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In this paper, dynamic stiffness and flexibility for circular membranes are analytically derivedusing an efficient mixed-part dual boundary element method (BEM). We employ three approaches, thecomplex-valued BEM, the real-part and imaginary-part BEM, to determine the dynamic stiffness andflexibility. In the analytical formulation, the continuous system for a circular membrane is transformedinto a discrete system with a circulant matrix. Based on the properties of the circulant, the analyticalsolutions for the dynamic stiffness and flexibility are derived. In deriving the stiffness and flexibility, thespurious resonance is cancelled out. Numerical aspects are discussed and emphasized. The problem ofnumerical instability due to division by zero is avoided by choosing additional constraints from theinformation of real and imaginary parts in the dual formulation. For the overdetermined system, the leastsquares method is considered to determine the dynamic stiffness and flexibility. A general purposeprogram has been developed to test several examples including circular and square cases.