FUNÇÕES HOLOMORFAS UNIFICANDO SISTEMAS OSCILATÓRIOS MECÂNICOS E SISTEMAS ELÉTRICOS

The study of oscillatory systems constitutes one of the essential foundations of Classical Physics and modern engineering, manifesting itself both in the movement of mechanical structures subjected to dynamic loads and in the flow of electrons in electrical circuits. Traditionally, the description of these dynamics is based on Second-Order Ordinary Differential Equations (ODEs) solved in the domain of Real Numbers. However, this conventional approach often results in extensive and difficult algebraic manipulations that can obscure the underlying conceptual and physical understanding. From this perspective, the transition to the set of Complex Numbers and to the domain of Functions of a Complex Variable emerges as an essential pedagogical and methodological alternative, capable of converting complicated Differential Equations into simplified algebraic structures that highlight the cyclical nature and phases of wave phenomena. This article aims to present a unified mathematical interpretation for the Damped-Forced Harmonic Oscillator and RLC Electrical Circuits, demonstrating how the analyticity and holomorphy of Complex Functions provide a deeper understanding of the stability and dynamic behavior of these systems, ranging from transient to steady-state regimes. The applied methodology has a theoretical-computational character, structured in rigorous Mathematical Modeling integrated with Computational Simulation. Initially, the formulation of Second-Order ODEs for mechanical (Newton's Second Law) and electrical (Kirchhoff's Voltage Law) systems was established, applying the principle of electromechanical analogy to map the equivalent physical quantities. Subsequently, the variables were redefined in the complex plane using Holomorphic Functions. Numerical simulations were implemented in the Octave Scientific Computing Language, using the ode45 algorithm based on the variable-step Runge-Kutta method to solve the equations decomposed into first-order systems under varying damping scenarios. Additionally, the advanced Domain Coloring technique was employed for the visual mapping of holomorphic properties and transfer functions in the complex frequency plane (s-plane). Furthermore, the disruptive methodology of Andrzej Odrzywołek, which replaces traditional transcendental functions with the unified primitive operator EML, was tested. The results obtained accurately demonstrated the equivalence and perfect mathematical symmetry between mechanics and electricity through holomorphic modeling. The simulations, performed in the Octave environment, accurately captured the characteristic spiral trajectories of the transient regime converging to the elliptical Limit Cycle that defines the steady-state regime in Complex Phase Space. Topological mapping via Domain Coloring allowed for the immediate visual identification of system stability and conformal mapping properties based on the exact location of poles and zeros in the left Gaussian half-plane. Finally, the implementation of the unified primitive operator EML successfully reproduced sinusoidal dynamic behaviors, validating the simplification of the computational structure without compromising the original analytical symmetry. It is concluded that the use of Holomorphic Functions and complex analysis transcends the condition of a trivial calculation artifice, consolidating itself as an indispensable conceptual and physical unification tool in higher education, especially in Exact and Earth Sciences and Engineering courses. The theoretical approach integrated with modern computational visualization techniques demystifies mathematical abstraction and provides students and researchers with a robust and integrated framework for the analysis, monitoring, and prediction of the stability of complex dynamic systems.