What's New
About the Project
NIST
5 Gamma FunctionProperties

§5.14 Multidimensional Integrals

Let Vn be the simplex: t1+t2++tn1, tk0. Then for zk>0, k=1,2,,n+1,

5.14.1 Vnt1z1-1t2z2-1tnzn-1dt1dt2dtn=Γ(z1)Γ(z2)Γ(zn)Γ(1+z1+z2++zn),
5.14.2 Vn(1-k=1ntk)zn+1-1k=1ntkzk-1dtk=Γ(z1)Γ(z2)Γ(zn+1)Γ(z1+z2++zn+1).

Selberg-type Integrals

Let

5.14.3 Δ(t1,t2,,tn)=1j<kn(tj-tk).

Then

5.14.4 [0,1]nt1t2tm|Δ(t1,,tn)|2ck=1ntka-1(1-tk)b-1dtk=1(Γ(1+c))nk=1ma+(n-k)ca+b+(2n-k-1)ck=1nΓ(a+(n-k)c)Γ(b+(n-k)c)Γ(1+kc)Γ(a+b+(2n-k-1)c),

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

Secondly,

5.14.5 [0,)nt1t2tm|Δ(t1,,tn)|2ck=1ntka-1e-tkdtk=k=1m(a+(n-k)c)k=1nΓ(a+(n-k)c)Γ(1+kc)(Γ(1+c))n,

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

Thirdly,

5.14.6 1(2π)n/2(-,)n|Δ(t1,,tn)|2ck=1nexp(-12tk2)dtk=k=1nΓ(1+kc)(Γ(1+c))n,
c>-1/n.

Dyson’s Integral

5.14.7 1(2π)n[-π,π]n1j<kn|eiθj-eiθk|2bdθ1dθn=Γ(1+bn)(Γ(1+b))n,
b>-1/n.