
11-46 Calculus – Differentiation and Integration
Properties of Beta Function
For x > 0, y > 0, the beta function holds the following properties.
1. The beta function has the symmetry property, i.e. B(x, y) = B(y, x).
B x y t t dt
d
x y
x y
( , ) ( )
(sin ) (cos )
/
= −
∫
=
∫
− −
− −
1 1
0
1
2 1 2 1
0
2
1
2 θ θ θ
π
(by suubstituting t
d B y x
y x
=
=
∫
=
− −
sin )
(sin ) (cos ) ( , ).
/
2
2 1 2 1
0
2
2
θ
θ θ θ
π
2. B x y
x
x y
B x y( , ) ( , ).+ =
+
1
Using Eq. (11.18), we have
B x y
x y
x x y
x y x y
x
x y
x
( , )
( )
( ) ( )
( ) ( )
( )
+ =
+
+ +
=
+ +
=
+
1
1
Γ Γ
Γ
Γ Γ
Γ
Γ Γ
(( )
( )
( , ).
y
x y
x
x y
B x y
Γ +
=
+
3.
B x y B x y B x y( , ) ( , ) ( , ).
+ + + =
Consider
B x y B x y B x y B y x B
x
x y