Holomorphic Functions: Cauchy-Riemann Equations and Applications

This lecture covers the concept of holomorphic functions and their properties, focusing on the Cauchy-Riemann equations. The instructor begins by reviewing previous material on holomorphicity and introduces the significance of the Cauchy-Riemann equations in establishing the relationship between the real and imaginary parts of complex functions. The lecture includes examples demonstrating how to verify whether a function is holomorphic by checking these equations. The instructor emphasizes the implications of holomorphicity, such as the existence of derivatives and the behavior of complex functions under transformations. Visual aids are used to illustrate the geometric interpretation of holomorphic functions, including how they map circles in the complex plane. The lecture concludes with a discussion on the logarithm function and its properties, particularly its holomorphic nature except along the negative real axis. The instructor also introduces the concept of Taylor series and their relevance in complex analysis, setting the stage for future topics in the course.