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EspañolHURWITZ 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).
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.
| Contents |
| Examples |
| Properties |
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 half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion.
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