  The Fredholm equations for one-dimensional two-component Fermions with
repulsive and with attractive delta-function interactions are solved by an
asymptotic expansion for A) strong repulsion, B) weak repulsion, C) weak
attraction and D) strong attraction. Consequently, we obtain the first few
terms of the expansion of ground state energy for the Fermi gas with
polarization for these regimes. We also prove that the two sets of the Fredhom
equations for weakly repulsive and attractive interactions are identical as
long as the integration boundaries match each other between the two sides. Thus
the asymptotic expansions of the energies of the repulsive and attractive
Fermions are identical to all orders in this region. The identity of the
asymptotic expansions may not mean that the energy analytically connects.
