2 results
Search Results
Now showing 1 - 2 of 2
Item Open Access The Generalization of the Hurwitz Theorem(Suleyman Demirel University, 2010) Ender Dogan; Vilademir TenThis paper examines a generalized form of the classical Hurwitz theorem, originally established in 1895, which provides necessary and sufficient conditions for all roots of a real-coefficient polynomial to lie in the left half-plane of the complex plane. The study focuses on extending this theorem to broader regions, including semi-planes and bounded domains. The authors analyze polynomials with real coefficients and investigate how relationships among coefficients determine the localization of polynomial roots within specified regions of the complex plane. The project is divided into two main parts: a theoretical background covering complex numbers, polynomials, matrices, and determinants, and a second part presenting the proof of the generalized theorem along with its computational implementation. The results contribute to the stability analysis of differential equations and to broader applications in mathematical modeling and control theory.Item Unknown Generalization of the Hurwitz Theorem(Suleyman Demirel University, 2011) Murzabulatov MeiramThis paper presents the main theorem of the author’s diploma work, which provides a generalization of the classical Hurwitz theorem for real-coefficient polynomials. By constructing two auxiliary polynomials and establishing precise relations between their coefficients and the coefficients of the original polynomial, several lemmas are proven to support the main result. The theorem shows that the roots of a polynomial lie in specific half-planes or domains of the complex plane if and only if the associated transformed polynomials are Hurwitz polynomials. This connection extends the classical Hurwitz stability criterion to shifted and reflected regions of the complex plane. The paper also discusses a broader geometric problem involving arbitrary lines in the plane and demonstrates, through counterexamples, that a general solution does not exist for all orientations.