A transcendental equation occurs when we compute the dispersion relations of an electromagnetic waveguide, such as a planar multilayer waveguide. Without an initial guess, the Sakurai-Sugiura projection method (SSM) can obtain solutions to the transcendental equation in a region bounded by a contour integral path in the complex plane. In this paper, a criterion employing the condition number of eigenvalues as a simple index to distinguish physical solutions from spurious ones in the SSM is presented, and a transcendental equation of a multilayer waveguide obtained by the transfer matrix method is solved by the SSM. Numerical results show the usefulness of the index and good agreement with the results of the argument principle method and Newton's method.