Uniform Expansions of the Third Symmetric Standard Elliptic Integral

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.