Complex Analysis: Derivatives and Integrals

This lecture covers the fundamentals of complex analysis, focusing on complex derivatives and integrals. It begins with a recap of complex derivatives, explaining the limit definition and conditions for differentiability. The instructor discusses the implications of complex differentiability, including the Cauchy-Riemann equations. The lecture then transitions to complex integration, defining regular curves and complex line integrals. Several examples illustrate the integration of functions over curves, including the upper half-circle and full circle around the origin. The Cauchy theorem is introduced, detailing conditions for applying the theorem and its significance in complex analysis. The instructor provides proofs and outlines practical applications of the theorem, emphasizing the importance of holomorphic functions. The lecture concludes with an extension of the Cauchy theorem, discussing its implications in various contexts and reinforcing the concepts through examples. Overall, the lecture provides a comprehensive overview of complex analysis, equipping students with essential tools for further study in the field.