Beta Function

Author

Eric W. Weisstein
March 16, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Plots

Show[GraphicsArray[{
Plot3D[If[NumberQ[b=Beta[x,y]],b,0],{x,-3,3},{y,-3,3},PlotPoints->200,Mesh->False,
PlotRange->{-10,10},
AxesLabel->TraditionalForm/@{x,y,""},
PlotLabel->TraditionalForm[Beta[x,y]],
DisplayFunction->Identity
],
ContourPlot[If[NumberQ[b=Beta[x,y]],b,0],{x,-3,3},{y,-3,3},
PlotPoints->250,
ColorFunction->Hue,
FrameLabel->TraditionalForm/@{x,y},
PlotLabel->TraditionalForm[Beta[x,y]],
ContourLines->False,
DisplayFunction->Identity
]
}]]

Integral

Integrate[Exp[-u]u^m,{u,0,∞},Assumptions->Re[m]>-1]Integrate[Exp[-v]v^n,{v,0,∞},Assumptions->Re[n]>-1]

4Integrate[Exp[-x^2]x^(2m+1),{x,0,∞},Assumptions->Re[m]>-1]Integrate[Exp[-y^2]y^(2n+1),{y,0,∞},Assumptions->Re[n]>-1]

4Integrate[Exp[-(x^2+y^2)]x^(2m+1)y^(2n+1),{x,0,∞},{y,0,∞}]

4Integrate[Exp[-(x^2+y^2)]x^(2m+1)y^(2n+1),{x,0,∞},{y,0,∞},GenerateConditions->False]

Integrate[Exp[-(x^2+y^2)]Abs[x]^(2m+1)Abs[y]^(2n+1),{x,-∞,∞},{y,-∞,∞},GenerateConditions->False]

v5.0.1

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2},GenerateConditions->False]

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2}]

v5.1.1

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2},GenerateConditions->False]

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2}]

v6.0

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2},GenerateConditions->False]

Integrate[Exp[-r^2](((r Cos[θ]))^(2m+1))(((r Sin[θ]))^(2n+1))r,{r,0,∞},{θ,0,Pi/2}]

Numerical Checks

Integrate[Abs[Cos[θ]]^(2*m+1)*Abs[Sin[θ]]^(2*n+1),{θ,0,2*Pi}]/.{m->1.1,n->2.1}

With[{m=1.1,n=2.1},Integrate[Abs[Cos[θ]]^(2*m+1)*Abs[Sin[θ]]^(2*n+1),{θ,0,2*Pi}]]

With[{m=1.1,n=2.1},2Beta[m+1,n+1]]

FullSimplify[Integrate[Abs[Cos[θ]]^(2*m+1)*Abs[Sin[θ]]^(2*n+1),{θ,0,2*Pi},GenerateConditions->False]]

Integrate[((Cos[θ])^(2m+1))((Sin[θ])^(2n+1)),{θ,0,Pi/2},Assumptions->Re[m]>-1&&Re[n]>-1]

FunctionExpand[Beta[m+1,n+1]]

4Integrate[Exp[-r^2](r^(2m+2n+3)),{r,0,∞},Assumptions->Re[m]>-1&&Re[n]>-1]

Identities

(Beta[p, p]) (Beta[p+(1/2), p+(1/2)])==(π/((2^(4 p-1)) p))

FullSimplify[Beta[p,p]Beta[p+(1/2),p+(1/2)]-(π/((2^(4p-1))p))]

Β(p,q) Β(p+q,r)==Β(q,r) Β(q+r,p)

FullSimplify[Beta[p,q]Beta[p+q,r]-Beta[q,r]Beta[q+r,p]]

Gosper Identity

b[n_Integer?OddQ]:={Product[Beta[i/(2n+1)+a,i/(2n+1)+b],{i,0,2n}],(2n+1)^((2n+1)/2) Pi^n Beta[n,((b+a)(2n+1)+1)/2]Beta[a(2n+1),b(2n+1)]/(n-1)!}
b[n_Integer?EvenQ]:={Product[Beta[i/(2n)+a,i/(2n)+b],{i,0,2n-1}],
(n^n Pi^n Beta[n,2(b+a)n]Beta[2a n,2b n])/(2^(2(b+a)n-n-1)(n-1)!Beta[(b+a)n,(b+a+1)n])}

b[1]

b[2]

Table[Subtract@@b[n],{n,1,5}] /. {a->Random[Complex],b->Random[Complex]}

Limit[b[n],n->Infinity]

Limit[FullSimplify[(1/((-1+n)!))((((1+2 n))^((1/2) (1+2 n))) (π^n) Beta[n,(1/2) (1+(a+b) (1+2 n))] Beta[a (1+2 n),b (1+2 n)])],n->∞]

Product

FullSimplify[((x+y)/(x y)) Product[((1+((x+y)/k))/((1+(x/k))(1+(y/k)))),{k,∞}]]

FunctionExpand[Beta[x,y]]

Derivatives

FullSimplify[D[Beta[a,b],a]]

FullSimplify[D[Beta[a,b],b]]

FullSimplify[D[Beta[a,b],{a,2}]]

FullSimplify[D[Beta[a,b],{b,2}]]

FullSimplify[D[Beta[a,b],a,b]]