From Real Analysis to Complex Dynamics: Mathematical Foundations and Fractal Boundaries in Newton's Method

This paper explores the theoretical foundations and dynamic behavior of the Newton-Raphson method, transitioning from its classical application in real analysis to its complex generalization. We first analytically derive the method and discuss its convergence properties for simple roots in R. Subsequently, we extend the iteration scheme to the complex plane C to analyze the roots of polynomials. By framing the numerical method as a discrete dynamical system, we investigate the topological properties of its basins of attraction. We demonstrate mathematically that while the method converges predictably in the immediate vicinity of the roots, the boundaries separating these convergence basins exhibit complex, chaotic behavior, forming Julia sets known as Newton fractals. Finally, we analyze the convergence speed, compare higher-degree polynomials, and discuss the implications of this extreme sensitivity to initial conditions in the context of numerical error and iterative stability.