An efficient root-finder method for the cutoff wave-number resolution in a rectangular partially filled waveguide based upon the Cauchy integral method is presented. The great advantage that this method, valid for lossy dielectric and magnetic materials, has over others is that no initial seed is necessary for the localization of zeros, and also that no roots are lost inside the region under study, which guarantees that all modes are taken into account when the full-wave problem is solved.