Hurwitz polynomial

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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).

A polynomial is said to be Hurwitz if the following conditions are satisfied:

1. P(s) is real when s is real

2. The roots of P(s) have real parts which are zero or negative.

  • Note: Here P(s) is any polynomial in s.


A simple example of a Hurwitz polynomial is the following:

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


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.

1. All the poles and zeros of a function are in the left half plane or on its boundary the imaginary axis. 2. Any poles and zeroes on the imaginary axis are simple (have a multiplicity of one). 3. Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative. 4. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle). 5. there have no any missing term of 's'

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