A Pragmatic Interpretation of Residue Theorem and Laurent Series: A Rigorous Mathematical Proof for Conceptual Equivalence

This paper presents a rigorous exploration of the intersection between pragmatism and complex analysis, focusing on the philosophical principle that "if there is no practical difference, there is no conceptual difference." Using the Residue Theorem and Laurent series, we establish mathematical conditions under which two analytic functions that are indistinguishable through practical application are conceptually equivalent. By delving into complex analysis, we demonstrate that if the residues and winding numbers of two functions are identical across all singularities and contours, the functions themselves are analytically and conceptually identical. This mathematical approach offers a formal proof for the pragmatic principle, enhancing our understanding of both complex analysis and philosophical pragmatism.