A Unified Non-Recursive Framework for Partial Fraction Decomposition of Rational Functions with Arbitrary Real and Complex Exponents

This paper presents a unified analytical framework for the partial fraction decomposition of rational functions whose denominators involve arbitrary real and complex exponents and coefficients. Unlike classical treatments, which are restricted to polynomial denominators with non-negative integer powers and rely on recursive coefficient determination, the proposed framework yields explicit non-recursive closed-form expansions obtained through a single algebraic reduction identity. The termination condition of the expansion is characterized in terms of an ordering criterion on the real exponent parameters, and its algebraic validity over the complex domain is established independently of this ordering. The methodology is further extended to general polynomial denominators, dual binomial structures, and higher-order powers via repeated differentiation with respect to a scalar parameter combined with the Generalized Leibniz Rule. The results are illustrated by examples covering fractional, negative, and complex exponents, and verified symbolically using computer algebra systems.