  The Gaudin integral equation for the ground state of a one-dimensional
delta-function attractive spin-1/2 fermions is solved in the form of power
series. The first few terms of the asymptotic expansions for both strong and
weak coupling cases are calculated analytically. The physical quantities such
as the ground state energy are expressed in terms of a single dimensionless
parameter gamma =c/D, where c is the coupling constant and D is the number
density. The results agree with those obtained from the perturbation
calculations, which include the one in the classical electrostatics originally
by Kirchhoff. In the strong coupling limit, the connection to the solutions of
the Lieb-Liniger integral equation for the ground state of a one-dimensional
delta-function repulsive bose gas is shown explicitly.
