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In this paper, elliptical cracks and rectangular cracks embedded in a three-dimensional infinite transversely isotropic piezoelectric solid are analyzed under combined mechanical tension and electric fields. The hypersingular integral equation method is used to solve the mentioned problems. The unknown function in the hypersingular integral equations is approximated with a product of the fundamental density function and polynomials. The hypersingular integrals can be numerically evaluated by using a method of Taylor series expansion. Therefore, the hypersingular integral equations for the crack problems can be solved immediately. Finally, numerical examples of the stress and electric displacement intensity factors as well as the energy release rates for these crack configurations are presented. The numerical results demonstrate the present approach to be very efficient.