Beta function
E621094
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Beta function canonical | 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.
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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.
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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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| hasAlternativeName | Euler beta function NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| namedAfter | Greek letter Beta ⓘ |
| normalizes | Beta distribution density ⓘ |
| relatedTo |
Beta distribution
NERFINISHED
ⓘ
Dirichlet integrals NERFINISHED ⓘ Gamma function NERFINISHED ⓘ 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 NERFINISHED ⓘ statistics ⓘ |
| usedToExpress |
integrals involving powers of sine and cosine
ⓘ
integrals of rational functions of polynomials ⓘ moments of Beta distribution ⓘ |
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: Beta function Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euler beta integral