On polynomial root distribution with respect to a sector

This paper extends previous results of the same authors on the determination of the polynomial root distribution with respect to a sector by means of elementary vector analysis. Specifically, it is shown how the overall phase variation of any real or complex polynomial along the radii of a sector accounts for the number of roots inside and outside the sector. The method applies to both symmetric and asymmetric sectors with respect to the real axis. Its practical application only requires plotting a Nyquist-like diagram (a hodograph). The procedure proves particularly useful in the stability analysis of fractional-order systems. A pair of examples is worked out to show how the method operates.