Complex Analysis: Laurent Series and Residue Theorem

This lecture covers the concepts of Laurent series and the residue theorem in complex analysis. The instructor begins by introducing the notion of Laurent series, explaining its application in domains that are simply connected. The lecture progresses to the definition of singularities and poles, detailing how to express functions in terms of Laurent series around these points. The instructor provides examples to illustrate the calculation of residues, emphasizing the importance of these residues in evaluating complex integrals. The discussion includes theorems related to residues and their applications in integration, particularly in the context of Cauchy's integral theorem. The lecture concludes with practical examples demonstrating how to compute residues for various functions, reinforcing the theoretical concepts presented throughout the session. This comprehensive overview equips students with the necessary tools to tackle problems involving complex functions and their singularities, preparing them for further studies in complex analysis and its applications.