Digital Library of Mathematical Functions
The Gamma Function -- Multidimensional Integral

### Multidimensional Integrals

Let V n be the simplex: t 1 + t 2 + … + t n ≤ 1 , t k ≥ 0 . Then for ℜ z k > 0 , k = 1 , 2 , … , n + 1 ,

∫ V n t 1 z 1 - 1 t 2 z 2 - 1 ⋯ t n z n - 1 ⅆ t 1 ⅆ t 2 ⋯ ⅆ t n = Γ ( z 1 ) Γ ( z 2 ) ⋯ Γ ( z n ) Γ ( 1 + z 1 + z 2 + … + z n )
∫ V n ( 1 - ∑ k = 1 n t k ) z n + 1 - 1 ∏ k = 1 n t k z k - 1 ⅆ t k = Γ ( z 1 ) Γ ( z 2 ) ⋯ Γ ( z n + 1 ) Γ ( z 1 + z 2 + … + z n + 1 )

##### Selberg-type Integrals

Δ ( t 1 , t 2 , … , t n ) = ∏ 1 ≤ j < k ≤ n ( t j - t k )

Then

∫ [ 0 , 1 ] n t 1 t 2 ⋯ t m | Δ ( t 1 , … , t n ) | 2 c ∏ k = 1 n t k a - 1 ( 1 - t k ) b - 1 ⅆ t k = 1 ( Γ ( 1 + c ) ) n ∏ k = 1 m a + ( n - k ) c a + b + ( 2 n - k - 1 ) c ∏ k = 1 n Γ ( a + ( n - k ) c ) Γ ( b + ( n - k ) c ) Γ ( 1 + k c ) Γ ( a + b + ( 2 n - k - 1 ) c )

provided that ℜ a , ℜ b > 0 , ℜ c > - min ( 1 n , ℜ a ( n - 1 ) , ℜ b ( n - 1 ) )

Secondly,

∫ [ 0 , ∞ ) n t 1 t 2 ⋯ t m | Δ ( t 1 , … , t n ) | 2 c ∏ k = 1 n t k a - 1 ⅇ - t k ⅆ t k = ∏ k = 1 m ( a + ( n - k ) c ) ∏ k = 1 n Γ ( a + ( n - k ) c ) Γ ( 1 + k c ) ( Γ ( 1 + c ) ) n

when ℜ a > 0 , ℜ c > - min ( 1 n , ℜ a ( n - 1 ) ) .

Thirdly,

1 ( 2 π ) n 2 ∫ ( - ∞ , ∞ ) n | Δ ( t 1 , … , t n ) | 2 c ∏ k = 1 n exp ( - 1 2 t k 2 ) ⅆ t k = ∏ k = 1 n Γ ( 1 + k c ) ( Γ ( 1 + c ) ) n

##### Dyson's Integral

1 ( 2 π ) n ∫ [ - π , π ] n ∏ 1 ≤ j < k ≤ n | ⅇ ⅈ θ j - ⅇ ⅈ θ k | 2 b ⅆ θ 1 ⋯ ⅆ θ n = Γ ( 1 + b n ) ( Γ ( 1 + b ) ) n
ℜ b > 1 n .