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19 Elliptic IntegralsSymmetric Integrals

§19.25 Relations to Other Functions

Contents

§19.25(i) Legendre’s Integrals as Symmetric Integrals

Let k2=1-k2 and c=csc2ϕ. Then

19.25.1 K(k) =RF(0,k2,1),
E(k) =2RG(0,k2,1),
E(k) =13k2(RD(0,k2,1)+RD(0,1,k2)),
K(k)-E(k) =k2D(k)
=13k2RD(0,k2,1),
E(k)-k2K(k) =13k2k2RD(0,1,k2).
19.25.2 Π(α2,k)-K(k)=13α2RJ(0,k2,1,1-α2).
19.25.3 Π(α2,k)=12πR-12(12,-12,1;k2,1,1-α2),

with Cauchy principal value

19.25.4 Π(α2,k)=-13(k2/α2)RJ(0,1-k2,1,1-(k2/α2)),
-<k2<1<α2.
19.25.5 F(ϕ,k) =RF(c-1,c-k2,c),
19.25.6 F(ϕ,k)k =13kRD(c-1,c,c-k2).
19.25.7 E(ϕ,k)=2RG(c-1,c-k2,c)-(c-1)RF(c-1,c-k2,c)-(c-1)(c-k2)/c,
19.25.8 E(ϕ,k)=R-12(12,-12,32;c-1,c-k2,c),
19.25.9 E(ϕ,k)=RF(c-1,c-k2,c)-13k2RD(c-1,c-k2,c),
19.25.10 E(ϕ,k)=k2RF(c-1,c-k2,c)+13k2k2RD(c-1,c,c-k2)+k2(c-1)/(c(c-k2)),
c>k2,
19.25.11 E(ϕ,k)=-13k2RD(c-k2,c,c-1)+(c-k2)/(c(c-1)),
ϕ12π.

Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of RD; see (19.21.7). All terms on the right-hand sides are nonnegative when k20, 0k21, or 1k2c, respectively.

19.25.12 E(ϕ,k)k=-13kRD(c-1,c-k2,c).
19.25.13 D(ϕ,k)=13RD(c-1,c-k2,c).
19.25.14 Π(ϕ,α2,k)-F(ϕ,k)=13α2RJ(c-1,c-k2,c,c-α2),
19.25.15 Π(ϕ,α2,k)=R-12(12,12,-12,1;c-1,c-k2,c,c-α2).

If α2>c, then the Cauchy principal value is

19.25.16 Π(ϕ,α2,k)=-13ω2RJ(c-1,c-k2,c,c-ω2)+(c-1)(c-k2)(α2-1)(1-ω2)RC(c(α2-1)(1-ω2),(α2-c)(c-ω2)),
ω2=k2/α2.

The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). For example, if we write (19.25.5) as

19.25.17 F(ϕ,k)=RF(x,y,z),

with

19.25.18 (x,y,z)=(c-1,c-k2,c),

then the five nontrivial permutations of x,y,z that leave RF invariant change k2 (=(z-y)/(z-x)) into 1/k2, k2, 1/k2, -k2/k2, -k2/k2, and sinϕ (=(z-x)/z) into ksinϕ, -itanϕ, -iktanϕ, (ksinϕ)/1-k2sin2ϕ, -iksinϕ/1-k2sin2ϕ. Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions).

The three changes of parameter of Π(ϕ,α2,k) in §19.7(iii) are unified in (19.21.12) by way of (19.25.14).

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

§19.25(iii) Symmetric Integrals as Legendre’s Integrals

Assume 0xyz, x<z, and p>0. Let

19.25.23 ϕ =arccosx/z
=arcsin(z-x)/z,
k =z-yz-x,
α2 =z-pz-x,

with α0. Then

19.25.24 (z-x)1/2RF(x,y,z)=F(ϕ,k),
19.25.25 (z-x)3/2RD(x,y,z)=(3/k2)(F(ϕ,k)-E(ϕ,k)),
19.25.26 (z-x)3/2RJ(x,y,z,p)=(3/α2)(Π(ϕ,α2,k)-F(ϕ,k)),
19.25.27 2(z-x)-1/2RG(x,y,z)=E(ϕ,k)+(cotϕ)2F(ϕ,k)+(cotϕ)1-k2sin2ϕ.

§19.25(iv) Theta Functions

For relations of symmetric integrals to theta functions, see §20.9(i).

§19.25(v) Jacobian Elliptic Functions

For the notation see §§22.2, 22.15, and 22.16(i).

With 0k21 and p,q,r any permutation of the letters c,d,n, define

19.25.28 Δ(p,q)=ps2(u,k)-qs2(u,k)=-Δ(q,p),

which implies

19.25.29 Δ(n,d) =k2,
Δ(d,c) =k2,
Δ(n,c) =1.

If cs2(u,k)0, then

19.25.30 am(u,k)=RC(cs2(u,k),ns2(u,k)),
19.25.31 u=RF(ps2(u,k),qs2(u,k),rs2(u,k));

compare (19.25.35) and (20.9.3).

19.25.32 arcps(x,k)=RF(x2,x2+Δ(q,p),x2+Δ(r,p)),
19.25.33 arcsp(x,k)=xRF(1,1+Δ(q,p)x2,1+Δ(r,p)x2),
19.25.34 arcpq(x,k)=wRF(x2,1,1+Δ(r,q)w),
w=(1-x2)/Δ(q,p),

where we assume 0x21 if x=sn, cn, or cd; x21 if x=ns, nc, or dc; x real if x=cs or sc; kx1 if x=dn; 1x1/k if x=nd; x2k2 if x=ds; 0x21/k2 if x=sd.

For the use of R-functions with Δ(p,q) in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008).

Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of RF(x,y,z). See (19.29.19), Carlson (2005), and (22.15.11), and compare with Abramowitz and Stegun (1964, Eqs. (17.4.41)–(17.4.52)). For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9).

§19.25(vi) Weierstrass Elliptic Functions

For the notation see §23.2.

19.25.35 z=RF((z)-e1,(z)-e2,(z)-e3),

provided that

19.25.36 (z)-ej\(-,0],
j=1,2,3,

and the left-hand side does not vanish for more than one value of j. Also,

19.25.37 ζ(z)+z(z)=2RG((z)-e1,(z)-e2,(z)-e3).

In (19.25.38) and (19.25.39) j,k, is any permutation of the numbers 1,2,3.

19.25.38 ωj=RF(0,ej-ek,ej-e),
19.25.39 ηj+ωjej=2RG(0,ej-ek,ej-e).

Lastly,

19.25.40 z=σ(z)RF(σ12(z),σ22(z),σ32(z)),

where

19.25.41 σj(z)=exp(-ηjz)σ(z+ωj)/σ(ωj),
j=1,2,3.

§19.25(vii) Hypergeometric Function

19.25.42 F12(a,b;c;z)=R-a(b,c-b;1-z,1),
19.25.43 R-a(b1,b2;z1,z2)=z2-aF12(a,b1;b1+b2;1-(z1/z2)).

For these results and extensions to the Appell function F116.13) and Lauricella’s function FD see Carlson (1963). (F1 and FD are equivalent to the R-function of 3 and n variables, respectively, but lack full symmetry.)