Digital Library of Mathematical Functions
The Gamma Function -- Beta Function

### Beta Function

In this section all fractional powers have their principal values, except where noted otherwise. In the next 4 equations it is assumed ℜ a > 0 and ℜ b > 0 .

##### Euler's Beta Integral

B ( a , b ) = ∫ 0 1 t a - 1 ( 1 - t ) b - 1 ⅆ t = Γ ( a ) Γ ( b ) Γ ( a + b )
∫ 0 π 2 sin 2 a - 1 θ cos 2 b - 1 θ ⅆ θ = 1 2 B ( a , b )
∫ 0 ∞ t a - 1 ⅆ t ( 1 + t ) a + b = B ( a , b )
∫ 0 1 t a - 1 ( 1 - t ) b - 1 ( t + z ) a + b ⅆ t = B ( a , b ) ( 1 + z ) - a z - b

with | ph z | < π and the integration path along the real axis.

∫ 0 π 2 ( cos t ) a - 1 cos ( b t ) ⅆ t = π 2 a 1 a B ( 1 2 ( a + b + 1 ) , 1 2 ( a - b + 1 ) )
ℜ a > 0 ,
∫ 0 π ( sin t ) a - 1 ⅇ ⅈ b t ⅆ t = π 2 a - 1 ⅇ ⅈ π b 2 a B ( 1 2 ( a + b + 1 ) , 1 2 ( a - b + 1 ) )
ℜ a > 0 ,
∫ 0 ∞ cosh ( 2 b t ) ( cosh t ) 2 a ⅆ t = 4 a - 1 B ( a + b , a - b )
ℜ a > | ℜ b | .
1 2 π ∫ - ∞ ∞ ⅆ t ( w + ⅈ t ) a ( z - ⅈ t ) b = ( w + z ) 1 - a - b ( a + b - 1 ) B ( a , b )
ℜ ( a + b ) > 1 , ℜ w > 0 , ℜ z > 0

The fractional powers have their principal values when w > 0 and z > 0 , and are continued via continuity.

1 2 π ⅈ ∫ c - ∞ ⅈ c + ∞ ⅈ t - a ( 1 - t ) - 1 - b ⅆ t = 1 b B ( a , b )
0 < c < 1 , ℜ ( a + b ) > 0
1 2 π ⅈ ∫ 0 ( 1 + ) t a - 1 ( t - 1 ) b - 1 ⅆ t = sin ( π b ) π B ( a , b )
ℜ a > 0 ,
t -plane. Contour for first loop integral for the beta function.

In the next two equations the fractional powers are continuous on the integration paths and take their principal values at the beginning.

1 ⅇ 2 π ⅈ a - 1 ∫ ∞ ( 0 + ) t a - 1 ( 1 + t ) - a - b ⅆ t = B ( a , b )

when ℜ b > 0 , a is not an integer and the contour cuts the real axis between - 1 and the origin.

t -plane. Contour for second loop integral for the beta function.

##### Pochhammer's Integral

When a , b ∈ ℂ

∫ P ( 1 + , 0 + , 1 - , 0 - ) t a - 1 ( 1 - t ) b - 1 ⅆ t = - 4 ⅇ π ⅈ ( a + b ) sin ( π a ) sin ( π b ) B ( a , b )

where the contour starts from an arbitrary point P in the interval ( 0 , 1 ) ,circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to P . It can always be deformed into the contour shown here.

t -plane. Contour for Pochhammer's integral.