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19 Elliptic IntegralsSymmetric Integrals

§19.20 Special Cases

  1. §19.20(i) RF(x,y,z)
  2. §19.20(ii) RG(x,y,z)
  3. §19.20(iii) RJ(x,y,z,p)
  4. §19.20(iv) RD(x,y,z)
  5. §19.20(v) Ra(𝐛;𝐳)

§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) =x1/2,
RF(λx,λy,λz) =λ1/2RF(x,y,z),
RF(x,y,y) =RC(x,y),
RF(0,y,y) =12πy1/2,
RF(0,0,z) =.

The first lemniscate constant is given by

19.20.2 01dt1t4=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)=R14(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) =x3/2,
RJ(λx,λy,λz,λp) =λ3/2RJ(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)=3xp(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) =3py(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,
limp0RJ(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(px)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 (px)(py)(pz) 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)=(pz)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+(1w)z,
w =zxz+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)=(pz)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) =x3/2,
RD(λx,λy,λz) =λ3/2RD(x,y,z),
RD(0,y,y) =34πy3/2,
RD(0,0,z) =.
19.20.19 RD(x,y,z)3x1/2y1/2z1/2,
19.20.20 RD(x,y,y)=32(yx)(RC(x,y)xy),
xy, y0,
19.20.21 RD(x,x,z)=3zx(RC(z,x)1z),
xz, xz0.

The second lemniscate constant is given by

19.20.22 01t2dt1t4=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)=R34(54,12;a2,xy),

§19.20(v) Ra(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 Rc(𝐛;𝐳)=j=1nzjbj,
19.20.26 Ra(𝐛;𝐳)=j=1nzjbjRa(𝐛;𝒛𝟏),
a+a=c, 𝒛𝟏=(z11,,zn1).

See also (19.16.11) and (19.16.19).