### abstract

- The order of the pole singularities encountered in spectral method of moments formulations for source-free periodic problems is investigated. The solution of the source-free problem is often obtained by searching for the zeros of the Z matrix determinant using an iterative algorithm. During this process, pole singularities of the determinant are encountered and may cause numerical instability. In order to cancel the poles, their order must be known. A rigorous proof of the pole singularity order in the Z matrix determinant is given. The proof is general and holds for any problem which is periodic in at least one of the spatial directions. This knowledge enables to cancel the poles by an appropriate fixed factor with a simple routine.