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19 Elliptic IntegralsSymmetric Integrals

§19.16 Definitions

Contents

§19.16(i) Symmetric Integrals

19.16.1 RF(x,y,z)=120ts(t),
19.16.2 RJ(x,y,z,p)=320ts(t)(t+p),
19.16.3 RG(x,y,z)=14π02π0π(xsin2θcos2ϕ+ysin2θsin2ϕ+zcos2θ)12×sinθθϕ,

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

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

In (19.16.1) and (19.16.2), 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). In (19.16.3) x, y, z0. It should be noted that the integrals (19.16.1)–(19.16.3) 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)=320ts(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 320tj=15t+xj.

§19.16(ii) R-a(b;z)

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.16.3), are special cases of a multivariate hypergeometric function

19.16.8 R-a(b;z)=R-a(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 R-a(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 R-a(b;z) was denoted by R(a;b;z).

19.16.9 R-a(b;z)=1B(a,a)0ta-1j=1n(t+zj)-bjt=1B(a,a)0ta-1j=1n(1+tzj)-bjt,
a,a>0, zj\(-,0],

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

19.16.10 a=-a+j=1nbj.
19.16.11 R-a(b;λz) =λ-aR-a(b;z),
R-a(b;x1) =x-a,
1=(1,,1).

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

19.16.12 R-a(b1,,b4;c-1,c-k2,c,c-α2)=2(sin2ϕ)1-aB(a,a)0ϕ(sinθ)2a-1(sin2ϕ-sin2θ)a-1(cosθ)1-2b1×(1-k2sin2θ)-b2(1-α2sin2θ)-b4θ,

where

19.16.13 c =csc2ϕ;
a,a >0;
b3 =a+a-b1-b2-b4.

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

§19.16(iii) Various Cases of R-a(b;z)

R-a(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) =R-12(12,12,12;x,y,z),
19.16.15 RD(x,y,z) =R-32(12,12,32;x,y,z),
19.16.16 RJ(x,y,z,p) =R-32(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) =R-12(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 R-a(b1,,bn;0,z2,,zn)=B(a,a-b1)B(a,a)R-a(b2,,bn;z2,,zn),
a+a>0, a>b1.

Thus

19.16.20 RF(0,y,z) =12πR-12(12,12;y,z),
19.16.21 RD(0,y,z) =34πR-32(12,32;y,z),
19.16.22 RJ(0,y,z,p) =34πR-32(12,12,1;y,z,p),
19.16.23 RG(0,y,z) =14πR12(12,12;y,z)
=14πzR-12(-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 R-a(b;z)=z1a-b1B(b1,a-b1)0tb1-1(t+z1)-aR-a(b;0,t+z2,,t+zn)t,
a>b1, a+a>b1>0.