Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

We consider the standard symmetric elliptic integral R_F(x,y,z) for complex x , y , z . We derive convergent expansions of R_F(x,y,z) in terms of elementary functions that hold uniformly for one of the three variables x , y or z in closed subsets (possibly unbounded) of ℂ∖(-∞ ,0] . The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.