Central Beta Function

Author

Eric W. Weisstein
February 1, 2009

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

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

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

Initialization

β[z_?NumericQ]:=Beta[z,z]

Plots

<<Utilities`Plot`

Real Axis

PlotDiscontinuous[β[x],{x,-3,-2,-1,0,3},
PlotRange->{Automatic,{-30,30}},
VerticalAsymptotes->Range[-3,0],AxesLabel->TraditionalForm/@{x,HoldForm[β[x]]}]

Plots in the Complex Plane

ComplexPlot3D[β[z],{z,-5,5},PlotPoints->100]

ComplexContourPlot[β[z],{z,-5,5}]//Timing

Integer Values

Table[β[n],{n,10}]

Denominator[%]

Table[(n-1)!^2/(2n-1)!,{n,10}]

Table[(2n)!/(2(n-1)!n!),{n,10}]

FullSimplify[(2n)!/(2(n-1)!n!),n∈Integers]

Products

(2/p) (∏_n^∞)((n (n+2 p))/(((n+p))^2))==(Beta[p, p])

(2/p)Product[((n(n+2p))/(((n+p))^2)),{n,∞}]

FullSimplify[%]

FullSimplify[%-Beta[p,p]]