  In this paper, we present the rigorous expression of the ground state energy
and study the phase transition of the three-dimensional homogeneous electron
gas by the eigenfunctional theory. The ground state energy is decided
completely by the pair distribution function $ g(r)$, which is, by definition,
strictly a positive function. But when the density decreases, the electronic
correction effect becomes strong from weak, the previous theories basing on
one-particle approximation, such as: RPA, Hubbard and STLS ,\cite{4,9} can't
insure $g(r)$ positive always, which implies that they are becoming invalid and
overestimate the ground state energy. The eigenfunctional theory has a
significant improvement over them, under the linear approximation in solving
the equation of the phase field, the $g(r)$ obtained by the eigenfunctional
theory is always positive and can well satisfy the normalization integral
($n_{0}\int d^3r[g(r)-1]=-1$) which is one of the important features of $g(r)$.
After obtaining $g(r)$ and calculating the ground state energy of the electron
gas both in paramagnetic and ferromagnetic by $g(r)$, we observe a continuous
phase transition from paramagnetic to the ferromagnetic occurring at
$r_s=19.9\pm0.8$. This can be indirectly supported by the observation of a
ferromagnetic state in doped hexaboride $(Ca_{1-x}La_xB_b)$,\cite{42} and is
close to the result of G.Ortiz et.al ($r_s=20\pm5$).\cite{12}
