We consider the third symmetric standard elliptic integral \(R_J(x,y,z,p)\) 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 \(R_J(x,y,z,p)\) 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 \(R_F(x,y,z)\) and \(R_D(x,y,z)\) . These expansions hold uniformly for one or two of the variables in large closed unbounded subsets of \(\mathbb {C}\setminus (-\infty ,0]\) . These expansions are accompanied by error bounds, and their accuracy and uniform features are illustrated by means of some numerical experiments.
