The Hurwitz(p, z) function determines whether the the polynomial  has all its zeros strictly in the left half plane.

A polynomial is a Hurwitz polynomial if all its roots are in the left half plane.

The parameter  is a polynomial with complex coefficients. The polynomial may have symbolic parameters, which evalc and Hurwitz assume to be real.  The paraconjugate   of  is defined as the polynomial whose roots are the roots of  reflected across the imaginary axis.

The parameter 's', if specified, is a name to which the sequence of partial fractions of the Stieltjes continued fraction of  will be assigned. The first element of the sequence returned in 's' is special. If it is of higher degree than  in ,  is not Hurwitz. If it is of the form , where ,  is not Hurwitz, either. If each subsequent polynomial in the sequence returned is of the form , where , then  is a Hurwitz polynomial.

This is useful if  has symbolic coefficients. You can decide the ranges of the coefficients that make  Hurwitz.

If the Hurwitz function can use the previous rules to determine that  is Hurwitz, it returns true. If it can decide that  is not Hurwitz, it returns false. Otherwise, it returns FAIL.

The parameter 'g', if specified, is a name to which the gcd of  and its paraconjugate   will be assigned. The zeros of this gcd are precisely the zeros of  which are symmetrical under reflection across the imaginary axis.

If the gcd is  while the sequence of partial fractions is empty, the conditions for being a Hurwitz polynomial are trivially satisfied. A manual check is recommended, though a warning is returned only if infolevel[Hurwitz] >= 1.

Levinson, Norman, and Redheffer, Raymond M. Complex Variables. Holden-Day, 1970.