ABSTRACT
This chapter considers the Laplace transform, Fourier series, the Fourier transform, and generating functions. It details the Fourier series of the sawtooth function. The Fourier transform exhibits a close relationship with convolution. The chapter discusses the relationship between Fourier series and Laplace transforms. The Laplace transform is often used in engineering when solving ordinary differential equations. The chapter relates the method of generating functions in solving constant coefficient linear recurrences with the method of the Laplace transform in solving constant coefficient linear ODE. The role of the Laplace transform in solving ODE is precisely analogous to the role of generating functions in solving recurrences. For Laplace transforms, the time and frequency domains have different flavors.
