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19 Elliptic IntegralsSymmetric Integrals

§19.23 Integral Representations

In (19.23.1)–(19.23.3) we assume y>0 and z>0.

19.23.1 RF(0,y,z)=0π/2(ycos2θ+zsin2θ)-1/2dθ,
19.23.2 RG(0,y,z)=120π/2(ycos2θ+zsin2θ)1/2dθ,
19.23.3 RD(0,y,z)=30π/2(ycos2θ+zsin2θ)-3/2sin2θdθ.
19.23.4 RF(0,y,z)=2π0π/2RC(y,zcos2θ)dθ=2π0RC(ycosh2t,z)dt.
19.23.5 RF(x,y,z)=2π0π/2RC(x,ycos2θ+zsin2θ)dθ,
y>0, z>0,
19.23.6 4πRF(x,y,z)=02π0πsinθdθdϕ(xsin2θcos2ϕ+ysin2θsin2ϕ+zcos2θ)1/2,

where x, y, and z have positive real parts—except that at most one of them may be 0.

In (19.23.7)–(19.23.10) one or more of the variables may be 0 if the integral converges. In (19.23.8) n=2, and in (19.23.9) n=3. Also, in (19.23.8) and (19.23.10) B denotes the beta function (§5.12).

19.23.7 RG(x,y,z)=1401t+xt+yt+z(xt+x+yt+y+zt+z)tdt,
x,y,z\(-,0].
19.23.8 R-a(b;z)=2B(b1,b2)0π/2(z1cos2θ+z2sin2θ)-a(cosθ)2b1-1(sinθ)2b2-1dθ,
b1,b2>0; z1,z2>0.

With l1,l2,l3 denoting any permutation of sinθcosϕ, sinθsinϕ, cosθ,

19.23.9 R-a(b;z)=4Γ(b1+b2+b3)Γ(b1)Γ(b2)Γ(b3)0π/20π/2(j=13zjlj2)-aj=13lj2bj-1sinθdθdϕ,
bj>0, zj>0.
19.23.10 R-a(b;z)=1B(a,a)01ua-1(1-u)a-1j=1n(1-u+uzj)-bjdu,
a,a>0; a+a=j=1nbj; zj\(-,0].

For generalizations of (19.16.3) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).