Incomplete Beta Function

Author

Eric W. Weisstein
April 30, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/IncompleteBetaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IncompleteBetaFunction.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Definition

Beta[z,a,b]//FunctionExpand

Beta[1,a,b]//FunctionExpand

Derivative and Integral

Dervative

D[Beta[z,a,b],z]

Integrate[%,z]

FullSimplify[%]

FullSimplify[%-Beta[z,a,b]]

Integral

Integrate[Beta[z,a,b],z]

D[%,z]

Integrals

(∫_0^z)(u^(a-1)) (((1-u))^(b-1))du==(Subscript[Β, z])(a,b)

Integrate[(u^(a-1))(((1-u))^(b-1)),{u,0,z}]

FullSimplify[%]

FullSimplify[(((z^a) Hypergeometric2F1[a,1-b,1+a,z])/a)]

FunctionExpand[(((z^a) Hypergeometric2F1[a,1-b,1+a,z])/a)]

FullSimplify[(((z^a) Hypergeometric2F1[a,1-b,1+a,z])/a)-Beta[z,a,b]]

∫(x^p) (((1-x))^q)dx==(Subscript[Β, x])(p+1,q+1)

Integrate[(x^p)(((1-x))^q),x]

FullSimplify[%]

FullSimplify[(((x^(1+p)) Hypergeometric2F1[1+p,-q,2+p,x])/(1+p))-Beta[x,p+1,q+1]]

FullSimplify[(x^(1+p)) Gamma[1+p] Hypergeometric2F1Regularized[1+p,-q,2+p,x]-Beta[x,p+1,q+1]]

{Beta[x,p+1,q+1],(x^(1+p)) Gamma[1+p] Hypergeometric2F1Regularized[1+p,-q,2+p,x]}/.{p->1.1,q->1.2,x->2.3}

Identities

(Subscript[Β, 1])(a,b)==Β(a,b)

FullSimplify[Beta[1,a,b]]

FullSimplify[%]

FullSimplify[Beta[1,a,b]-Beta[a,b]]

((z^a)/a) (Subscript[, 2])(Subscript[F, 1])(a,1-b;a+1;z)==(Subscript[Β, z])(a,b)

FullSimplify[(((z^a) Hypergeometric2F1[a,1-b,1+a,z])/a)]

((z^a)/a) (Subscript[, 1])(Subscript[F, 1])(1-b;a+1;z)==(Subscript[Β, z])(a,b)

{(((z^a) Hypergeometric1F1[1-b,1+a,z])/a),Beta[z,a,b]}/.{z->1.3,a->5,b->3}

(z^a)Sum[((Pochhammer[1-b,n])/(n!a Pochhammer[1+a,n]))(z^n),{n,0,∞}]

FullSimplify[(((z^a) Hypergeometric1F1[1-b,1+a,z])/a)-Beta[z,a,b]]

FullSimplify[(((z^a) Hypergeometric1F1[1-b,1+a,z])/a)-(((z^a) Hypergeometric1F1[1-b,1+a,z])/a)]

(z^a) Γ(a) (Subscript[, 2])(Subscript[(OverTilde[F]), 1])(a,1-b;a+1;z)==(Subscript[Β, z])(a,b)

((x (Subscript[Β, (x^4)])((1/4),(1/2)))/(4 ((x^4)^(1/4))))==F((sin^(-1))(x)|-1)

FullSimplify[(x Beta[x^4,1/4,1/2])/(4 (x^4)^(1/4))-EllipticF[ArcSin[x],-1]]

FunctionExpand[%]

Sums

(z^a) (∑_(n=0)^∞)(((z^n) (Pochhammer[1-b, n]))/(n! (a+n)))==(Beta3[z, a, b])

Product[k-b,{k,n}]

Product[a+k,{k,0,n}]

FullSimplify[%]

(z^a)Sum[((Pochhammer[1-b,n])/(n! (a+n)))(z^n),{n,0,∞}]

FullSimplify[%]

(z^a) (∑_(n=0)^∞)(( a! (n-b)!)/(a (-b)! n! (a+n)!))(z^n)==(Subscript[Β, z])(a,b)

(z^a)Sum[((a! (-b+n)!)/(a (-b)! n! (a+n)!))(z^n),{n,0,∞}]

FullSimplify[%]

Series

SeriesCoefficient[Beta[z,a,b]/z^a,{z,0,n}]

SeriesCoefficient[((Hypergeometric1F1[1-b,1+a,z])/a),{z,0,n}]//InputForm

Sum[((a! (-b+n)!)/(a (-b)! n! (a+n)!))(z^n),{n,0,∞}]

GeneratingFunction[((a! (-b+n)!)/(a (-b)! n! (a+n)!)),n,z]

FullSimplify[%]

GeneratingFunction[((Pochhammer[1-b,n])/(n! Pochhammer[a,n+1])),n,z]