By expanding the circular function into a linear superposition of a set of complex Gaussian functions, the approximative analytic expression for the far-field of Gaussian beam with phase aberrations diffracted by a circular aperture is obtained. For different parameters, the diffraction patterns of the expression of the Fraunhofer diffraction are identical, indicating the correctness of the expression. For its relatively simple form, the calculation becomes more convenient. Based on these, the far-field divergence of Gaussian beam with phase aberrations is completely studied. The divergence of the certain beam does not change with the transmission distance, but with the beam-waist radius and the phase aberrations under the limit of circular aperture, and the result indicates that two parameters are the key factors affecting the divergence.