Abstract We consider the third symmetric standard elliptic integral RJ (x, y, z, p) R J (x, y, z, p) for complex values of its variables. By homogeneity arguments, this function is indeed a function of only three variables, and we derive two different integral representations of RJ (x, y, z, p) R J (x, y, z, p) which only involve three variables. Both integral representations are suitable for the analysis introduced in Lopez, Pagola and Palacios, 2021 to derive uniform expansions of parametric integrals. Using this theory, we derive six convergent expansions of this function in terms of elementary functions; two of these expansions also involve the other two symmetric standard elliptic integrals RF (x, y, z) R F (x, y, z) and RD (x, y, z) R D (x, y, z). These expansions hold uniformly for one or two of the variables in large closed unbounded subsets of C (-, 0] C \ (- ∞, 0 ]. These expansions are accompanied by error bounds, and their accuracy and uniform features are illustrated by means of some numerical experiments.
Bujanda et al. (Wed,) studied this question.