Complex Analysis: Laurent Series and Residue Theorem

This lecture covers the completion of the chapter on Laurent series, focusing on the residue theorem and its applications in complex integral calculus. The instructor begins by reviewing the previous results related to determining the order of poles in functions expressed as fractions. The lecture explains how to identify poles and calculate residues using Laurent series expansions. The instructor provides a detailed example involving the function f(z) = 1/(z sin z), analyzing its behavior around the pole at zero. The process of isolating the residue is demonstrated through a systematic approach of eliminating singularities and deriving necessary terms. The lecture transitions into the definition of complex integration, emphasizing the importance of parametrization of curves in the complex plane. The instructor introduces the residue theorem, illustrating how to compute integrals of holomorphic functions around singularities. The session concludes with practical examples, reinforcing the concepts of residues and their significance in evaluating complex integrals, setting the stage for further exploration in complex analysis.