  Free-electron lasers play an increasing role in science, from generating
unique femtosecond X- ray pulses for single short recording of the protein
structures to amplifying feeble interactions in advanced cooling systems for
high-energy hadron colliders. While modern Free-electron laser codes can
describe their amplification mechanism, a deep analytical understanding of the
mechanism is of extreme importance for a number of applications. Mode
competition, their growth rates and amplification bandwidth are among the most
important parameters of a free-electron laser. A dispersion relation, which
defines these important characteristics, can be solved analytically only for a
very few simple cases. In this letter we show that for a typical bell-shape
energy distribution in electron beam there is no more that one growing mode. We
also derive an analytical expression which determines the bandwidth of the
free-electron laser.
