Estudio Analítico, Dinámico y Topológico de la Ecuación híbrida Trascendente ax^i + bx +c= 0 en el Plano Complejo.

ABSTRACTThis paper addresses the systematic, analytical, and topological study of the hybrid algebraic-transcendental equation a x^i+ b x + c = 0, where the coefficients a, b, c ∈ ℝ (with a ≠ 0), x is the complex unknown, and i = √(−1) is the imaginary unit.Due to the presence of the pure complex exponent i, the function exhibits a multivalued character governed by the branchesof the complex logarithm. The exact analytical formulation is developed using the exponential representation of thevariable, decomposing the equation into a coupled system of transcendental equations in terms of the modulus r and the rgument θ. Furthermore, the Newton-Raphson iterative method is extended to the complex plane for numerical root localization, and the topology of the basins of attraction is thoroughly analyzed, demonstrating the emergence of pathological fractal boundaries of high dimension. Finally , connections with mathematical physics and log-periodic scaling laws are discussed.Keywords: Complex analysis, transcendental functions, imaginary exponent, Newton-Raphson method, basins of attraction, fractal geometry .