Chapter 12 - Fourier Series

Abstract

This chapter deals with Fourier series. It defines the frequency, angular frequency, and period of a sinusoidal signal. It discusses wave superposition, standing and traveling waves, nodes, and the intensity of an oscillation. It shows how to expand arbitrary periodic wave distributions in Fourier series, giving formulas for the coefficients (Fourier coefficients). Fourier sine and cosine series are introduced; also series using the functions exp(inx). Dirichlet’s theorem on convergence of Fourier series and Parseval’s theorem are introduced, and examples show the use of Parseval’s theorem in evaluating summations. Conditions for integrating or differentiating Fourier series are discussed. Included are applications to sound waves and to functions with finite discontinuities either in the function itself or in its first derivative (e.g., square or triangular waves, ramp functions). It is shown how to work with Fourier series using symbolic computation in Maple and Mathematica.