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19 Elliptic IntegralsSymmetric Integrals

§19.26 Addition Theorems

Contents

§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=λ-2((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+ηζ-ξηξηζ+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+ηζ-ξη=ξζ+ηζ-ξη.) Equivalent forms of (19.26.2) are given by

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

and

19.26.6 (λμ-xy-xz-yz)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(γ-δ,γ),

where

19.26.10 γ =p(p+λ)(p+μ),
δ =(p-x)(p-y)(p-z).

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 =λ-2(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,α2-θ)+RC(β2,β2-θ)=RC(σ2,σ2-θ),
σ=(αβ+θ)/(α+β),

where 0<γ2-θ<γ2 for γ=α,β,σ, except that σ2-θ can be 0, and

19.26.14 (p-y)RC(x,p)+(q-y)RC(x,q)=(η-ξ)RC(ξ,η),
x0, y0; p,q\{0},

where

19.26.15 (p-x)(q-x) =(y-x)2,
ξ =y2/x,
η =pq/x,
η-ξ =p+q-2y.

§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)-x-y-z.
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),
β2-α2 =(p-x)(p-y)(p-z),

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+ηζ-ξη)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,x2-θ)=2RC(s2,s2-θ),
s=x+x2-θ, θx2 or s2.