Algebraic phase unwrapping and zero distribution of polynomial for continuous-time systems

An analytic solution is provided to the symbolic algebra-based computational problem for the unwrapped phase (that can be uniquely expressed as an integral involving itself and its derivative) of a continuous-time linear time-invariant system whose characteristic polynomial has coefficients belonging to the algebraically closed field of complex numbers. This solution is based on the use of the classical Cauchy indices. Application and adaptation of this analytic solution to an arbitrary univariate polynomial, yields its zero distribution with respect to the unbounded imaginary axis in the complex plane. Importantly, the algorithm that yields this zero distribution is designed to enforce the nonoccurrence of singular cases and can be implemented to any desired accuracy by rational operations