The asymptotic expressions for the gain in the free electron lasers (FELs), which operate in the strong-pump Compton regime and in the collective Raman regime, and which employ a sheet electron beam and a cylindrical electron beam, are presented. For each case we identify a coupling parameter that measures the strength of the interaction. Large values of the coupling parameter correspond to strong optical guiding, while small values correspond to large diffraction. When the coupling parameter is small, the scaling of the gain is different from that of the one-dimensional theory because of diffraction losses. The particular scaling of the gain when diffraction losses are large depends both on the regime of operation and on the beam geometry. We give asymptotic expressions for the gain when diffraction is large for each case. This linear analysis is valid when the signal is small and is useful mainly when the gain is high prior to saturation.