Ender DoganVilademir Ten2025-12-122025-12-122010Ender Dogan, Dr. Vilademir Ten/ The Generalization of the Hurwitz Theorem/ Suleyman Demirel University/ СДУ хабаршысы, 15(2).https://repository.sdu.edu.kz/handle/123456789/2306This 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.enAttribution-NonCommercial-ShareAlike 4.0 InternationalHurwitz theorempolynomial rootsstabilitycomplex planereal coefficientsdifferential equationssemi-plane conditions.Article