  Obtaining exact solutions of the spherically symmetric general relativistic
gravitational field equations describing the interior structure of an isotropic
fluid sphere is a long standing problem in theoretical and mathematical
physics. The usual approach to this problem consists mainly in the numerical
investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity
equations, which describes the hydrostatic stability of the dense stars. In the
present paper we introduce an alternative approach for the study of the
relativistic fluid sphere, based on the relativistic mass equation, obtained by
eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation.
Despite its apparent complexity, the relativistic mass equation can be solved
exactly by using a power series representation for the mass, and the Cauchy
convolution for infinite power series. We obtain exact series solutions for
general relativistic dense astrophysical objects described by the linear
barotropic and the polytropic equations of state, respectively. For the
polytropic case we obtain the exact power series solution corresponding to
arbitrary values of the polytropic index $n$. The explicit form of the solution
is presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and
$n=1/5$, respectively. The case of $n=3$ is also considered. In each case the
exact power series solution is compared with the exact numerical solutions,
which are reproduced by the power series solutions truncated to seven terms
only. The power series representations of the geometric and physical properties
of the linear barotropic and polytropic stars are also obtained.
