Analytic Solutions of Univariate Complex-Power Equations in Transcendental Differential Algebraic Closure

This paper establishes a rigorous transcendental differential algebraic closure framework for solving univariate complex-power equations. We prove that for equations of the form Ni=0aixri = 0 with complex exponents ri ∈ C and coefficients ai ∈ C, all solutions can be expressed in closed form within a well-defined transcendental differential algebraic closure KC. This closure extends the classical differential algebraic closure by explicitly adjoining exponential, logarithmic, and necessary special functions adapted to the complex domain, including the generalized Lambert W function with complex branches and hypergeometric functions.We provide a complete, constructive proof via transformation to exponential polynomials, derive explicit solution formulas using generalized Lambert W functions with complex parameter branches and their hypergeometric series expansions, and present an efficient, adaptive-precision numerical algorithm handling complex arithmetic and branch selection. Extensive numerical validation on equations involving complex exponents like i, π + i√2, and transcendental complex numbers demonstrates residuals below 10−30 using 256-bit precision.Our work reconciles with the Abel-Ruffini theorem and extends the framework of transcendental closures to the complex domain, showing that while general transcendental equations have no algebraic solutions, explicit analytic solutions exist in appropriately extended transcendental closures that account for multi-valuedness and branch structure.