The β function: Beta

6.8.16 The β function: Beta

The β function is defined by

β(x,y)=

∫

t^x−1 (1−t)^y−1 =

Γ(x)*Γ(y)

Γ(x+y)

This is defined for x and y positive reals (to ensure the convergence of the integral) and by extension for x and y if they are not negative integers.
Remarkable values:

β(1,1)=1, β(n,1)=

, β(n,2)=

n(n+1)

The Beta command computes the β function.

Beta takes two arguments:
a and b, real numbers.
Beta(a,b) returns the value of the β(a,b).

Examples.

Input:
Beta(5,2)
Output:

1

30
Input:
Beta(x,y)
Output:

Γ ⎛
⎝ x ⎞
⎠ Γ ⎛
⎝ y ⎞
⎠

Γ ⎛
⎝ x+y ⎞
⎠
Input:
Beta(5.1,2.2)
Output:
0.0242053671402