# Hurwitz polynomial

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Hurwitz polynomial

In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. One sometimes uses the term Hurwitz polynomial simply as a (real or complex) polynomial with all zeros in the left-half plane (i.e., a Hurwitz stable polynomial).

Examples

A simple example of a Hurwitz polynomial is the following:

:$x^2 + 2x + 1.$

The only real solution is −1, as it factors to:

:$\left(x+1\right)^2.$

Properties

For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion.A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique.

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