Based on the generalized Huygens-Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical expressions of standard and elegant Laguerre-Gaussian beams passing through a paraxial ABCD optical system with a hard-edged aperture are derived. On the basis of the truncated second-order moments method in the cylindrical coordinate system and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, an approximate method for calculating the generalized beam propagation factor is proposed. Closed-form expressions for the generalized beam propagation factor of a truncated standard and elegant Laguerre-Gaussian beam are derived that depend on the orders of Laguerre polynomial and the beam truncation parameter. Some typical numerical examples are given and compared. It is shown that the results obtained by using the obtained analytical method accord with those by using the numerical integration method.© (2004) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.