓘ moments of Beta distribution ⓘ

How these facts were elicited

Referenced by (3)

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

arithmetic–geometric mean identities → usesConcept → Beta function ⓘ

Selberg integral → generalizationOf → Euler beta integral ⓘ

linked to: Beta function

Beta function → hasAlternativeName → Euler beta function ⓘ