  The transformation theory of the Appell $F_2(a,b_1,b_2;c_1,c_2;x,y)$ double
hypergeometric function is used to obtain a set of series representations of
$F_2$ which provide an efficient way to evaluate $F_2$ for real values of its
arguments $x$ and $y$ and generic complex values of its parameters $a,b_1, b_2,
c_1$ and $c_2$ (i.e. in the nonlogarithmic case). This study rests on a
classical approach where the usual double series representation of $F_2$ and
other double hypergeometric series that appear in the intermediate steps of the
calculations are written as infinite sums of one variable hypergeometric
series, such as the Gauss $_2F_1$ or the $_3F_2$, various linear
transformations of the latter being then applied to derive known and new
formulas. Using the three well-known Euler transformations of $F_2$ on these
results allows us to obtain a total of 44 series which form the basis of the
Mathematica package AppellF2, dedicated to the evaluation of $F_2$. A brief
description of the package and of the numerical analysis that we have performed
to test it are also presented.
