About the Project
19 Elliptic IntegralsSymmetric Integrals

§19.20 Special Cases


§19.20(i) RF(x,y,z)

In this subsection, and also §§19.20(ii)19.20(v), the variables of all R-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated.

19.20.1 RF(x,x,x) =x-1/2,
RF(λx,λy,λz) =λ-1/2RF(x,y,z),
RF(x,y,y) =RC(x,y),
RF(0,y,y) =12πy-1/2,
RF(0,0,z) =.

The first lemniscate constant is given by

19.20.2 01dt1-t4=RF(0,1,2)=(Γ(14))24(2π)1/2=1.31102 87771 46059 90523.

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. The general lemniscatic case is

19.20.3 RF(x,a,y)=R-14(34,12;a2,xy),

§19.20(ii) RG(x,y,z)

19.20.4 RG(x,x,x) =x1/2,
RG(λx,λy,λz) =λ1/2RG(x,y,z),
RG(0,y,y) =14πy1/2,
RG(0,0,z) =12z1/2,
19.20.5 2RG(x,y,y)=yRC(x,y)+x.

§19.20(iii) RJ(x,y,z,p)

19.20.6 RJ(x,x,x,x) =x-3/2,
RJ(λx,λy,λz,λp) =λ-3/2RJ(x,y,z,p),
RJ(x,y,z,z) =RD(x,y,z),
RJ(0,0,z,p) =,
RJ(x,x,x,p) =RD(p,p,x)=3x-p(RC(x,p)-1x),
xp, xp0.
19.20.7 RJ(x,y,z,p)+,
p0+ or 0-; x,y,z>0.
19.20.8 RJ(0,y,y,p) =3π2(yp+py),
RJ(0,y,y,-q) =-3π2y(y+q),
RJ(x,y,y,p) =3p-y(RC(x,y)-RC(x,p)),
RJ(x,y,y,y) =RD(x,y,y).
19.20.9 RJ(0,y,z,±yz)=±32yzRF(0,y,z).
19.20.10 limp0+pRJ(0,y,z,p) =3π2yz,
limp0-RJ(0,y,z,p) =-RD(0,y,z)-RD(0,z,y)=-6yzRG(0,y,z).
19.20.11 RJ(0,y,z,p)=32pzln(16zy)-3pRC(z,p)+O(ylny),
y0+; p (0) real.
19.20.12 limp±pRJ(x,y,z,p)=3RF(x,y,z).
19.20.13 2(p-x)RJ(x,y,z,p)=3RF(x,y,z)-3xRC(yz,p2),

where x,y,z may be permuted.

When the variables are real and distinct, the various cases of RJ(x,y,z,p) are called circular (hyperbolic) cases if (p-x)(p-y)(p-z) is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If x,y,z are permuted so that 0x<y<z, then the Cauchy principal value of RJ is given by

19.20.14 (q+z)RJ(x,y,z,-q)=(p-z)RJ(x,y,z,p)-3RF(x,y,z)+3(xyzxy+pq)1/2RC(xy+pq,pq),

valid when

19.20.15 q >0,
p =z(x+y+q)-xyz+q,


19.20.16 p =wy+(1-w)z,
w =z-xz+q,
0 <w<1.

Since x<y<p<z, p is in a hyperbolic region. In the complete case (x=0) (19.20.14) reduces to

19.20.17 (q+z)RJ(0,y,z,-q)=(p-z)RJ(0,y,z,p)-3RF(0,y,z),
p=z(y+q)/(z+q), w=z/(z+q).

§19.20(iv) RD(x,y,z)

19.20.18 RD(x,x,x) =x-3/2,
RD(λx,λy,λz) =λ-3/2RD(x,y,z),
RD(0,y,y) =34πy-3/2,
RD(0,0,z) =.
19.20.19 RD(x,y,z)3x-1/2y-1/2z-1/2,
19.20.20 RD(x,y,y)=32(y-x)(RC(x,y)-xy),
xy, y0,
19.20.21 RD(x,x,z)=3z-x(RC(z,x)-1z),
xz, xz0.

The second lemniscate constant is given by

19.20.22 01t2dt1-t4=13RD(0,2,1)=(Γ(34))2(2π)1/2=0.59907 01173 67796 10371.

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. Compare (19.20.2). The general lemniscatic case is

19.20.23 RD(x,y,a)=R-34(54,12;a2,xy),

§19.20(v) R-a(b;z)

Define c=j=1nbj. Then

19.20.24 R0(b;z) =1,
RN(b;z) =N!(c)NTN(b,z),

where TN is defined by (19.19.1). Also,

19.20.25 R-c(b;z)=j=1nzj-bj,
19.20.26 R-a(b;z)=j=1nzj-bjR-a(b;z-1),
a+a=c, z-1=(z1-1,,zn-1).

See also (19.16.11) and (19.16.19).