The work is an extensive study of the M-solutions (singular at ) which is one of the three different types of radial positive solutions of the general Matukuma equation , , , . We apply an asymptotic expansion method which begins with rough estimate and improves the accuracy step by step to the desired extend. Comparing with the case when (see Batt and Li, 2010 [5]), such asymptotic expansions require more delicate study.