An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

We derive new convergent expansions of the symmetric standard elliptic integral $$R_D(x,y,z)$$ , for $$x, y,z\in {\mathbb {C}}{\setminus }(-\infty ,0]$$ , in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which $$R_D(x,y,z)$$ is a particular case. It turns out that this parametric integral is an integral representation of the Appell function $$F_1(a;b,c;a+1;x,y)$$ . Therefore, as a byproduct, we deduce convergent expansions of $$F_1(a;b,c;a+1;x,y)$$ . We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.