Residue Theorem: Cauchy's Integral Formula and Applications

This lecture discusses the residue theorem and its applications in complex analysis. The instructor begins by reviewing the concept of residues and their significance in evaluating complex integrals. They explain how to compute residues using Laurent series and demonstrate the theorem's power through various examples, including integrals around singularities. The instructor emphasizes the importance of Cauchy's theorem, which states that if a function is holomorphic inside a closed curve, the integral over that curve is zero. They provide corollaries of the residue theorem, including the integral formula of Cauchy, which relates integrals of functions with poles to their residues. The lecture includes detailed examples, illustrating how to apply these concepts to compute integrals involving different types of singularities. The instructor also highlights the relationship between the residue theorem and Cauchy's theorem, clarifying their roles in complex analysis. The session concludes with a discussion on the implications of these theorems in mathematical theory and practice.