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In the analysis of scattering by perfectly conducting cylinders with polygonal cross-section by means of surface integral operator formulations, fast convergence can be achieved by expanding the surface current density on each side with basis functions factorizing the correct behavior of the fields on the wedges. Usually, the factorized edge behavior is chosen to be coincident with the first order behavior prescribed by Meixner's theory. However, it could not be the correct one and, consequently, the convergence of the method becomes increasingly slow as the theoretical behavior differs from the real one. This phenomenon is particularly notable when one or more of the predicted singularities unexpectedly disappear. To overcome this problem, in this work the analysis of TM scattering is made by introducing a new expansion devised so that only the first two terms are responsible for the reconstruction of the singularities while the remaining part of the expansion factorizes the second order edge behavior. Actually, the proposed expansion outperforms the one introduced by the authors in a previous work factorizing the first order edge behavior even when this is the correct one.