Laplace Transform: Definition and Examples

This lecture introduces the Laplace transform as a tool to analyze functions, connecting it to the Fourier transform. The instructor explains the formal definition of the Laplace transform, its convergence properties, and provides examples of calculating the Laplace transform for constant and exponential functions. The lecture emphasizes the importance of the convergence abscissa in determining the validity of the Laplace transform calculation, showcasing how it differs from the Fourier transform. Through detailed mathematical derivations, the instructor illustrates the step-by-step process of finding the Laplace transform for different types of functions, highlighting the conditions for convergence and the resulting transformed functions.