Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $\mathop{F_{D}\/}\nolimits$ (Carlson (1961b)). The function $\mathop{R_{-a}\/}\nolimits\!\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_% {n}\right)$ (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in $\mathop{F_{D}\/}\nolimits$, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation.
Symmetry in $x,y,z$ of $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$, and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ 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.