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19 Elliptic IntegralsSymmetric Integrals

§19.16 Definitions

  1. §19.16(i) Symmetric Integrals
  2. §19.16(ii) Ra(𝐛;𝐳)
  3. §19.16(iii) Various Cases of Ra(𝐛;𝐳)

§19.16(i) Symmetric Integrals

19.16.1 RF(x,y,z)=120dts(t),
19.16.2 RJ(x,y,z,p)=320dts(t)(t+p),
19.16.2_5 RG(x,y,z)=1401s(t)(xt+x+yt+y+zt+z)tdt.
19.16.3 Moved to (19.23.6_5).

where p (0) is a real or complex constant, and

19.16.4 s(t)=t+xt+yt+z.

In (19.16.1)–(19.16.2_5), x,y,z(,0] except that one or more of x,y,z may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when p is real and negative. See also (19.20.14). It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that RF(1,1,1)=RJ(1,1,1,1)=RG(1,1,1)=1.

A fourth integral that is symmetric in only two variables is defined by

19.16.5 RD(x,y,z)=RJ(x,y,z,z)=320dts(t)(t+z),

with the same conditions on x, y, z as for (19.16.1), but now z0.

Just as the elementary function RC(x,y)19.2(iv)) is the degenerate case

19.16.6 RC(x,y)=RF(x,y,y),

and RD is a degenerate case of RJ, so is RJ a degenerate case of the hyperelliptic integral,

19.16.7 320dtj=15t+xj.

§19.16(ii) Ra(𝐛;𝐳)

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function

19.16.8 Ra(𝐛;𝐳)=Ra(b1,,bn;z1,,zn),

which is homogeneous and of degree a in the z’s, and unchanged when the same permutation is applied to both sets of subscripts 1,,n. Thus Ra(𝐛;𝐳) is symmetric in the variables zj and z if the parameters bj and b are equal. The R-function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 Ra(𝐛;𝐳) was denoted by R(a;𝐛;𝐳).

19.16.9 Ra(𝐛;𝐳)=1B(a,a)0ta1j=1n(t+zj)bjdt=1B(a,a)0ta1j=1n(1+tzj)bjdt,
b1++bn>a>0, bj, zj(,0],

where B(x,y) is the beta function (§5.12) and

19.16.10 a=a+j=1nbj.
19.16.11 Ra(𝐛;λ𝐳) =λaRa(𝐛;𝐳),
Ra(𝐛;x𝟏) =xa,

When n=4 a useful version of (19.16.9) is given by

19.16.12 Ra(b1,,b4;c1,ck2,c,cα2)=2(sin2ϕ)1aB(a,a)0ϕ(sinθ)2a1(sin2ϕsin2θ)a1(cosθ)12b1×(1k2sin2θ)b2(1α2sin2θ)b4dθ,


19.16.13 c =csc2ϕ;
a,a >0;
b3 =a+ab1b2b4.

For generalizations and further information, especially representation of the R-function as a Dirichlet average, see Carlson (1977b).

§19.16(iii) Various Cases of Ra(𝐛;𝐳)

Ra(𝐛;𝐳) is an elliptic integral iff the z’s are distinct and exactly four of the parameters a,a,b1,,bn are half-odd-integers, the rest are integers, and none of a, a, a+a is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of a and a is either 12 or 1 and each bj is 12. The only cases that are integrals of the third kind are those in which at least one bj is a positive integer. All other elliptic cases are integrals of the second kind.

19.16.14 RF(x,y,z) =R12(12,12,12;x,y,z),
19.16.15 RD(x,y,z) =R32(12,12,32;x,y,z),
19.16.16 RJ(x,y,z,p) =R32(12,12,12,1;x,y,z,p),
19.16.17 RG(x,y,z) =R12(12,12,12;x,y,z),
19.16.18 RC(x,y) =R12(12,1;x,y).

(Note that RC(x,y) is not an elliptic integral.)

When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an R-function with one less variable:

19.16.19 Ra(b1,,bn;0,z2,,zn)=B(a,ab1)B(a,a)Ra(b2,,bn;z2,,zn),
a+a>0, a>b1.


19.16.20 RF(0,y,z) =12πR12(12,12;y,z),
19.16.21 RD(0,y,z) =34πR32(12,32;y,z),
19.16.22 RJ(0,y,z,p) =34πR32(12,12,1;y,z,p),
19.16.23 RG(0,y,z) =14πR12(12,12;y,z)=14πzR12(12,32;y,z).

The last R-function has a=a=12.

Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral:

19.16.24 Ra(𝐛;𝐳)=z1ab1B(b1,ab1)0tb11(t+z1)aRa(𝐛;0,t+z2,,t+zn)dt,
a>b1, a+a>b1>0.