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19 Elliptic IntegralsLegendre’s Integrals

§19.7 Connection Formulas


§19.7(i) Complete Integrals of the First and Second Kinds

Legendre’s Relation


19.7.2 K(ik/k) =kK(k),
K(-ik/k) =kK(k),
E(ik/k) =(1/k)E(k),
E(-ik/k) =(1/k)E(k).
19.7.3 K(1/k) =k(K(k)iK(k)),
K(1/k) =k(K(k)±iK(k)),
E(1/k) =(1/k)(E(k)±iE(k)-k2K(k)ik2K(k)),
E(1/k) =(1/k)(E(k)iE(k)-k2K(k)±ik2K(k)),

where upper signs apply if k2>0 and lower signs if k2<0. This dichotomy of signs (missing in several references) is due to Fettis (1970).

§19.7(ii) Change of Modulus and Amplitude

See also (19.2.10).

Reciprocal-Modulus Transformation

19.7.4 F(ϕ,k1) =kF(β,k),
E(ϕ,k1) =(E(β,k)-k2F(β,k))/k,
Π(ϕ,α2,k1) =kΠ(β,k2α2,k),
k1=1/k, sinβ=k1sinϕ1.

Provided the functions in these identities are correctly analytically continued in the complex β-plane, then the identities will also hold in the complex β-plane.

Imaginary-Modulus Transformation

19.7.5 F(ϕ,ik) =κF(θ,κ),
E(ϕ,ik) =(1/κ)(E(θ,κ)-κ2(sinθcosθ)(1-κ2sin2θ)-1/2),
Π(ϕ,α2,ik) =(κ/α12)(κ2F(θ,κ)+κ2α2Π(θ,α12,κ)),


19.7.6 κ =k1+k2,
κ =11+k2,
sinθ =1+k2sinϕ1+k2sin2ϕ,
α12 =α2+k21+k2.

Imaginary-Argument Transformation

With sinhϕ=tanψ,

19.7.7 F(iϕ,k) =iF(ψ,k),
E(iϕ,k) =i(F(ψ,k)-E(ψ,k)+(tanψ)1-k2sin2ψ),
Π(iϕ,α2,k) =i(F(ψ,k)-α2Π(ψ,1-α2,k))/(1-α2).

For two further transformations of this type see Erdélyi et al. (1953b, p. 316).

§19.7(iii) Change of Parameter of Π(ϕ,α2,k)

There are three relations connecting Π(ϕ,α2,k) and Π(ϕ,ω2,k), where ω2 is a rational function of α2. If k2 and α2 are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)).

The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π(ϕ,α2,k) when α2>csc2ϕ (see (19.6.5) for the complete case). Let c=csc2ϕα2. Then

19.7.8 Π(ϕ,α2,k)+Π(ϕ,ω2,k)=F(ϕ,k)+cRC((c-1)(c-k2),(c-α2)(c-ω2)),

Since k2c we have α2ω2c; hence α2>c implies ω2<1c.

The second relation maps each hyperbolic region onto itself and each circular region onto the other:

19.7.9 (k2-α2)Π(ϕ,α2,k)+(k2-ω2)Π(ϕ,ω2,k)=k2F(ϕ,k)-α2ω2c-1RC(c(c-k2),(c-α2)(c-ω2)),

The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other:

19.7.10 (1-α2)Π(ϕ,α2,k)+(1-ω2)Π(ϕ,ω2,k)=F(ϕ,k)+(1-α2-ω2)c-k2RC(c(c-1),(c-α2)(c-ω2)),