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In [[mathematics]], a '''Hurwitz polynomial''', named after [[Adolf Hurwitz]], is a [[polynomial]] whose coefficients are positive [[real number]]s and whose [[root of a function|roots]] ([[zero (complex analysis)|zeros]]) are located in the left half-plane of the [[complex number|complex plane]] or on the ''jω'' axis, that is, the real part of every root is zero or negative.<ref name="Kuo">{{cite book 
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  | last = Kuo
  | first = Franklin F.
  | authorlink =
  | coauthors =
  | title = Network Analysis and Synthesis, 2nd Ed.
  | publisher = John Wiley & Sons
  | date = 1966
  | location =
  | pages = 295-296
  | url =
  | doi =
  | id =
  | isbn = 0471511188}}</ref>  The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the axis (i.e., a Hurwitz [[stable polynomial]]).<ref name=" Weisstein">{{cite web
  | last =  Weisstein
  | first = Eric W
  | title = Hurwitz polynomial
  | work = Wolfram Mathworld
  | publisher = Wolfram Research
  | date = 1999
  | url = http://mathworld.wolfram.com/HurwitzPolynomial.html
  | format =
  | doi =
  | accessdate = July 3, 2013}}</ref><ref name="Reddy">{{cite conference
  | first = Hari C.
  | last = Reddy
  | title = Theory of two-dimensional Hurwitz polynomials
  | booktitle = The Circuits and Filters Handbook, 2nd Ed.
  | pages = 260-263
  | publisher = CRC Press
  | date = 2002
  | location =
  | url = http://books.google.com/books?id=SmDImt1zHXkC&pg=PA262&dq=hurwitz+polynomial
  | doi =
  | id = ISBN 1420041401
  | accessdate = July 3, 2013}}</ref>


A polynomial function ''P''(''s'') of a [[complex variable]] ''s'' is said to be Hurwitz if the following conditions are satisfied:
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:1. ''P''(''s'') is real when ''s'' is real.
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:2. The roots of ''P''(''s'') have real parts which are zero or negative.


Hurwitz polynomials are important in [[control system|control systems theory]], because they represent the [[Characteristic polynomial#Characteristic equation|characteristic equations]] of [[Stability theory|stable]] [[linear system]]s. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the [[Routh-Hurwitz stability criterion]].
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== Examples ==
<ul>
 
 
A simple example of a Hurwitz polynomial is the following:
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:<math>x^2 + 2x + 1.</math>
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The only real solution is &minus;1, as it factors to
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:<math>(x+1)^2.</math>
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== Properties ==
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For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. A necessary and sufficient condition that a polynomial is Hurwitz is that it pass the [[Routh-Hurwitz stability criterion]]. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.
</ul>
 
The properties of Hurwitz polynomials are:
 
# All the [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeros]] are in the left half plane or on its boundary, the imaginary axis.
# Any poles and zeros on the imaginary axis are simple (have a multiplicity of one).
# 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.
# 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).
# The polynomial should not have missing powers of s.
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Hurwitz Polynomial}}
[[Category:Polynomials]]
 
{{mathanalysis-stub}}

Revision as of 19:34, 10 February 2014

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