Complex Analysis: Holomorphic Functions and Cauchy-Riemann Equations

This lecture covers the fundamentals of complex analysis, focusing on holomorphic functions and the Cauchy-Riemann equations. The instructor begins with a review of complex numbers and their properties, including the definitions of real and imaginary parts. The discussion progresses to the concept of holomorphic functions, emphasizing the conditions under which a function is considered holomorphic. The Cauchy-Riemann equations are introduced as necessary criteria for differentiability in the complex plane. The instructor provides examples to illustrate these concepts, including the exponential and logarithmic functions. The lecture also addresses the continuity and differentiability of complex functions, highlighting the significance of these properties in complex analysis. Throughout the session, the instructor engages with students, encouraging questions and clarifying complex topics. The lecture concludes with a summary of the key points discussed, setting the stage for further exploration of complex functions in subsequent sessions.