Complex Analysis: Laplace and Fourier Transforms

This lecture covers complex analysis, focusing on the Laplace transform and Fourier series. It begins with a review of trigonometric series and orthogonality relations, emphasizing the properties of sine and cosine functions. The instructor discusses the heat equation, providing insights into its solutions and uniqueness. The lecture also addresses the resolution of differential equations using Laplace transforms, illustrating how to derive explicit formulas for temperature distribution over time. The connection between Fourier series and partial differential equations is explored, highlighting the application of these mathematical tools in solving real-world problems. The lecture concludes with a discussion on the uniqueness of solutions to the heat equation and the continuity of the derived functions, reinforcing the importance of these concepts in mathematical analysis.