Analytic Continuation of a Generalized One-Parameter Power Iteration System to the Complex Plane

This paper investigates the analytic continuation of a generalized one-parameter power iteration system u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}, where s is a complex parameter. Building on previous works [1, 2], we prove that for almost all initial values u_0 ∈ C \ {0}, the iteration converges to a unique fixed point L_s satisfying the balance equation ln L_s + π/(s(s-1)) L_s = 0. By introducing the Lambert W function [3], we obtain the closed-form solution: L_s = s(s-1)/π · W_0( π/(s(s-1)) ), s ∈ C \ {0,1} where W_0 denotes the principal branch, corresponding to the physically realized attracting fixed point. This expression provides an analytic continuation of the system from its original domain s > 1 to the entire complex plane (excluding the singular points s = 0,1). We prove that the continued fixed point satisfies the symmetry L_{1-s} = L_s, and derive complete expressions for the triple-space duality and coupled system. Numerical verification confirms that the principal branch selection agrees with iteration results, while other branches correspond to repelling fixed points. These results reveal deep connections between power iteration systems and the Lambert W function, providing new paradigms for complex dynamical systems [4, 5, 6, 7, 8].