What's New
About the Project
NIST
19 Elliptic IntegralsLegendre’s Integrals

§19.11 Addition Theorems

Contents

§19.11(i) General Formulas

19.11.1 F(θ,k)+F(ϕ,k)=F(ψ,k),
19.11.2 E(θ,k)+E(ϕ,k)=E(ψ,k)+k2sinθsinϕsinψ.

Here

19.11.3 sinψ =(sinθcosϕ)Δ(ϕ)+(sinϕcosθ)Δ(θ)1-k2sin2θsin2ϕ,
Δ(θ) =1-k2sin2θ.

Also,

19.11.4 cosψ =cosθcosϕ-(sinθsinϕ)Δ(θ)Δ(ϕ)1-k2sin2θsin2ϕ,
tan(12ψ) =(sinθ)Δ(ϕ)+(sinϕ)Δ(θ)cosθ+cosϕ.

Lastly,

19.11.5 Π(θ,α2,k)+Π(ϕ,α2,k)=Π(ψ,α2,k)-α2RC(γ-δ,γ),

where

19.11.6 γ =((csc2θ)-α2)((csc2ϕ)-α2)((csc2ψ)-α2),
δ =α2(1-α2)(α2-k2).

§19.11(ii) Case ψ=π/2

19.11.7 F(ϕ,k)=K(k)-F(θ,k),
19.11.8 E(ϕ,k)=E(k)-E(θ,k)+k2sinθsinϕ,

where

19.11.9 tanθ=1/(ktanϕ).
19.11.10 Π(ϕ,α2,k)=Π(α2,k)-Π(θ,α2,k)-α2RC(γ-δ,γ),

where

19.11.11 γ =(1-α2)((csc2θ)-α2)((csc2ϕ)-α2),
δ =α2(1-α2)(α2-k2).

§19.11(iii) Duplication Formulas

If ϕ=θ in §19.11(i) and Δ(θ) is again defined by (19.11.3), then

19.11.12 F(ψ,k)=2F(θ,k),
19.11.13 E(ψ,k)=2E(θ,k)-k2sin2θsinψ,
19.11.14 sinψ=(sin2θ)Δ(θ)/(1-k2sin4θ),
19.11.15 cosψ =(cos(2θ)+k2sin4θ)/(1-k2sin4θ),
tan(12ψ) =(tanθ)Δ(θ),
sinθ =(sinψ)/(1+cosψ)(1+Δ(ψ)),
cosθ =(cosψ)+Δ(ψ)1+Δ(ψ),
tanθ =tan(12ψ)1+cosψ(cosψ)+Δ(ψ),
19.11.16 Π(ψ,α2,k)=2Π(θ,α2,k)+α2RC(γ-δ,γ),
19.11.17 γ =((csc2θ)-α2)2((csc2ψ)-α2),
δ =α2(1-α2)(α2-k2).