The Hurwitz(p, z) function determines whether the polynomial $p\left(z\right)$ 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 $p$ is a polynomial with complex coefficients. The polynomial may have symbolic parameters, which evalc and Hurwitz assume to be real.  The paraconjugate  $p^{\*}$ of $p$ is defined as the polynomial whose roots are the roots of $p$ 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 $\frac{\left(p-p^{\&ast;}\right)}{\left(p\&plus;p^{\&ast;}\right)}$ will be assigned. The first element of the sequence returned in 's' is special. If it is of higher degree than $1$ in $z$, $p$ is not Hurwitz. If it is of the form $bz+a$, where $\mathrm{\Re }\left(a\right)\ne 0\mathbf{or}b<0$, $p$ is not Hurwitz, either. If each subsequent polynomial in the sequence returned is of the form $bz+a$, where $\mathrm{\Re }\left(a\right)=0\mathbf{and}0<b$, then $p$ is a Hurwitz polynomial.

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

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

The parameter 'g', if specified, is a name to which the gcd of $p$ and its paraconjugate  $p^{\&ast;}$ will be assigned. The zeros of this gcd are precisely the zeros of $p$ which are symmetrical under reflection across the imaginary axis.

If the gcd is $1$ 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.

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\&colon;$

$\mathrm{p1}≔z^2+z+1$

$\mathrm{p1}≔z^2+z+1$

$\mathrm{Hurwitz}\left(\mathrm{p1}\&comma;z\right)$

$\mathrm{true}$

$\mathrm{p2}≔3z^3+2z^2+z+c$

$\mathrm{p2}≔3z^3+2z^2+c+z$

$\mathrm{Hurwitz}\left(\mathrm{p2}\&comma;z\&comma;'\mathrm{s2}'\&comma;'\mathrm{g2}'\right)$

$\mathrm{FAIL}$

$\mathrm{s2}$

$\left[\frac{3z}{2}\&comma;-\frac{4z}{3c-2}\&comma;-\frac{3z}{2}+\frac{z}{c}\right]$

$\mathrm{g2}$

$1$

$\mathrm{p3}≔4z^4+z^3+z^2+c$

$\mathrm{p3}≔4z^4+z^3+z^2+c$

$\mathrm{Hurwitz}\left(\mathrm{p3}\&comma;z\&comma;'\mathrm{s3}'\&comma;'\mathrm{g3}'\right)$

$\mathrm{FAIL}$

$\mathrm{s3}$

$\left[0\&comma;4z\&comma;z\&comma;-\frac{z}{c}\&comma;-z\right]$

$\mathrm{p4}≔z^5+5z^4+4z^3+3z^2+2z+c$

$\mathrm{p4}≔z^5+5z^4+4z^3+3z^2+c+2z$

$\mathrm{Hurwitz}\left(\mathrm{p4}\&comma;z\&comma;'\mathrm{s4}'\&comma;'\mathrm{g4}'\right)$

$\mathrm{FAIL}$

$\mathrm{s4}$

$\left[\frac{z}{5}\&comma;\frac{25z}{17}\&comma;\frac{289z}{5\left(5c+1\right)}\&comma;-\frac{{\left(5c+1\right)}^{2}z}{17\left(c^2+48c-2\right)}\&comma;\frac{\left(-c^2-48c+2\right)z}{\left(5c+1\right)c}\right]$

$\mathrm{p5}≔\mathrm{p2}+\mathrm{I}d$

$\mathrm{p5}≔3z^3+2z^2+c+z+\mathrm{I}d$

$\mathrm{Hurwitz}\left(\mathrm{p5}\&comma;z\&comma;'\mathrm{s5}'\&comma;'\mathrm{g5}'\right)$

$\mathrm{FAIL}$

$\mathrm{s5}$

$\left[\frac{3z}{2}\&comma;-\frac{4z}{3c-2}-\frac{8\mathrm{I}d}{{\left(3c-2\right)}^2}\&comma;-\frac{{\left(3c-2\right)}^{3}z}{2\left(9c^3-12c^2-8d^2+4c\right)}+\frac{\mathrm{I}d{\left(3c-2\right)}^2}{9c^3-12c^2-8d^2+4c}\right]$

$\mathrm{p6}≔\mathrm{expand}\left(\left(x-1\right)\left(x^2+2\right)\left(x-c\right)\right)$

$\mathrm{p6}≔-c{x}^3+x^4+c{x}^2-x^3-2cx+2x^2+2c-2x$

$\mathrm{Hurwitz}\left(\mathrm{p6}\&comma;x\&comma;'\mathrm{s6}'\&comma;'\mathrm{g6}'\right)$

$\mathrm{false}$

$\mathrm{g6}$

$x^2+2$

$\mathrm{p7}≔x+\mathrm{sqrt}\left(2\right)$

$\mathrm{p7}≔x+\sqrt{2}$

$\mathrm{Hurwitz}\left(\mathrm{p7}\&comma;x\right)$

$\mathrm{true}$

$\mathrm{p8}≔x^3+c{x}^2+\left(c^2-1\right)x+1$

$\mathrm{p8}≔x^3+c{x}^2+\left(c^2-1\right)x+1$

$\mathrm{Hurwitz}\left(\mathrm{p8}\&comma;x\&comma;'\mathrm{s8}'\&comma;'\mathrm{g8}'\right)$

$\mathrm{FAIL}$

$\mathrm{s8}$

$\left[\frac{x}{c}\&comma;\frac{c^{2}x}{c^3-c-1}\&comma;c^{2}x-x-\frac{x}{c}\right]$

$\mathrm{p9}≔\mathrm{expand}\left(\left(c{z}^2+1\right)\left(z+1\right)\left(z^2+2z+2\right)\right)$

$\mathrm{p9}≔c{z}^5+3c{z}^4+4c{z}^3+2c{z}^2+z^3+3z^2+4z+2$

$\mathrm{Hurwitz}\left(\mathrm{p9}\&comma;z\&comma;'\mathrm{s9}'\&comma;'\mathrm{g9}'\right)$

$\mathrm{FAIL}$

$\mathrm{s9}$

$\mathrm{g9}$

$c{z}^2+1$

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