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19 Elliptic IntegralsSymmetric Integrals

§19.26 Addition Theorems

Contents
  1. §19.26(i) General Formulas
  2. §19.26(ii) Case x=0
  3. §19.26(iii) Duplication Formulas

§19.26(i) General Formulas

In this subsection, and also §§19.26(ii) and 19.26(iii), we assume that λ,x,y,z are positive, except that at most one of x,y,z can be 0.

19.26.1 RF(x+λ,y+λ,z+λ)+RF(x+μ,y+μ,z+μ)=RF(x,y,z),

where μ>0 and

19.26.2 x+μ=λ2((x+λ)yz+x(y+λ)(z+λ))2,

with corresponding equations for y+μ and z+μ obtained by permuting x,y,z. Also,

19.26.3 z=ξζ+ηζξηξηζ+ξηζ,

where

19.26.4 (ξ,η,ζ) =(x+λ,y+λ,z+λ),
(ξ,η,ζ) =(x+μ,y+μ,z+μ),

with x and y obtained by permuting x, y, and z. (Note that ξζ+ηζξη=ξζ+ηζξη.) Equivalent forms of (19.26.2) are given by

19.26.5 μ=λ2(xyz+(x+λ)(y+λ)(z+λ))2λxyz,

and

19.26.6 (λμxyxzyz)2=4xyz(λ+μ+x+y+z).

Also,

19.26.7 RD(x+λ,y+λ,z+λ)+RD(x+μ,y+μ,z+μ)=RD(x,y,z)3z(z+λ)(z+μ),
19.26.8 2RG(x+λ,y+λ,z+λ)+2RG(x+μ,y+μ,z+μ)=2RG(x,y,z)+λRF(x+λ,y+λ,z+λ)+μRF(x+μ,y+μ,z+μ)+λ+μ+x+y+z.
19.26.9 RJ(x+λ,y+λ,z+λ,p+λ)+RJ(x+μ,y+μ,z+μ,p+μ)=RJ(x,y,z,p)3RC(γδ,γ),

where

19.26.10 γ =p(p+λ)(p+μ),
δ =(px)(py)(pz).

Lastly,

19.26.11 RC(x+λ,y+λ)+RC(x+μ,y+μ)=RC(x,y),

where λ>0, y>0, x0, and

19.26.12 x+μ =λ2(x+λy+x(y+λ))2,
y+μ =(y(y+λ)/λ2)(x+x+λ)2.

Equivalent forms of (19.26.11) are given by

19.26.13 RC(α2,α2θ)+RC(β2,β2θ)=RC(σ2,σ2θ),
σ=(αβ+θ)/(α+β),

where 0<γ2θ<γ2 for γ=α,β,σ, except that σ2θ can be 0, and

19.26.14 (py)RC(x,p)+(qy)RC(x,q)=(ηξ)RC(ξ,η),
x0, y0; p,q{0},

where

19.26.15 (px)(qx) =(yx)2,
ξ =y2/x,
η =pq/x,
ηξ =p+q2y.

§19.26(ii) Case x=0

If x=0, then λμ=yz. For example,

19.26.16 RF(λ,y+λ,z+λ)=RF(0,y,z)RF(μ,y+μ,z+μ),
λμ=yz.

An equivalent version for RC is

19.26.17 αRC(β,α+β)+βRC(α,α+β)=π/2,
α,β(,0), α+β>0.

§19.26(iii) Duplication Formulas

19.26.18 RF(x,y,z)=2RF(x+λ,y+λ,z+λ)=RF(x+λ4,y+λ4,z+λ4),

where

19.26.19 λ=xy+yz+zx.
19.26.20 RD(x,y,z)=2RD(x+λ,y+λ,z+λ)+3z(z+λ).
19.26.21 2RG(x,y,z)=4RG(x+λ,y+λ,z+λ)λRF(x,y,z)xyz.
19.26.22 RJ(x,y,z,p)=2RJ(x+λ,y+λ,z+λ,p+λ)+3RC(α2,β2),

where

19.26.23 α =p(x+y+z)+xyz,
β =p(p+λ),
β±α =(p±x)(p±y)(p±z),
β2α2 =(px)(py)(pz),

either upper or lower signs being taken throughout.

The equations inverse to z+λ=(z+x)(z+y) and the two other equations obtained by permuting x,y,z (see (19.26.19)) are

19.26.24 z=(ξζ+ηζξη)2/(4ξηζ),
(ξ,η,ζ)=(x+λ,y+λ,z+λ),

and two similar equations obtained by exchanging z with x (and ζ with ξ), or z with y (and ζ with η).

Next,

19.26.25 RC(x,y)=2RC(x+λ,y+λ),
λ=y+2xy.

Equivalent forms are given by (19.22.22). Also,

19.26.26 RC(x2,y2)=RC(a2,ay),
a=(x+y)/2, x0, y>0,

and

19.26.27 RC(x2,x2θ)=2RC(s2,s2θ),
s=x+x2θ, θx2 or s2.