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 halfplane 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 lefthalf 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.
Examples
A simple example of a Hurwitz polynomial is the following:
The only real solution is −1, as it factors to:
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 halfplane, it is necessary and sufficient that the polynomial in question pass the RouthHurwitz 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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