This paper presents the generalisation of the complex-source-point method for all the ‘series A’ and ‘series B’ Laguerre-Gaussian beams. When complex values are assigned to the source co-ordinates in the field expressions of point multipoles, the resulting fields have paraxially the form of a Laguerre-Gaussian beam. An application of this technique is found in deducing a generalised resonant formula for the spherical open resonator with a ‘large aperture’. The result is more accurate than the conventional one in order of magnitudes. Moreover, the small difference in resonant frequencies between the ‘series A’ and the ‘series B’ modes of the same order when l > 0 confirms the nonexistence of the commonly accepted linearly polarised modes with angular dependence. The theoretical pedictions are substantiated by experimental work.