Lecture
This lecture focuses on the applications of the Residue theorem in complex analysis, particularly in evaluating integrals that relate to real analysis. The instructor begins by discussing the integral of a rational function, specifically 1/(x² + 1), and demonstrates how this integral evaluates to π. The lecture progresses to a more general case involving a quotient of polynomials, emphasizing the conditions under which the Residue theorem can be applied. The instructor explains the significance of singular points and how they relate to the integral's evaluation. A detailed derivation of a general formula for integrals is presented, including the use of contour integration techniques. The lecture also covers the behavior of integrals as the radius approaches infinity and the importance of estimating integrals over curves. The session concludes with examples that illustrate the practical applications of these concepts, including the computation of residues and their implications for Fourier transforms and differential equations.