A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di?erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di?erent from the previous research too, that the orthogonal properties of the material characteristic matrices A and B are no longer necessary in obtaining the POP of EEF in cracked piezoelectric materials. Of the greatest signi?cance is that the method presented in this paper can be widely extended to treat many kinds of problems concerning path- independent integrals with multi-variables. A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di? Erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di? erent from the previous research too, that the orthogonal properties of the material characteristic matrices A and B are no longer necessary in obtaining the POP of EEF in the cracked piezoelectric materials. Of the greatest signi? cance is that the method presented in this paper can be widely extended to to treat many kinds of problems concerning path- independent integrals with multi-variables.