  The FEL gain formulas for non-resonant case are studied. For the
mono-energetic and non-resonant electron beam, the exact expression of the
solution of the FEL characteristic cubic equation is obtained with a form much
more simple than that in the literatures, and the gain length as the function
of the detuning parameter is explicitly given, then the gain for different
detuning parameter and from low to high can be easily calculated. A simplified
approximation formula is also given for the exponential gain calculation in the
non-resonant case. For the case of the electron beam with an energy spread, the
solution of the characteristic cubic equation is given explicitly for
rectangular energy distribution and Lorentz distribution, respectively.
Moreover the explicit expression also can be used for the solution of the
characteristic cubic equation including the impact of the space charge. The
transition from the low gain to the high gain is analyzed. The variations of
the gain bandwidth and of the detuning parameter for the maximum gain are
demonstrated. The applicable ranges of the small signal gain formula and the
exponential gain formula are analyzed.
