Routh-Hurwitz theorem

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.

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 ${\displaystyle P_{0}(y)}$ (a polynomial of degree n) and ${\displaystyle P_{1}(y)}$ (a nonzero polynomial of degree strictly less than n) by ${\displaystyle f(iy)=P_{0}(y)+iP_{1}(y)}$, respectively the real and imaginary parts of f on the imaginary line.

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