In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative.
[2][3] A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied: Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems.
This follows directly from the quadratic formula: where, if the discriminant b2−4ac is less than zero, then the polynomial will have two complex-conjugate solutions with real part −b/2a, which is negative for positive a and b.
A necessary and sufficient condition that a polynomial is Hurwitz is that it passes 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.