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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 Vnt1z11t2z21tnzn1dt1dt2dtn=Γ(z1)Γ(z2)Γ(zn)Γ(1+z1+z2++zn),
5.14.2 Vn(1k=1ntk)zn+11k=1ntkzk1dtk=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(tjtk).

Then

5.14.4 [0,1]nt1t2tm|Δ(t1,,tn)|2ck=1ntka1(1tk)b1dtk=1(Γ(1+c))nk=1ma+(nk)ca+b+(2nk1)ck=1nΓ(a+(nk)c)Γ(b+(nk)c)Γ(1+kc)Γ(a+b+(2nk1)c),

provided that a, b>0, c>min(1/n,a/(n1),b/(n1)).

Secondly,

5.14.5 [0,)nt1t2tm|Δ(t1,,tn)|2ck=1ntka1etkdtk=k=1m(a+(nk)c)k=1nΓ(a+(nk)c)Γ(1+kc)(Γ(1+c))n,

when a>0, c>min(1/n,a/(n1)).

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|eiθjeiθk|2bdθ1dθn=Γ(1+bn)(Γ(1+b))n,
b>1/n.