Unified Analytic Solution of Transcendental Equations in Differential Algebraic Closure

This paper establishes a rigorous differential algebraic framework for solving analytic transcendental equations, extending the methodology previously developed for polynomial equations. We construct a well-defined transcendental differential algebraic closure K(M) trans for any truncation order M and prove that the roots of the corresponding truncated equation fM(x) = 0 can be analytically expressed within this closure. The solution formula generalizes the polynomial case, featuring critical point analysis, Taylor series truncation, combinatorial correction terms, and spectral decomposition. We provide constructive proofs, derive combinatorial expressions for the correction coefficients γ(M) m , and present a detailed O(M2) algorithm. Extensive numerical validation across diverse transcendental equations demonstrates high-precision accuracy (residuals of the truncated equation typically below 10−28). This work reconciles with classical impossibility results by demonstrating that explicit analytic solutions exist in the appropriately extended differential algebraic closure K(M) trans, even for equations unsolvable in elementary functions.