# The β function: Beta

### 5.8.16 The β function: Beta

The β function is defined by

β(x,y)=

∫

1
0

tx−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)=

1
n

, β(n,2)=

1
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