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The frequency spectrum of a piezoelectric hollow resonator cylinder is studied by using Legendre polynomials. Formulation is given that allows to take into account the electric source. Normal frequencies and electric input impedance are obtained easily regardless of the type of modes, thickness extensional (TE), thickness shear (TS), edge (E) or radial (R). These quantities are calculated as a function of the diameter to thickness ratio and the results are compared against other calculations from an analytical approach. The numerical results for harmonic and modal analyses such as normalized electric input impedance, the resonance and antiresonance frequencies are presented and compared with the analytical results in order to check the accuracy and range of applicability of the proposed approach. Agreement is obtained.