Digital Library of Mathematical Functions
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19 Elliptic IntegralsSymmetric Integrals

§19.28 Integrals of Elliptic Integrals

In (19.28.1)–(19.28.3) we assume σ>0. Also, B again denotes the beta function (§5.12).

19.28.1 01tσ-1RF(0,t,1)t =12(B(σ,12))2,
19.28.2 01tσ-1RG(0,t,1)t =σ4σ+2(B(σ,12))2,
19.28.3 01tσ-1(1-t)RD(0,t,1)t=34σ+2(B(σ,12))2.
19.28.4 01tσ-1(1-t)c-1R-a(b1,b2;t,1)t=Γ(c)Γ(σ)Γ(σ+b2-a)Γ(σ+c-a)Γ(σ+b2),
c=b1+b2>0, σ>max(0,a-b2).

In (19.28.5)–(19.28.9) we assume x,y,z, and p are real and positive.

19.28.5 zRD(x,y,t)t=6RF(x,y,z),
19.28.6 01RD(x,y,v2z+(1-v2)p)v=RJ(x,y,z,p).
19.28.7 0RJ(x,y,z,r2)r=32πRF(xy,xz,yz),
19.28.8 0RJ(tx,y,z,tp)t=6pRC(p,x)RF(0,y,z).
19.28.9 0π/2RF(sin2θcos2(x+y),sin2θcos2(x-y),1)θ=RF(0,cos2x,1)RF(0,cos2y,1),
19.28.10 0RF((ac+bd)2,(ad+bc)2,4abcdcosh2z)z=12RF(0,a2,b2)RF(0,c2,d2),
a,b,c,d>0.

See also (19.16.24). To replace a single component of z in R-a(b;z) by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)).