Symmetry in c, d, n of Jacobian elliptic functions

AbstractThe relation connecting the symmetric elliptic integral RF with the Jacobian elliptic functions is symmetric in the first three of the four letters c, d, n, and s that are used in ordered pairs to name the 12 functions. A symbol Δ(p,q)=ps2(u,k)−qs2(u,k), p,q∈{c,d,n}, is independent of u and allows formulas for differentiation, bisection, duplication, and addition to remain valid when c, d, and n are permuted. The five transformations of first order, which change the argument and modulus of the functions, take a unified form in which they correspond to the five nontrivial permutations of c, d, and n. There are 18 transformations of second order (including Landen's and Gauss's transformations) comprising three sets of six. The sets are related by permutations of the original functions cs, ds, and ns, and there are only three sets because each set is symmetric in two of these. The six second-order transformations in each set are related by first-order transformations of the transformed functions, and all 18 take a unified form. All results are derived from properties of RF without invoking Weierstrass functions or theta functions