A Unified Framework for Partial Fraction Decomposition: Novel Analytical Formulas and Solutions to Open Structural Challenges
Year: 2026
Type: preprint
Abstract: Partial fraction decomposition (PFD) serves as a core analytical mechanism within advanced calculus, theoretical physics, and computer algebra systems (CAS). Despite its mature development, classical methodology remains significantly constrained when managing multi-term multinomial denominators, continuous complex dynamic exponents, dual-binomial coupled fields, and high-multiplicity parameter derivatives. This paper introduces a structured investigation into seven primary computational and structural boundaries found in algebraic analysis. Concurrently, we present the structural application of Saif Raad’s identities—specifically spanning non- recursive multinomial networks and continuous complex parameter maps. We demonstrate that substituting this unified framework directly reduces variable polynomial time complexities to a deterministic O(1) constant layout, resolving the historical combinatorial and arithmetic explosions intrinsic to traditional iterative reduction profiles.
Author: Lazim, Saif R.
Institution: University of Basrah
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207823796
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