A Deep Investigation of the Residue Theorem via Exponential and Laurent Series Structures

This paper presents a deep and formal investigation of the residue theorem in complex analysis through the lens of exponential and Laurent series structures. We systematically prove eight advanced properties relating contour integrals, Laurent expansions, and exponential forms, drawing connections to transcendence, algebraic structures, and the nature of mathematical instability. The analysis highlights the broader mathematical landscape of residues, transcendental functions, and the algebraic field of Laurent series, offering both theoretical rigor and foundational insights.