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In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz-stable. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz. It is used in the Routh–Hurwitz stability criterion. ==Notations== Let ''f''(''z'') be a polynomial (with complex coefficients) of degree ''n'' with no roots on the imaginary line (i.e. the line ''Z'' = ''ic'' where ''i'' is the imaginary unit and ''c'' is a real number). Let us define (a polynomial of degree ''n'') and (a nonzero polynomial of degree strictly less than ''n'') by , respectively the real and imaginary parts of ''f'' on the imaginary line. Furthermore, let us denote by: * ''p'' the number of roots of ''f'' in the left half-plane (taking into account multiplicities); * ''q'' the number of roots of ''f'' in the right half-plane (taking into account multiplicities); * the variation of the argument of ''f''(''iy'') when ''y'' runs from −∞ to +∞; * ''w''(''x'') is the number of variations of the generalized Sturm chain obtained from and by applying the Euclidean algorithm; * is the Cauchy index of the rational function ''r'' over the real line. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Routh–Hurwitz theorem」の詳細全文を読む スポンサード リンク
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