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19 Elliptic IntegralsSymmetric Integrals

§19.21 Connection Formulas

  1. §19.21(i) Complete Integrals
  2. §19.21(ii) Incomplete Integrals
  3. §19.21(iii) Change of Parameter of RJ

§19.21(i) Complete Integrals

Legendre’s relation (19.7.1) can be written

19.21.1 RF(0,z+1,z)RD(0,z+1,1)+RD(0,z+1,z)RF(0,z+1,1)=3π/(2z),

The case z=1 shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is π/4.

19.21.2 3RF(0,y,z)=zRD(0,y,z)+yRD(0,z,y).
19.21.3 6RG(0,y,z)=yz(RD(0,y,z)+RD(0,z,y))=3zRF(0,y,z)+z(yz)RD(0,y,z).

The complete cases of RF and RG have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if 0<phz<π, and lower signs if π<phz<0:

19.21.4 RF(0,z1,z)=RF(0,1z,1)iRF(0,z,1),
19.21.5 2RG(0,z1,z)=2RG(0,1z,1)±i2RG(0,z,1)+(z1)RF(0,1z,1)izRF(0,z,1).

Let y, z, and p be positive and distinct, and permute y and z to ensure that y does not lie between z and p. The complete case of RJ can be expressed in terms of RF and RD:

19.21.6 (rp/z)RJ(0,y,z,p)=(r1)RF(0,y,z)RD(p,rz,z)+RD(0,y,z)RF(p,rz,z),

If 0<p<z and y=z+1, then as p0 (19.21.6) reduces to Legendre’s relation (19.21.1).

§19.21(ii) Incomplete Integrals

RD(x,y,z) is symmetric only in x and y, but either (nonzero) x or (nonzero) y can be moved to the third position by using

19.21.7 (xy)RD(y,z,x)+(zy)RD(x,y,z)=3RF(x,y,z)3y1/2x1/2z1/2,

or the corresponding equation with x and y interchanged.

19.21.8 RD(y,z,x)+RD(z,x,y)+RD(x,y,z) =3x1/2y1/2z1/2,
19.21.9 xRD(y,z,x)+yRD(z,x,y)+zRD(x,y,z) =3RF(x,y,z).
19.21.10 2RG(x,y,z)=zRF(x,y,z)13(xz)(yz)RD(x,y,z)+x1/2y1/2z1/2,

Because RG is completely symmetric, x,y,z can be permuted on the right-hand side of (19.21.10) so that (xz)(yz)0 if the variables are real, thereby avoiding cancellations when RG is calculated from RF and RD (see §19.36(i)).

19.21.11 6RG(x,y,z)=3(x+y+z)RF(x,y,z)x2RD(y,z,x)=x(y+z)RD(y,z,x),

where both summations extend over the three cyclic permutations of x,y,z.

Connection formulas for Ra(𝐛;𝐳) are given in Carlson (1977b, pp. 99, 101, and 123–124).

§19.21(iii) Change of Parameter of RJ

Let x,y,z be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter p of RJ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). The latter case allows evaluation of Cauchy principal values (see (19.20.14)).

19.21.12 (px)RJ(x,y,z,p)+(qx)RJ(x,y,z,q)=3RF(x,y,z)3RC(ξ,η),


19.21.13 (px)(qx) =(yx)(zx),
ξ =yz/x,
η =pq/x,

and x,y,z may be permuted. Also,

19.21.14 ηξ=p+qyz=(py)(pz)px=(qy)(qz)qx=(py)(qy)xy=(pz)(qz)xz.

For each value of p, permutation of x,y,z produces three values of q, one of which lies in the same region as p and two lie in the other region of the same type. In (19.21.12), if x is the largest (smallest) of x,y, and z, then p and q lie in the same region if it is circular (hyperbolic); otherwise p and q lie in different regions, both circular or both hyperbolic. If x=0, then ξ=η= and RC(ξ,η)=0; hence

19.21.15 pRJ(0,y,z,p)+qRJ(0,y,z,q)=3RF(0,y,z),