Laplace Transforms: Applications and Convergence Properties

This lecture covers the fundamentals of Laplace transforms, focusing on their definition and properties. The instructor explains the concept of the abscissa of convergence and provides theorems related to the holomorphic nature of Laplace transforms. Several examples illustrate the application of Laplace transforms to piecewise continuous functions, demonstrating how to compute the transforms and analyze their convergence. The lecture also discusses the significance of the Laplace transform in solving ordinary differential equations and its applications in various problems, including the Candy problem and the Cauchy problem. The instructor emphasizes the importance of understanding the conditions under which the Laplace transform is defined and the implications of these conditions for practical applications. The session concludes with exercises that reinforce the concepts presented, allowing students to apply their knowledge in a structured manner.