  We present our investigation of the study of two variable hypergeometric
series, namely Appell $F_{1}$ and $F_{3}$ series, and obtain a comprehensive
list of its analytic continuations enough to cover the whole real $(x,y)$
plane, except on their singular loci. We also derive analytic continuations of
their 3-variable generalization, the Lauricella $F_{D}^{(3)}$ series and the
Lauricella-Saran $F_{S}^{(3)}$ series, leveraging the analytic continuations of
$F_{1}$ and $F_{3}$, which ensures that the whole real $(x,y,z)$ space is
covered, except on the singular loci of these functions. While these studies
are motivated by the frequent occurrence of these multivariable hypergeometric
functions in Feynman integral evaluation, they can also be used whenever they
appear in other branches of mathematical physics. To facilitate their practical
use, we provide four packages: $\texttt{AppellF1.wl}$, $\texttt{AppellF3.wl}$,
$\texttt{LauricellaFD.wl}$, and $\texttt{LauricellaSaranFS.wl}$ in
$\textit{MATHEMATICA}$. These packages are applicable for generic as well as
non-generic values of parameters, keeping in mind their utilities in the
evaluation of the Feynman integrals. We explicitly present various physical
applications of these packages in the context of Feynman integral evaluation
and compare the results using other means as well. Various
$\textit{MATHEMATICA}$ notebooks demonstrating different numerical results are
also provided along with this paper.
