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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Observed surface forms (1)

Surface form	Occurrences
Euler beta integral	1

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 ⓘ

Referenced by (2)

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

arithmetic–geometric mean identities → usesConcept → Beta function ⓘ

Selberg integral → generalizationOf → Beta function ⓘ

this entity surface form: Euler beta integral