19 Elliptic IntegralsSymmetric Integrals19.14 Reduction of General Elliptic Integrals19.16 Definitions

Elliptic integrals are special cases of a particular multivariate
hypergeometric function called *Lauricella’s* ${F}_{D}$
(Carlson (1961b)). The function
${R}_{-a}\left({b}_{1},{b}_{2},\mathrm{\dots},{b}_{n};{z}_{1},{z}_{2},\mathrm{\dots},{z}_{n}\right)$
(Carlson (1963)) reveals the full permutation symmetry that is
partially hidden in ${F}_{D}$, and leads to symmetric standard integrals
that simplify many aspects of theory, applications, and numerical computation.

Symmetry in $x,y,z$ of ${R}_{F}\left(x,y,z\right)$, ${R}_{G}\left(x,y,z\right)$, and ${R}_{J}\left(x,y,z,p\right)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral.

Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)).

Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. (Compare (19.14.4)–(19.14.10) with (19.29.19), and see the last paragraph of §19.29(i) and the text following (19.29.15).) These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)).

For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). For example, the computation of depolarization factors for solid ellipsoids is simplified considerably; compare (19.33.7) with Cronemeyer (1991).