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 RF (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\(-infinity, 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.