Residue Theorem: Applications in Complex Analysis

This lecture covers the residue theorem and its applications in complex analysis, focusing on calculating integrals using residues. The instructor begins by introducing the theorem and its significance in evaluating complex integrals. The discussion includes various examples demonstrating how to apply the residue theorem to compute integrals over closed contours. The lecture emphasizes the importance of identifying singularities and determining residues at these points. The instructor explains the process of calculating residues for different types of singularities, including poles of various orders. The lecture also addresses practical problems where the residue theorem can simplify complex integral calculations, particularly in the context of holomorphic functions. The instructor provides step-by-step examples, illustrating how to set up integrals and apply the theorem effectively. Throughout the lecture, the instructor encourages students to engage with the material by solving related problems, reinforcing their understanding of the residue theorem's applications in complex analysis.