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In this paper, we present a numerical computation of the energy levels and the corresponding wave functions in a microwave resonator using the multipole expansion technique. The approach permits closed form, fast, and robust solutions of the Helmholtz equation (and the Schrodinger equation for two-dimensional systems) in an important class of wave chaos problem. In particular, wave functions inside the billiard are expressed in terms of a simple expansion of Hankel functions. The implementation of the approach is described, and the classical bowtie cavity is considered as a case study to demonstrate the versatility and efficiency of the method. To validate the accuracy, the field distribution and the eigenvalues calculated using this approach are compared to the solution obtained by boundary integral method. The case when the cavity contains objects (perfect electric conductors and/or dielectrics) is also presented and discussed.