A Unified Framework for Non-Recursive Partial Fraction Decomposition and Higher-Order Expansions

AbstractThis paper presents a systematically organized analytical framework for the partial fraction decomposition of rational functions involving specific algebraic structures and their general cases. Novel mathematical theorems and formulas are established and rigorously proved to simplify complex rational expressions into elementary partial fractions, facilitating integration processes.Furthermore, the study extends to develop generalization formulas for general powers and higher-order expansions, providing a comprehensive approach for handling complex denominators with arbitrary exponents. The theoretical results are demonstrated with concrete examples verified for accuracy.