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19 Elliptic IntegralsSymmetric Integrals

§19.21 Connection Formulas


§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(y-z)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,z-1,z)=RF(0,1-z,1)iRF(0,z,1),
19.21.5 2RG(0,z-1,z)=2RG(0,1-z,1)±i2RG(0,z,1)+(z-1)RF(0,1-z,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)=(r-1)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 (x-y)RD(y,z,x)+(z-y)RD(x,y,z)=3RF(x,y,z)-3y1/2x-1/2z-1/2,

or the corresponding equation with x and y interchanged.

19.21.8 RD(y,z,x)+RD(z,x,y)+RD(x,y,z) =3x-1/2y-1/2z-1/2,
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(x-z)(y-z)RD(x,y,z)+3x1/2y1/2z-1/2,

Because RG is completely symmetric, x,y,z can be permuted on the right-hand side of (19.21.10) so that (x-z)(y-z)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 R-a(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 (p-x)RJ(x,y,z,p)+(q-x)RJ(x,y,z,q)=3RF(x,y,z)-3RC(ξ,η),


19.21.13 (p-x)(q-x) =(y-x)(z-x),
ξ =yz/x,
η =pq/x,

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

19.21.14 η-ξ=p+q-y-z=(p-y)(p-z)p-x=(q-y)(q-z)q-x=(p-y)(q-y)x-y=(p-z)(q-z)x-z.

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),