Root distribution of a polynomial in subregions of complex plane

Modified processes are proposed for directly treating singularities in the Routh array, and procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. The extended Sturm test is developed for determining the root distribution on some specified lines. Moreover, to enhance the Routh stability theorem, the extended Routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed