Editing Hurwitz polynomial (section)

{{short description|Polynomial whose complex roots have non-positive real parts}}

In [[mathematics]], a '''Hurwitz polynomial''' (named after German mathematician  [[Adolf Hurwitz]]) is a [[polynomial]]  whose [[root of a function|roots]] (zeros) are located in the left half-plane of the [[complex plane]] or on the [[imaginary axis]], that is, the [[complex number|real part]] of every root is zero or negative.<ref name="Kuo">{{cite book   
  | last = Kuo
  | first = Franklin F.
  | title = Network Analysis and Synthesis, 2nd Ed.
  | publisher = John Wiley & Sons
  | date = 1966
  | pages = 295–296
  | isbn = 0471511188}}</ref>  Such a polynomial must have [[coefficient]]s that are positive [[real number]]s. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary 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
  | access-date = July 3, 2013}}</ref><ref name="Reddy">{{cite conference
  | first = Hari C.
  | last = Reddy
  | title = Theory of two-dimensional Hurwitz polynomials
  | book-title = The Circuits and Filters Handbook, 2nd Ed.
  | pages = 260–263
  | publisher = CRC Press
  | date = 2002
  | url = https://books.google.com/books?id=SmDImt1zHXkC&dq=hurwitz+polynomial&pg=PA262
  | isbn = 1420041401
  | access-date = July 3, 2013}}</ref>

A polynomial function {{math|''P''(''s'')}} of a [[complex variable]] {{mvar|s}} is said to be Hurwitz if the following conditions are satisfied:

# {{math|''P''(''s'')}} is real when {{mvar|s}} is real.
# The roots of {{math|''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]].