Digital Library of Mathematical Functions
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19 Elliptic IntegralsLegendre’s Integrals

§19.4 Derivatives and Differential Equations

Contents

§19.4(i) Derivatives

19.4.1 K(k)k =E(k)-k2K(k)kk2,
(E(k)-k2K(k))k =kK(k),
19.4.2 E(k)k =E(k)-K(k)k,
(E(k)-K(k))k =-kE(k)k2,
19.4.3 2E(k)k2=-1kK(k)k=k2K(k)-E(k)k2k2,
19.4.4 Π(α2,k)k=kk2(k2-α2)(E(k)-k2Π(α2,k)).
19.4.5 F(ϕ,k)k=E(ϕ,k)-k2F(ϕ,k)kk2-ksinϕcosϕk21-k2sin2ϕ,
19.4.6 E(ϕ,k)k=E(ϕ,k)-F(ϕ,k)k,
19.4.7 Π(ϕ,α2,k)k=kk2(k2-α2)(E(ϕ,k)-k2Π(ϕ,α2,k)-k2sinϕcosϕ1-k2sin2ϕ).

§19.4(ii) Differential Equations

Let Dk=/k. Then

19.4.8 (kk2Dk2+(1-3k2)Dk-k)F(ϕ,k)=-ksinϕcosϕ(1-k2sin2ϕ)3/2,
19.4.9 (kk2Dk2+k2Dk+k)E(ϕ,k)=ksinϕcosϕ1-k2sin2ϕ.

If ϕ=π/2, then these two equations become hypergeometric differential equations (15.10.1) for K(k) and E(k). An analogous differential equation of third order for Π(ϕ,α2,k) is given in Byrd and Friedman (1971, 118.03).