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19 Elliptic IntegralsSymmetric Integrals

§19.25 Relations to Other Functions

  1. §19.25(i) Legendre’s Integrals as Symmetric Integrals
  2. §19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
  3. §19.25(iii) Symmetric Integrals as Legendre’s Integrals
  4. §19.25(iv) Theta Functions
  5. §19.25(v) Jacobian Elliptic Functions
  6. §19.25(vi) Weierstrass Elliptic Functions
  7. §19.25(vii) Hypergeometric Function

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

Let k2=1k2 and c=csc2ϕ with 0ϕπ/2. 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,k2,1,1α2).
19.25.3 Π(α2,k)=12πR12(12,12,1;k2,1,1α2),

with Cauchy principal value

19.25.4 Π(α2,k)=13(k2/α2)RJ(0,1k2,1,1(k2/α2)),
19.25.5 F(ϕ,k) =RF(c1,ck2,c),
19.25.6 F(ϕ,k)k =13kRD(c1,c,ck2).
19.25.7 E(ϕ,k)=2RG(c1,ck2,c)(c1)RF(c1,ck2,c)c1ck2/c,
19.25.8 E(ϕ,k)=R12(12,12,32;c1,ck2,c),
19.25.9 E(ϕ,k)=RF(c1,ck2,c)13k2RD(c1,ck2,c),
19.25.10 E(ϕ,k)=k2RF(c1,ck2,c)+13k2k2RD(c1,c,ck2)+k2c1/(cck2),
19.25.11 E(ϕ,k)=13k2RD(ck2,c,c1)+ck2/(cc1),

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(c1,ck2,c).
19.25.13 D(ϕ,k)=13RD(c1,ck2,c).
19.25.14 Π(ϕ,α2,k)F(ϕ,k)=13α2RJ(c1,ck2,c,cα2),
19.25.15 Π(ϕ,α2,k)=R12(12,12,12,1;c1,ck2,c,cα2).

If α2>c, then the Cauchy principal value is

19.25.16 Π(ϕ,α2,k)=13ω2RJ(c1,ck2,c,cω2)+(c1)(ck2)(α21)(1ω2)RC(c(α21)(1ω2),(α2c)(cω2)),

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),


19.25.18 (x,y,z)=(c1,ck2,c),

then the five nontrivial permutations of x,y,z that leave RF invariant change k2 (=(zy)/(zx)) into 1/k2, k2, 1/k2, k2/k2, k2/k2, and sinϕ (=(zx)/z) into ksinϕ, itanϕ, iktanϕ, (ksinϕ)/1k2sin2ϕ, iksinϕ/1k2sin2ϕ. 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

Let r=1/x2. Then

19.25.19 cel(kc,p,a,b) =aRF(0,kc2,1)+13(bpa)RJ(0,kc2,1,p),
19.25.20 el1(x,kc) =RF(r,r+kc2,r+1),
19.25.21 el2(x,kc,a,b) =ael1(x,kc)+13(ba)RD(r,r+kc2,r+1),
19.25.22 el3(x,kc,p) =el1(x,kc)+13(1p)RJ(r,r+kc2,r+1,r+p).

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

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

19.25.23 ϕ =arccosx/z=arcsin(zx)/z,
k =zyzx,
α2 =zpzx,

with α0. Then

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

§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),

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, (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 and §23.3. Let 𝕃 be a lattice for the Weierstrass elliptic function (z). Then

19.25.35 z+2ω=±RF((z)e1,(z)e2,(z)e3),

for some 2ω𝕃, provided that z satisfies

19.25.36 (z)ej(,0],

The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which (z)ej<0, for some j. Also,

19.25.37 ζ(z+2ω)+(z+2ω)(z)=±2RG((z)e1,(z)e2,(z)e3),

in which the sign and the ω are the same as in (19.25.35).

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,ejek,eje),
19.25.39 ζ(ωj)+ωjej=2RG(0,ejek,eje),

for some 2ωj𝕃 and (ωj)=ej.


19.25.40 z+2ω=±σ(z)RF(σ12(z),σ22(z),σ32(z)),

for some 2ω𝕃, where

19.25.41 σj(z)=exp(ηjz)σ(z+ωj)/σ(ωj),

in which 2ω1 and 2ω3 are generators for the lattice 𝕃, ω2=ω1ω3, and ηj=ζ(ωj) (see (23.2.12)). The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which σj2(z)<0, for some j.

§19.25(vii) Hypergeometric Function

19.25.42 F12(a,b;c;z)=Ra(b,cb;1z,1),
19.25.43 Ra(b1,b2;z1,z2)=z2aF12(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.)