Generalizations of the Hermite-Biehler theorem
The authors generalize the Hermite-Biehler theorem which characterizes the Hurwitz stability of a real polynomial \(\delta(s).\) Let \(\delta(s)\) be a real polynomial of degree \(n\) and let \(l\) and \(r\) denote the numbers of roots in the left half and the right half of the complex plane, respectively. The signature \(l-r\) of \(\delta(s)\) determines \(l\) and \(r\) since \(l+r=n.\) Let \(i\) be the square root of \(-1\) and let \(\omega\) be a real number. Then \(\delta(i\omega)=p(\omega)+iq(\omega).\) The authors find a formula expressing the signature of \(\delta(s)\) involving \(p\) and \(q\) and zeros of these functions.
scientific article; zbMATH DE number 1419863