Advanced Applications of the Residue Theorem in Complex Analysis: Derivatives, Loop Integrals, and Root Structures

This comprehensive study delves into the advanced applications of the Residue Theorem within the realm of complex analysis, with a particular emphasis on the interplay between derivatives at singular points and loop integrals surrounding other singularities. By examining functions characterized by multiple isolated singularities, including removable singularities, poles of various orders, and essential singularities, we elucidate how negative power terms influence both loop integrals and derivative computations. Furthermore, the paper explores the intricate relationships between these mathematical constructs and the distribution of roots in complex functions, highlighting scenarios where transcendental solutions emerge due to the presence of essential singularities. Through rigorous mathematical derivations, illustrative examples, and comprehensive discussions, this work extends the foundational understanding of the Residue Theorem, providing deeper insights into its applications and implications in complex function theory.