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.

Examples [

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.

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 zeros 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 zeros 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'but it possible after the testing the prf stability

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